Subgroups — C₆₇×#61664 #30740915

Order 17152.

2136 conjugacy classes of subgroups.

The subgroup lattice has 578 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×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3329	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3303	C₂	1	4	yes	maximal, complemented
5	D₄×SD₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3272	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3206	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3200	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3106	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3010	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	complemented
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
14	C₂²×SD₁₆×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	—
18	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
19	#547×C₆₇	—	2	4	no	—
20	#547×C₆₇	C₂²	1	4	yes	complemented
21	D₄²×C₆₇	—	2	1	no	—
22	D₄²×C₆₇	C₂²	1	4	yes	—
23	#536×C₆₇	—	2	2	no	—
24	#536×C₆₇	C₂²	1	2	yes	complemented
25	#535×C₆₇	—	2	4	no	—
26	#535×C₆₇	C₂²	1	4	yes	—
27	#532×C₆₇	—	2	1	no	—
28	#532×C₆₇	C₂²	1	2	yes	—
29	#532×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
31	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#492×C₆₇	C₂²	1	4	yes	—
34	#487×C₆₇	C₂²	1	4	yes	complemented
35	#485×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	#468×C₆₇	C₂²	1	8	yes	complemented
40	#465×C₆₇	C₂²	1	8	yes	complemented
41	#461×C₆₇	C₂²	1	4	yes	complemented
42	#461×C₆₇	C₂²	1	4	yes	complemented
43	#460×C₆₇	C₂²	1	2	yes	—
44	#459×C₆₇	C₂²	1	2	yes	—
45	#450×C₆₇	C₂²	1	8	yes	complemented
46	#449×C₆₇	C₂²	1	4	yes	complemented
47	#449×C₆₇	C₂²	1	8	yes	complemented
48	#448×C₆₇	C₂²	1	8	yes	complemented
49	#448×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	#419×C₆₇	C₂²	1	4	yes	complemented
56	#416×C₆₇	C₂²	1	4	yes	complemented
57	#408×C₆₇	C₂²	1	4	yes	complemented
58	2₋¹⁺⁴×C₆₇	—	4	4	no	—
59	2₋¹⁺⁴×C₆₇	—	4	4	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	4	no	—
62	C₂×#41×C₆₇	—	2	2	no	—
63	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×#41×C₆₇	C₂³	1	4	yes	—
65	C₂×#41×C₆₇	C₂³	1	2	yes	—
66	C₂²×Q₈×C₆₇	—	2	2	no	—
67	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
68	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×D₄×C₆₇	—	2	4	no	—
70	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
71	AGL(1,ℤ/8)×C₆₇	—	4	8	no	—
72	C₂×SD₁₆×C₆₇	—	4	8	no	—
73	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
76	C₂×D₈×C₆₇	C₂³	1	4	yes	—
77	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
79	C₄⋊Q₈×C₆₇	—	2	2	no	—
80	C₄⋊Q₈×C₆₇	—	2	4	no	—
81	C₄⋊Q₈×C₆₇	—	2	1	no	—
82	C₄⋊Q₈×C₆₇	—	2	1	no	—
83	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
85	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
86	#127×C₆₇	—	2	4	no	—
87	#127×C₆₇	D₄	1	1	yes	characteristic
88	#127×C₆₇	C₂³	1	2	yes	—
89	#124×C₆₇	—	2	4	no	—
90	#124×C₆₇	—	2	2	no	—
91	#124×C₆₇	—	2	8	no	—
92	#124×C₆₇	—	2	2	no	—
93	#124×C₆₇	D₄	1	2	yes	—
94	#123×C₆₇	—	2	4	no	—
95	#123×C₆₇	—	2	8	no	—
96	C₂²⋊Q₈×C₆₇	—	2	8	no	—
97	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
98	#121×C₆₇	—	2	4	no	—
99	#121×C₆₇	—	2	4	no	—
100	#121×C₆₇	—	2	2	no	—
101	#121×C₆₇	—	2	4	no	—
102	#121×C₆₇	D₄	1	4	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	8	no	—
108	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
109	C₄×D₄×C₆₇	—	2	4	no	—
110	C₄×D₄×C₆₇	—	2	2	no	—
111	C₄×D₄×C₆₇	—	2	2	no	—
112	C₄×D₄×C₆₇	C₂³	1	2	yes	—
113	C₄×D₄×C₆₇	C₂³	1	4	yes	—
114	C₄×D₄×C₆₇	C₂³	1	4	yes	—
