Subgroups — C₆₇×#61661 #30740912

Order 17152.

2272 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3549	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3325	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3305	C₂	1	4	yes	maximal, complemented
5	D₄×D₈×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3274	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3205	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3199	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3106	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3008	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	complemented
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
14	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
15	H(3,𝔽₄)×C₆₇	—	2	2	no	—
16	#555×C₆₇	—	2	1	no	—
17	#555×C₆₇	C₂²	1	4	yes	complemented
18	#550×C₆₇	C₂²	1	4	yes	complemented
19	D₄²×C₆₇	—	2	1	no	—
20	D₄²×C₆₇	—	2	4	no	—
21	D₄²×C₆₇	C₂²	1	4	yes	complemented
22	D₄²×C₆₇	C₂²	1	4	yes	complemented
23	#535×C₆₇	—	2	2	no	—
24	#535×C₆₇	—	2	4	no	—
25	#535×C₆₇	C₂²	1	4	yes	complemented
26	#535×C₆₇	C₂²	1	2	yes	complemented
27	#532×C₆₇	—	2	1	no	—
28	#532×C₆₇	C₂²	1	1	yes	characteristic
29	#532×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#127×C₆₇	—	2	1	no	—
31	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#493×C₆₇	C₂²	1	4	yes	complemented
34	#486×C₆₇	C₂²	1	4	yes	complemented
35	#482×C₆₇	C₂²	1	8	yes	complemented
36	#482×C₆₇	C₂²	1	4	yes	complemented
37	#477×C₆₇	C₂²	1	2	yes	complemented
38	#476×C₆₇	C₂²	1	2	yes	complemented
39	#469×C₆₇	C₂²	1	8	yes	complemented
40	#466×C₆₇	C₂²	1	8	yes	complemented
41	#460×C₆₇	C₂²	1	2	yes	complemented
42	#459×C₆₇	C₂²	1	4	yes	complemented
43	#459×C₆₇	C₂²	1	2	yes	complemented
44	#459×C₆₇	C₂²	1	4	yes	complemented
45	#449×C₆₇	C₂²	1	8	yes	complemented
46	#449×C₆₇	C₂²	1	4	yes	complemented
47	#447×C₆₇	C₂²	1	8	yes	complemented
48	#447×C₆₇	C₂²	1	8	yes	complemented
49	#447×C₆₇	C₂²	1	8	yes	complemented
50	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
51	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
52	#436×C₆₇	C₂²	1	4	yes	complemented
53	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
54	#431×C₆₇	C₂²	1	2	yes	complemented
55	#418×C₆₇	C₂²	1	4	yes	complemented
56	#416×C₆₇	C₂²	1	4	yes	complemented
57	#408×C₆₇	C₂²	1	4	yes	complemented
58	H(5,𝔽₂)×C₆₇	—	4	4	no	—
59	H(5,𝔽₂)×C₆₇	—	4	4	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×#41×C₆₇	C₂³	1	4	yes	—
64	C₂×#41×C₆₇	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	4	no	—
67	C₂²×D₄×C₆₇	—	2	4	no	—
68	C₂²×D₄×C₆₇	—	2	4	no	—
69	C₂²×D₄×C₆₇	—	2	2	no	—
70	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
71	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
72	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
73	AGL(1,ℤ/8)×C₆₇	—	4	8	no	—
74	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×D₈×C₆₇	—	4	8	no	—
76	C₂×D₈×C₆₇	C₂³	1	4	yes	—
77	C₂×D₈×C₆₇	C₂³	1	8	yes	—
78	C₂×D₈×C₆₇	C₂³	1	8	yes	—
79	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
81	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
83	#127×C₆₇	—	2	2	no	—
84	#127×C₆₇	—	2	4	no	—
85	#127×C₆₇	—	2	4	no	—
86	#127×C₆₇	—	2	1	no	—
87	#127×C₆₇	—	2	1	no	—