115	#117×C₆₇	—	2	4	no	—
116	#117×C₆₇	—	2	1	no	—
117	#117×C₆₇	C₂³	1	1	yes	characteristic
118	#117×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
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	#103×C₆₇	C₂³	1	8	yes	—
129	#103×C₆₇	C₂³	1	8	yes	—
130	#102×C₆₇	C₂³	1	4	yes	—
131	#102×C₆₇	C₂³	1	4	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	#41×C₆₇	—	2	2	no	—
138	#41×C₆₇	—	2	2	no	—
139	#41×C₆₇	—	2	8	no	—
140	#41×C₆₇	—	4	2	no	—
141	#41×C₆₇	—	4	2	no	—
142	#41×C₆₇	—	4	4	no	—
143	#41×C₆₇	—	4	8	no	—
144	#41×C₆₇	—	4	16	no	—
145	#41×C₆₇	—	8	4	no	—
146	C₂×Q₈×C₆₇	—	2	4	no	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	—	2	2	no	—
149	C₂×Q₈×C₆₇	—	2	8	no	—
150	C₂×Q₈×C₆₇	—	4	4	no	—
151	C₂×Q₈×C₆₇	—	4	2	no	—
152	C₂×Q₈×C₆₇	—	4	2	no	—
153	C₂×Q₈×C₆₇	—	4	8	no	—
154	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
155	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
156	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₂×D₄×C₆₇	—	2	1	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	8	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	2	no	—
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	—	2	4	no	—
164	C₂×D₄×C₆₇	—	4	4	no	—
165	C₂×D₄×C₆₇	—	4	4	no	—
166	C₂×D₄×C₆₇	—	8	8	no	—
167	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
168	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
170	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	2	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	SD₁₆×C₆₇	—	2	8	no	—
182	SD₁₆×C₆₇	—	4	8	no	—
183	SD₁₆×C₆₇	—	4	8	no	—
184	D₈×C₆₇	—	2	8	no	—
185	M₁₆×C₆₇	—	4	4	no	—
186	C₂×C₈×C₆₇	—	4	4	no	—
187	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
188	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	8	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₆₇	—	2	1	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	2	no	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
205	C₄²×C₆₇	—	2	1	no	—
206	C₄²×C₆₇	—	2	2	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂⁴	1	2	yes	—
211	C₂³×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	8	4	no	—
214	C₂³×C₆₇	—	8	8	no	—
215	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
216	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₆₇	—	2	1	no	—
218	Q₈×C₆₇	—	2	4	no	—
219	Q₈×C₆₇	—	2	4	no	—
220	Q₈×C₆₇	—	4	2	no	—
221	Q₈×C₆₇	—	4	8	no	—
222	Q₈×C₆₇	—	4	4	no	—
223	Q₈×C₆₇	—	8	2	no	—
224	Q₈×C₆₇	H(5,𝔽₂)	1	2	yes	—
225	D₄×C₆₇	—	2	4	no	—
226	D₄×C₆₇	—	2	8	no	—
227	D₄×C₆₇	—	2	1	no	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	4	2	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	8	4	no	—
235	D₄×C₆₇	—	8	2	no	—
236	D₄×C₆₇	—	8	8	no	—
237	D₄×C₆₇	—	8	4	no	—
238	D₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	4	8	no	—
250	C₂×C₄×C₆₇	—	4	8	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	—	4	4	no	—
253	C₂×C₄×C₆₇	—	4	8	no	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
259	C₈×C₆₇	—	2	4	no	—
260	C₈×C₆₇	—	2	4	no	—
261	C₂²×C₆₇	—	2	1	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	8	no	—
266	C₂²×C₆₇	—	4	2	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	4	no	—
269	C₂²×C₆₇	—	4	4	no	—
270	C₂²×C₆₇	—	16	8	no	—
271	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
272	C₄×C₆₇	—	2	1	no	—
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	2	4	no	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	4	4	no	—
277	C₄×C₆₇	—	4	2	no	—
278	C₄×C₆₇	—	4	4	no	—
279	C₄×C₆₇	#577	1	4	yes	—
280	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
281	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
282	#61664	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂×C₆₇	—	4	1	no	—
284	C₂×C₆₇	—	4	4	no	—
285	C₂×C₆₇	—	4	2	no	—
286	C₂×C₆₇	—	8	4	no	—
287	C₂×C₆₇	#3484	1	1	yes	characteristic
288	C₂×C₆₇	C₂×#577	1	2	yes	—
289	#3549	C₂×C₆₇	1	1	yes	characteristic, complemented
290	#3499	C₂×C₆₇	1	1	yes	characteristic, complemented
291	#3329	C₂×C₆₇	1	4	yes	complemented
292	#3303	C₂×C₆₇	1	4	yes	complemented
293	D₄×SD₁₆	C₂×C₆₇	1	4	yes	complemented
294	#3272	C₂×C₆₇	1	4	yes	complemented
295	#3206	C₂×C₆₇	1	2	yes	complemented
296	#3200	C₂×C₆₇	1	4	yes	complemented
297	#3106	C₂×C₆₇	1	2	yes	complemented
298	#3010	C₂×C₆₇	1	4	yes	complemented
299	#2979	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₆₇	#61664	1	1	yes	characteristic, direct product factor, Sylow
301	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	complemented
302	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
303	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
304	H(3,𝔽₄)	—	2	2	no	—
305	#555	—	2	1	no	—
306	#555	C₂²×C₆₇	1	4	yes	—
307	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
308	#547	—	2	4	no	—
309	#547	C₂²×C₆₇	1	4	yes	complemented
310	D₄²	—	2	1	no	—
311	D₄²	C₂²×C₆₇	1	4	yes	—
312	#536	—	2	2	no	—
313	#536	C₂²×C₆₇	1	2	yes	complemented
314	#535	—	2	4	no	—
315	#535	C₂²×C₆₇	1	4	yes	—
316	#532	—	2	1	no	—
317	#532	C₂²×C₆₇	1	2	yes	—
318	#532	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×C₄⋊Q₈	—	2	1	no	—
320	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
321	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic, complemented
322	#492	C₂²×C₆₇	1	4	yes	—
323	#487	C₂²×C₆₇	1	4	yes	complemented
324	#485	C₂²×C₆₇	1	8	yes	complemented
325	#482	C₂²×C₆₇	1	4	yes	complemented
326	#477	C₂²×C₆₇	1	2	yes	complemented
327	#476	C₂²×C₆₇	1	2	yes	complemented
328	#468	C₂²×C₆₇	1	8	yes	complemented
329	#465	C₂²×C₆₇	1	8	yes	complemented
330	#461	C₂²×C₆₇	1	4	yes	complemented
331	#461	C₂²×C₆₇	1	4	yes	complemented
332	#460	C₂²×C₆₇	1	2	yes	—
333	#459	C₂²×C₆₇	1	2	yes	—
334	#450	C₂²×C₆₇	1	8	yes	complemented
335	#449	C₂²×C₆₇	1	4	yes	complemented
336	#449	C₂²×C₆₇	1	8	yes	complemented
337	#448	C₂²×C₆₇	1	8	yes	complemented
338	#448	C₂²×C₆₇	1	8	yes	complemented
339	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
340	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
341	#436	C₂²×C₆₇	1	4	yes	complemented
342	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
343	#431	C₂²×C₆₇	1	2	yes	complemented
344	#419	C₂²×C₆₇	1	4	yes	complemented
345	#416	C₂²×C₆₇	1	4	yes	complemented
346	#408	C₂²×C₆₇	1	4	yes	complemented
347	2₋¹⁺⁴	—	4	4	no	—
348	2₋¹⁺⁴	—	4	4	no	—
349	C₂×#41	—	2	2	no	—
350	C₂×#41	—	2	4	no	—
351	C₂×#41	—	2	2	no	—
352	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
353	C₂×#41	C₂³×C₆₇	1	4	yes	—
354	C₂×#41	C₂³×C₆₇	1	2	yes	—
355	C₂²×Q₈	—	2	2	no	—
356	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
357	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
358	C₂²×D₄	—	2	4	no	—
359	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
360	AGL(1,ℤ/8)	—	4	8	no	—
361	C₂×SD₁₆	—	4	8	no	—
362	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
363	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
364	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
365	C₂×D₈	C₂³×C₆₇	1	4	yes	—
366	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
367	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	—	2	4	no	—
370	C₄⋊Q₈	—	2	1	no	—
371	C₄⋊Q₈	—	2	1	no	—
372	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