88	#127×C₆₇	D₄	1	1	yes	characteristic
89	#127×C₆₇	C₂³	1	4	yes	—
90	#127×C₆₇	C₂³	1	2	yes	—
91	#124×C₆₇	—	2	2	no	—
92	#124×C₆₇	—	2	4	no	—
93	#124×C₆₇	—	2	8	no	—
94	#124×C₆₇	—	2	2	no	—
95	#124×C₆₇	D₄	1	2	yes	—
96	#121×C₆₇	—	2	4	no	—
97	#121×C₆₇	—	2	2	no	—
98	#121×C₆₇	—	2	4	no	—
99	#121×C₆₇	—	2	4	no	—
100	#121×C₆₇	—	2	8	no	—
101	#121×C₆₇	D₄	1	4	yes	—
102	#121×C₆₇	C₂³	1	8	yes	—
103	#121×C₆₇	C₂³	1	4	yes	—
104	#121×C₆₇	C₂³	1	8	yes	—
105	#120×C₆₇	—	2	4	no	—
106	#120×C₆₇	—	2	4	no	—
107	#120×C₆₇	—	2	4	no	—
108	#120×C₆₇	—	2	8	no	—
109	#120×C₆₇	—	2	8	no	—
110	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
111	C₄×D₄×C₆₇	—	2	2	no	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	C₂³	1	4	yes	—
115	C₄×D₄×C₆₇	C₂³	1	2	yes	—
116	C₄×D₄×C₆₇	C₂³	1	4	yes	—
117	#117×C₆₇	—	2	4	no	—
118	#117×C₆₇	—	2	1	no	—
119	#117×C₆₇	C₂³	1	1	yes	characteristic
120	#117×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄²×C₆₇	—	2	1	no	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	#107×C₆₇	C₂³	1	4	yes	—
124	#106×C₆₇	C₂³	1	4	yes	—
125	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
126	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
127	#103×C₆₇	C₂³	1	4	yes	—
128	#102×C₆₇	C₂³	1	8	yes	—
129	#102×C₆₇	C₂³	1	4	yes	—
130	#102×C₆₇	C₂³	1	4	yes	—
131	#102×C₆₇	C₂³	1	8	yes	—
132	#102×C₆₇	C₂³	1	4	yes	—
133	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
134	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
135	C₄×C₈×C₆₇	C₂³	1	4	yes	—
136	C₂⁴×C₆₇	—	2	4	no	—
137	C₂⁴×C₆₇	—	2	2	no	—
138	C₂⁴×C₆₇	—	2	4	no	—
139	#41×C₆₇	—	2	2	no	—
140	#41×C₆₇	—	2	2	no	—
141	#41×C₆₇	—	2	8	no	—
142	#41×C₆₇	—	4	2	no	—
143	#41×C₆₇	—	4	2	no	—
144	#41×C₆₇	—	4	4	no	—
145	#41×C₆₇	—	4	8	no	—
146	#41×C₆₇	—	8	4	no	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	8	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	8	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	2	no	—
160	C₂×D₄×C₆₇	—	2	2	no	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	2	4	no	—
163	C₂×D₄×C₆₇	—	4	4	no	—
164	C₂×D₄×C₆₇	—	4	2	no	—
165	C₂×D₄×C₆₇	—	4	2	no	—
166	C₂×D₄×C₆₇	—	4	4	no	—
167	C₂×D₄×C₆₇	—	4	4	no	—
168	C₂×D₄×C₆₇	—	4	8	no	—
169	C₂×D₄×C₆₇	—	4	16	no	—
170	C₂×D₄×C₆₇	—	8	4	no	—
171	C₂×D₄×C₆₇	—	8	8	no	—
172	C₂×D₄×C₆₇	—	8	8	no	—
173	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
178	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	—	2	1	no	—
180	C₂²×C₄×C₆₇	—	2	2	no	—
181	C₂²×C₄×C₆₇	—	2	4	no	—
182	C₂²×C₄×C₆₇	—	2	4	no	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
186	SD₁₆×C₆₇	—	2	8	no	—
187	D₈×C₆₇	—	2	8	no	—
188	D₈×C₆₇	—	4	8	no	—
189	D₈×C₆₇	—	4	8	no	—
190	M₁₆×C₆₇	—	4	4	no	—
191	C₂×C₈×C₆₇	—	4	4	no	—
192	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
193	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	2	2	no	—
196	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂²⋊C₄×C₆₇	—	2	1	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	—	2	8	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
207	C₄²×C₆₇	—	2	1	no	—
208	C₄²×C₆₇	—	2	2	no	—
209	C₄²×C₆₇	—	2	4	no	—
210	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₆₇	C₂⁴	1	2	yes	—
213	C₂³×C₆₇	—	2	1	no	—
214	C₂³×C₆₇	—	2	4	no	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	—	4	8	no	—