373	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
374	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
375	#127	—	2	4	no	—
376	#127	D₄×C₆₇	1	1	yes	characteristic
377	#127	C₂³×C₆₇	1	2	yes	—
378	#124	—	2	4	no	—
379	#124	—	2	2	no	—
380	#124	—	2	8	no	—
381	#124	—	2	2	no	—
382	#124	D₄×C₆₇	1	2	yes	—
383	#123	—	2	4	no	—
384	#123	—	2	8	no	—
385	C₂²⋊Q₈	—	2	8	no	—
386	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	#121	—	2	2	no	—
390	#121	—	2	4	no	—
391	#121	D₄×C₆₇	1	4	yes	—
392	#121	C₂³×C₆₇	1	4	yes	—
393	#121	C₂³×C₆₇	1	8	yes	—
394	#120	—	2	4	no	—
395	#120	—	2	4	no	—
396	#120	—	2	8	no	—
397	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	C₂³×C₆₇	1	2	yes	—
402	C₄×D₄	C₂³×C₆₇	1	4	yes	—
403	C₄×D₄	C₂³×C₆₇	1	4	yes	—
404	#117	—	2	4	no	—
405	#117	—	2	1	no	—
406	#117	C₂³×C₆₇	1	1	yes	characteristic
407	#117	C₂³×C₆₇	1	4	yes	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	—	2	4	no	—
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
412	#107	C₂³×C₆₇	1	4	yes	—
413	#106	C₂³×C₆₇	1	4	yes	—
414	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
415	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
416	#103	C₂³×C₆₇	1	8	yes	—
417	#103	C₂³×C₆₇	1	4	yes	—
418	#103	C₂³×C₆₇	1	8	yes	—
419	#102	C₂³×C₆₇	1	4	yes	—
420	#102	C₂³×C₆₇	1	4	yes	—
421	#102	C₂³×C₆₇	1	4	yes	—
422	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
423	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
424	C₄×C₈	C₂³×C₆₇	1	4	yes	—
425	C₂⁴	—	2	4	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	8	no	—
429	#41	—	4	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	4	no	—
432	#41	—	4	8	no	—
433	#41	—	4	16	no	—
434	#41	—	8	4	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	8	no	—
439	C₂×Q₈	—	4	4	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	—	4	8	no	—
443	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
444	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
445	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	4	no	—
455	C₂×D₄	—	8	8	no	—
456	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
457	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
458	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
459	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
460	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
468	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
470	SD₁₆	—	2	8	no	—
471	SD₁₆	—	4	8	no	—
472	SD₁₆	—	4	8	no	—
473	D₈	—	2	8	no	—
474	M₁₆	—	4	4	no	—
475	C₂×C₈	—	4	4	no	—
476	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
477	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
484	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	4	no	—
497	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
499	C₄²	C₂⁴×C₆₇	1	2	yes	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	8	4	no	—
503	C₂³	—	8	8	no	—
504	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
505	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
506	Q₈	—	2	1	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	2	4	no	—
509	Q₈	—	4	2	no	—
510	Q₈	—	4	8	no	—
511	Q₈	—	4	4	no	—
512	Q₈	—	8	2	no	—
513	Q₈	H(5,𝔽₂)×C₆₇	1	2	yes	—
514	D₄	—	2	4	no	—
515	D₄	—	2	8	no	—
516	D₄	—	2	1	no	—
517	D₄	—	2	4	no	—
518	D₄	—	2	4	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	8	no	—
522	D₄	—	4	4	no	—
523	D₄	—	8	4	no	—
524	D₄	—	8	2	no	—
525	D₄	—	8	8	no	—
526	D₄	—	8	4	no	—
527	D₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	—	4	4	no	—
543	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
545	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
546	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
547	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
548	C₈	—	2	4	no	—
549	C₈	—	2	4	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	8	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	4	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	16	8	no	—
560	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	4	no	—
565	C₄	—	4	4	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	4	no	—
568	C₄	#577×C₆₇	1	4	yes	—
569	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
570	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
571	C₂	—	4	4	no	—
572	C₂	—	4	1	no	—
573	C₂	—	4	2	no	—
574	C₂	—	8	4	no	—
575	C₂	C₆₇×#3484	1	1	yes	characteristic
576	C₂	C₂×#577×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×#3499	C₂	semidirect
3. C₆₇×#3329	C₂	semidirect
4. C₆₇×#3303	C₂	semidirect
5. D₄×SD₁₆×C₆₇	C₂	semidirect
6. C₆₇×#3272	C₂	semidirect
7. C₆₇×#3206	C₂	semidirect
8. C₆₇×#3200	C₂	semidirect
9. C₆₇×#3106	C₂	semidirect
10. C₆₇×#3010	C₂	semidirect
11. C₆₇×#2979	C₂	semidirect
12. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
14. C₂²×SD₁₆×C₆₇	C₂²	semidirect
18. D₄×Q₈×C₆₇	C₂²	semidirect
20. #547×C₆₇	C₂²	semidirect
24. #536×C₆₇	C₂²	semidirect
31. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
32. C₄×#41×C₆₇	C₂²	semidirect
34. #487×C₆₇	C₂²	semidirect
35. #485×C₆₇	C₂²	semidirect
36. #482×C₆₇	C₂²	semidirect
37. #477×C₆₇	C₂²	semidirect
38. #476×C₆₇	C₂²	semidirect
39. #468×C₆₇	C₂²	semidirect
40. #465×C₆₇	C₂²	semidirect
41. #461×C₆₇	C₂²	semidirect
42. #461×C₆₇	C₂²	semidirect
45. #450×C₆₇	C₂²	semidirect
46. #449×C₆₇	C₂²	semidirect
47. #449×C₆₇	C₂²	semidirect
48. #448×C₆₇	C₂²	semidirect
49. #448×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. #419×C₆₇	C₂²	semidirect
56. #416×C₆₇	C₂²	semidirect
57. #408×C₆₇	C₂²	semidirect
282. #61664	C₆₇	direct (N × H)
289. #3549	C₂×C₆₇	semidirect
290. #3499	C₂×C₆₇	semidirect
291. #3329	C₂×C₆₇	semidirect
292. #3303	C₂×C₆₇	semidirect
293. D₄×SD₁₆	C₂×C₆₇	semidirect
294. #3272	C₂×C₆₇	semidirect
295. #3206	C₂×C₆₇	semidirect
296. #3200	C₂×C₆₇	semidirect
297. #3106	C₂×C₆₇	semidirect
298. #3010	C₂×C₆₇	semidirect
299. #2979	C₂×C₆₇	semidirect
300. C₆₇	#61664	direct (N × H)
301. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
302. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
303. C₂²×SD₁₆	C₂²×C₆₇	semidirect
307. D₄×Q₈	C₂²×C₆₇	semidirect
309. #547	C₂²×C₆₇	semidirect
313. #536	C₂²×C₆₇	semidirect
320. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
321. C₄×#41	C₂²×C₆₇	semidirect
323. #487	C₂²×C₆₇	semidirect
324. #485	C₂²×C₆₇	semidirect
325. #482	C₂²×C₆₇	semidirect
326. #477	C₂²×C₆₇	semidirect
327. #476	C₂²×C₆₇	semidirect
328. #468	C₂²×C₆₇	semidirect
329. #465	C₂²×C₆₇	semidirect
330. #461	C₂²×C₆₇	semidirect
331. #461	C₂²×C₆₇	semidirect
334. #450	C₂²×C₆₇	semidirect
335. #449	C₂²×C₆₇	semidirect
336. #449	C₂²×C₆₇	semidirect
337. #448	C₂²×C₆₇	semidirect
338. #448	C₂²×C₆₇	semidirect
339. C₄×SD₁₆	C₂²×C₆₇	semidirect
340. C₄×D₈	C₂²×C₆₇	semidirect
341. #436	C₂²×C₆₇	semidirect
342. C₈×D₄	C₂²×C₆₇	semidirect
343. #431	C₂²×C₆₇	semidirect
344. #419	C₂²×C₆₇	semidirect
345. #416	C₂²×C₆₇	semidirect
346. #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×#3499	2
C₆₇×#3329	2
C₆₇×#3303	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3272	2
C₆₇×#3206	2
C₆₇×#3200	2
C₆₇×#3106	2
C₆₇×#3010	2
C₆₇×#2979	2
#61664	67

Subgroups — C67×#61664 #30740915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61664 #30740915