217	C₂³×C₆₇	—	4	4	no	—
218	C₂³×C₆₇	—	8	4	no	—
219	C₂³×C₆₇	—	8	4	no	—
220	C₂³×C₆₇	—	8	8	no	—
221	C₂³×C₆₇	—	8	8	no	—
222	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	1	no	—
225	Q₈×C₆₇	—	4	2	no	—
226	Q₈×C₆₇	—	8	2	no	—
227	Q₈×C₆₇	H(5,𝔽₂)	1	2	yes	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	8	no	—
230	D₄×C₆₇	—	2	1	no	—
231	D₄×C₆₇	—	2	4	no	—
232	D₄×C₆₇	—	2	4	no	—
233	D₄×C₆₇	—	2	4	no	—
234	D₄×C₆₇	—	2	4	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	4	8	no	—
238	D₄×C₆₇	—	4	8	no	—
239	D₄×C₆₇	—	4	4	no	—
240	D₄×C₆₇	—	4	4	no	—
241	D₄×C₆₇	—	4	8	no	—
242	D₄×C₆₇	—	8	4	no	—
243	D₄×C₆₇	—	8	4	no	—
244	D₄×C₆₇	—	8	2	no	—
245	D₄×C₆₇	—	8	8	no	—
246	D₄×C₆₇	—	8	8	no	—
247	D₄×C₆₇	—	8	4	no	—
248	D₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	1	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	4	8	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	—	4	8	no	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
266	C₈×C₆₇	—	2	4	no	—
267	C₈×C₆₇	—	2	4	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	8	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	4	8	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	16	8	no	—
280	C₂²×C₆₇	—	16	4	no	—
281	C₂²×C₆₇	—	16	8	no	—
282	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
283	C₄×C₆₇	—	2	1	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	4	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	4	2	no	—
288	C₄×C₆₇	—	4	4	no	—
289	C₄×C₆₇	#576	1	4	yes	—
290	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
291	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
292	#61661	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
293	C₂×C₆₇	—	4	1	no	—
294	C₂×C₆₇	—	4	4	no	—
295	C₂×C₆₇	—	4	2	no	—
296	C₂×C₆₇	—	8	4	no	—
297	C₂×C₆₇	—	8	4	no	—
298	C₂×C₆₇	#3484	1	1	yes	characteristic
299	C₂×C₆₇	C₂×#576	1	2	yes	—
300	#3549	C₂×C₆₇	1	1	yes	characteristic, complemented
301	#3498	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#3325	C₂×C₆₇	1	4	yes	complemented
303	#3305	C₂×C₆₇	1	4	yes	complemented
304	D₄×D₈	C₂×C₆₇	1	4	yes	complemented
305	#3274	C₂×C₆₇	1	4	yes	complemented
306	#3205	C₂×C₆₇	1	2	yes	complemented
307	#3199	C₂×C₆₇	1	4	yes	complemented
308	#3106	C₂×C₆₇	1	2	yes	complemented
309	#3008	C₂×C₆₇	1	4	yes	complemented
310	#2979	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	#61661	1	1	yes	characteristic, direct product factor, Sylow
312	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	complemented
313	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
314	C₂²×D₈	C₂²×C₆₇	1	4	yes	complemented
315	H(3,𝔽₄)	—	2	2	no	—
316	#555	—	2	1	no	—
317	#555	C₂²×C₆₇	1	4	yes	complemented
318	#550	C₂²×C₆₇	1	4	yes	complemented
319	D₄²	—	2	1	no	—
320	D₄²	—	2	4	no	—
321	D₄²	C₂²×C₆₇	1	4	yes	complemented
322	D₄²	C₂²×C₆₇	1	4	yes	complemented
323	#535	—	2	2	no	—
324	#535	—	2	4	no	—
325	#535	C₂²×C₆₇	1	2	yes	complemented
326	#535	C₂²×C₆₇	1	4	yes	complemented
327	#532	—	2	1	no	—
328	#532	C₂²×C₆₇	1	1	yes	characteristic
329	#532	C₂²×C₆₇	1	2	yes	complemented
330	C₂×#127	—	2	1	no	—
331	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
332	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic, complemented
333	#493	C₂²×C₆₇	1	4	yes	complemented
334	#486	C₂²×C₆₇	1	4	yes	complemented
335	#482	C₂²×C₆₇	1	4	yes	complemented
336	#482	C₂²×C₆₇	1	8	yes	complemented
337	#477	C₂²×C₆₇	1	2	yes	complemented
338	#476	C₂²×C₆₇	1	2	yes	complemented
339	#469	C₂²×C₆₇	1	8	yes	complemented
340	#466	C₂²×C₆₇	1	8	yes	complemented
341	#460	C₂²×C₆₇	1	2	yes	complemented
342	#459	C₂²×C₆₇	1	2	yes	complemented
343	#459	C₂²×C₆₇	1	4	yes	complemented
344	#459	C₂²×C₆₇	1	4	yes	complemented
345	#449	C₂²×C₆₇	1	4	yes	complemented
346	#449	C₂²×C₆₇	1	8	yes	complemented
347	#447	C₂²×C₆₇	1	8	yes	complemented
348	#447	C₂²×C₆₇	1	8	yes	complemented
349	#447	C₂²×C₆₇	1	8	yes	complemented
350	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
351	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
352	#436	C₂²×C₆₇	1	4	yes	complemented
353	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
354	#431	C₂²×C₆₇	1	2	yes	complemented
355	#418	C₂²×C₆₇	1	4	yes	complemented
356	#416	C₂²×C₆₇	1	4	yes	complemented
357	#408	C₂²×C₆₇	1	4	yes	complemented
358	H(5,𝔽₂)	—	4	4	no	—
359	H(5,𝔽₂)	—	4	4	no	—
360	C₂×#41	—	2	2	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
363	C₂×#41	C₂³×C₆₇	1	4	yes	—
364	C₂×#41	C₂³×C₆₇	1	2	yes	—
365	C₂²×D₄	—	2	2	no	—
366	C₂²×D₄	—	2	4	no	—
367	C₂²×D₄	—	2	4	no	—
368	C₂²×D₄	—	2	4	no	—
369	C₂²×D₄	—	2	2	no	—
370	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
371	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
372	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
373	AGL(1,ℤ/8)	—	4	8	no	—
374	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
375	C₂×D₈	—	4	8	no	—
376	C₂×D₈	C₂³×C₆₇	1	4	yes	—
377	C₂×D₈	C₂³×C₆₇	1	8	yes	—
378	C₂×D₈	C₂³×C₆₇	1	8	yes	—
379	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
380	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
381	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
382	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
383	#127	—	2	2	no	—
384	#127	—	2	4	no	—
385	#127	—	2	4	no	—
386	#127	—	2	1	no	—
387	#127	—	2	1	no	—
388	#127	D₄×C₆₇	1	1	yes	characteristic
389	#127	C₂³×C₆₇	1	4	yes	—
390	#127	C₂³×C₆₇	1	2	yes	—
391	#124	—	2	2	no	—
392	#124	—	2	4	no	—
393	#124	—	2	8	no	—
394	#124	—	2	2	no	—
395	#124	D₄×C₆₇	1	2	yes	—
396	#121	—	2	4	no	—
397	#121	—	2	2	no	—
398	#121	—	2	4	no	—
399	#121	—	2	4	no	—
400	#121	—	2	8	no	—
401	#121	D₄×C₆₇	1	4	yes	—
402	#121	C₂³×C₆₇	1	4	yes	—
403	#121	C₂³×C₆₇	1	8	yes	—
404	#121	C₂³×C₆₇	1	8	yes	—
405	#120	—	2	4	no	—
406	#120	—	2	4	no	—
407	#120	—	2	4	no	—
408	#120	—	2	8	no	—
409	#120	—	2	8	no	—
410	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	C₂³×C₆₇	1	2	yes	—
415	C₄×D₄	C₂³×C₆₇	1	4	yes	—
416	C₄×D₄	C₂³×C₆₇	1	4	yes	—
417	#117	—	2	4	no	—
418	#117	—	2	1	no	—
419	#117	C₂³×C₆₇	1	1	yes	characteristic
420	#117	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄²	—	2	1	no	—
422	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
423	#107	C₂³×C₆₇	1	4	yes	—
424	#106	C₂³×C₆₇	1	4	yes	—
425	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
426	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
427	#103	C₂³×C₆₇	1	4	yes	—
428	#102	C₂³×C₆₇	1	8	yes	—
429	#102	C₂³×C₆₇	1	4	yes	—
430	#102	C₂³×C₆₇	1	8	yes	—
431	#102	C₂³×C₆₇	1	4	yes	—
432	#102	C₂³×C₆₇	1	4	yes	—
433	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
434	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
435	C₄×C₈	C₂³×C₆₇	1	4	yes	—
436	C₂⁴	—	2	4	no	—
437	C₂⁴	—	2	2	no	—
438	C₂⁴	—	2	4	no	—
439	#41	—	2	2	no	—
440	#41	—	2	2	no	—
441	#41	—	2	8	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	4	4	no	—
445	#41	—	4	8	no	—
446	#41	—	8	4	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
449	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	2	no	—
465	C₂×D₄	—	4	2	no	—
466	C₂×D₄	—	4	4	no	—
467	C₂×D₄	—	4	4	no	—
468	C₂×D₄	—	4	8	no	—
469	C₂×D₄	—	4	16	no	—
470	C₂×D₄	—	8	4	no	—
471	C₂×D₄	—	8	8	no	—
472	C₂×D₄	—	8	8	no	—
473	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
474	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
475	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
476	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
477	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
478	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
486	SD₁₆	—	2	8	no	—
487	D₈	—	2	8	no	—
488	D₈	—	4	8	no	—
489	D₈	—	4	8	no	—
490	M₁₆	—	4	4	no	—
491	C₂×C₈	—	4	4	no	—
492	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
493	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
497	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
498	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	4	no	—
510	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
512	C₄²	C₂⁴×C₆₇	1	2	yes	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	4	8	no	—
517	C₂³	—	4	4	no	—
518	C₂³	—	8	4	no	—
519	C₂³	—	8	4	no	—
520	C₂³	—	8	8	no	—
521	C₂³	—	8	8	no	—
522	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
523	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	4	2	no	—
526	Q₈	—	8	2	no	—
527	Q₈	H(5,𝔽₂)×C₆₇	1	2	yes	—
528	D₄	—	2	4	no	—
529	D₄	—	2	8	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	4	no	—
532	D₄	—	2	4	no	—
533	D₄	—	2	4	no	—
534	D₄	—	2	4	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	8	no	—
538	D₄	—	4	8	no	—
539	D₄	—	4	4	no	—
540	D₄	—	4	4	no	—
541	D₄	—	4	8	no	—
542	D₄	—	8	4	no	—
543	D₄	—	8	4	no	—
544	D₄	—	8	2	no	—
545	D₄	—	8	8	no	—
546	D₄	—	8	8	no	—
547	D₄	—	8	4	no	—
548	D₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	4	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
563	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
566	C₈	—	2	4	no	—
567	C₈	—	2	4	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	8	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	8	no	—
576	C₂²	—	4	4	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	16	8	no	—
580	C₂²	—	16	4	no	—
581	C₂²	—	16	8	no	—
582	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
583	C₄	—	2	2	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	2	no	—
586	C₄	—	2	4	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	4	no	—
589	C₄	#576×C₆₇	1	4	yes	—
590	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
591	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
592	C₂	—	4	1	no	—
593	C₂	—	4	4	no	—
594	C₂	—	4	2	no	—
595	C₂	—	8	4	no	—
596	C₂	—	8	4	no	—
597	C₂	C₆₇×#3484	1	1	yes	characteristic
598	C₂	C₂×#576×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3549	C₂	semidirect
2. C₆₇×#3498	C₂	semidirect
3. C₆₇×#3325	C₂	semidirect
4. C₆₇×#3305	C₂	semidirect
5. D₄×D₈×C₆₇	C₂	semidirect
6. C₆₇×#3274	C₂	semidirect
7. C₆₇×#3205	C₂	semidirect
8. C₆₇×#3199	C₂	semidirect
9. C₆₇×#3106	C₂	semidirect
10. C₆₇×#3008	C₂	semidirect
11. C₆₇×#2979	C₂	semidirect
12. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
14. C₂²×D₈×C₆₇	C₂²	semidirect
17. #555×C₆₇	C₂²	semidirect
18. #550×C₆₇	C₂²	semidirect
21. D₄²×C₆₇	C₂²	semidirect
22. D₄²×C₆₇	C₂²	semidirect
25. #535×C₆₇	C₂²	semidirect
26. #535×C₆₇	C₂²	semidirect
29. #532×C₆₇	C₂²	semidirect
31. C₂×#127×C₆₇	C₂²	semidirect
32. C₄×#41×C₆₇	C₂²	semidirect
33. #493×C₆₇	C₂²	semidirect
34. #486×C₆₇	C₂²	semidirect
35. #482×C₆₇	C₂²	semidirect
36. #482×C₆₇	C₂²	semidirect
37. #477×C₆₇	C₂²	semidirect
38. #476×C₆₇	C₂²	semidirect
39. #469×C₆₇	C₂²	semidirect
40. #466×C₆₇	C₂²	semidirect
41. #460×C₆₇	C₂²	semidirect
42. #459×C₆₇	C₂²	semidirect
43. #459×C₆₇	C₂²	semidirect
44. #459×C₆₇	C₂²	semidirect
45. #449×C₆₇	C₂²	semidirect
46. #449×C₆₇	C₂²	semidirect
47. #447×C₆₇	C₂²	semidirect
48. #447×C₆₇	C₂²	semidirect
49. #447×C₆₇	C₂²	semidirect
50. C₄×SD₁₆×C₆₇	C₂²	semidirect
51. C₄×D₈×C₆₇	C₂²	semidirect
52. #436×C₆₇	C₂²	semidirect
53. C₈×D₄×C₆₇	C₂²	semidirect
54. #431×C₆₇	C₂²	semidirect
55. #418×C₆₇	C₂²	semidirect
56. #416×C₆₇	C₂²	semidirect
57. #408×C₆₇	C₂²	semidirect
292. #61661	C₆₇	direct (N × H)
300. #3549	C₂×C₆₇	semidirect
301. #3498	C₂×C₆₇	semidirect
302. #3325	C₂×C₆₇	semidirect
303. #3305	C₂×C₆₇	semidirect
304. D₄×D₈	C₂×C₆₇	semidirect
305. #3274	C₂×C₆₇	semidirect
306. #3205	C₂×C₆₇	semidirect
307. #3199	C₂×C₆₇	semidirect
308. #3106	C₂×C₆₇	semidirect
309. #3008	C₂×C₆₇	semidirect
310. #2979	C₂×C₆₇	semidirect
311. C₆₇	#61661	direct (N × H)
312. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
313. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
314. C₂²×D₈	C₂²×C₆₇	semidirect
317. #555	C₂²×C₆₇	semidirect
318. #550	C₂²×C₆₇	semidirect
321. D₄²	C₂²×C₆₇	semidirect
322. D₄²	C₂²×C₆₇	semidirect
325. #535	C₂²×C₆₇	semidirect
326. #535	C₂²×C₆₇	semidirect
329. #532	C₂²×C₆₇	semidirect
331. C₂×#127	C₂²×C₆₇	semidirect
332. C₄×#41	C₂²×C₆₇	semidirect
333. #493	C₂²×C₆₇	semidirect
334. #486	C₂²×C₆₇	semidirect
335. #482	C₂²×C₆₇	semidirect
336. #482	C₂²×C₆₇	semidirect
337. #477	C₂²×C₆₇	semidirect
338. #476	C₂²×C₆₇	semidirect
339. #469	C₂²×C₆₇	semidirect
340. #466	C₂²×C₆₇	semidirect
341. #460	C₂²×C₆₇	semidirect
342. #459	C₂²×C₆₇	semidirect
343. #459	C₂²×C₆₇	semidirect
344. #459	C₂²×C₆₇	semidirect
345. #449	C₂²×C₆₇	semidirect
346. #449	C₂²×C₆₇	semidirect
347. #447	C₂²×C₆₇	semidirect
348. #447	C₂²×C₆₇	semidirect
349. #447	C₂²×C₆₇	semidirect
350. C₄×SD₁₆	C₂²×C₆₇	semidirect
351. C₄×D₈	C₂²×C₆₇	semidirect
352. #436	C₂²×C₆₇	semidirect
353. C₈×D₄	C₂²×C₆₇	semidirect
354. #431	C₂²×C₆₇	semidirect
355. #418	C₂²×C₆₇	semidirect
356. #416	C₂²×C₆₇	semidirect
357. #408	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3549	2
C₆₇×#3498	2
C₆₇×#3325	2
C₆₇×#3305	2
D₄×D₈×C₆₇	2
C₆₇×#3274	2
C₆₇×#3205	2
C₆₇×#3199	2
C₆₇×#3106	2
C₆₇×#3008	2
C₆₇×#2979	2
#61661	67

Subgroups — C67×#61661 #30740912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61661 #30740912