Subgroups — C₆₇×#61079 #30740330

Order 17152.

2826 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3496	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3267	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3264	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3060	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3036	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3012	C₂	1	4	yes	maximal, complemented
9	C₂×#442×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	complemented
12	#576×C₆₇	—	2	16	no	—
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
15	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
16	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
17	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
18	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
20	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
21	#547×C₆₇	—	2	2	no	—
22	#547×C₆₇	C₂²	1	4	yes	complemented
23	#546×C₆₇	—	2	4	no	—
24	#546×C₆₇	C₂²	1	8	yes	complemented
25	D₄²×C₆₇	—	2	2	no	—
26	D₄²×C₆₇	C₂²	1	4	yes	complemented
27	#537×C₆₇	C₂²	1	4	yes	complemented
28	#534×C₆₇	—	2	4	no	—
29	#532×C₆₇	C₂²	1	1	yes	characteristic
30	#532×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#121×C₆₇	—	2	4	no	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
33	#497×C₆₇	C₂²	1	4	yes	complemented
34	#496×C₆₇	C₂²	1	4	yes	complemented
35	#470×C₆₇	C₂²	1	4	yes	complemented
36	#469×C₆₇	C₂²	1	4	yes	complemented
37	#465×C₆₇	C₂²	1	8	yes	complemented
38	#463×C₆₇	C₂²	1	8	yes	complemented
39	#461×C₆₇	C₂²	1	4	yes	complemented
40	#459×C₆₇	C₂²	1	4	yes	complemented
41	#452×C₆₇	C₂²	1	8	yes	complemented
42	#450×C₆₇	C₂²	1	8	yes	complemented
43	#449×C₆₇	C₂²	1	8	yes	complemented
44	#447×C₆₇	C₂²	1	8	yes	complemented
45	#442×C₆₇	C₂²	1	8	yes	complemented
46	#435×C₆₇	C₂²	1	8	yes	complemented
47	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
48	#423×C₆₇	C₂²	1	1	yes	characteristic, complemented
49	#417×C₆₇	C₂²	1	4	yes	complemented
50	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
51	#408×C₆₇	C₂²	1	2	yes	complemented
52	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
53	H(5,𝔽₂)×C₆₇	—	2	16	no	—
54	H(5,𝔽₂)×C₆₇	—	4	8	no	—
55	C₂×#41×C₆₇	—	2	4	no	—
56	C₂×#41×C₆₇	C₂³	1	4	yes	—
57	C₂×#41×C₆₇	C₂³	1	8	yes	—
58	C₂×#41×C₆₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₇	—	2	8	no	—
60	C₂²×D₄×C₆₇	—	2	4	no	—
61	C₂²×D₄×C₆₇	—	2	4	no	—
62	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	—	2	2	no	—
65	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
66	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
67	#135×C₆₇	—	2	16	no	—
68	#135×C₆₇	—	2	8	no	—
69	#135×C₆₇	—	4	4	no	—
70	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
73	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×D₈×C₆₇	—	2	16	no	—
76	C₂×D₈×C₆₇	D₄	1	8	yes	complemented
77	C₂×D₈×C₆₇	C₂³	1	8	yes	—
78	C₂×D₈×C₆₇	C₂³	1	2	yes	—
79	C₂×D₈×C₆₇	C₂³	1	2	yes	—
80	C₂×D₈×C₆₇	C₂³	1	4	yes	—
81	#131×C₆₇	—	2	8	no	—
82	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
84	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
85	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
88	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
89	#127×C₆₇	D₄	1	1	yes	characteristic
90	#127×C₆₇	C₂³	1	2	yes	—
91	#124×C₆₇	D₄	1	2	yes	—
92	#124×C₆₇	C₂³	1	4	yes	—
93	#123×C₆₇	—	2	8	no	—
94	#123×C₆₇	—	2	8	no	—
95	C₂²⋊Q₈×C₆₇	—	2	4	no	—
96	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
97	#121×C₆₇	—	2	8	no	—
98	#121×C₆₇	—	2	4	no	—
99	#121×C₆₇	—	2	4	no	—
100	#121×C₆₇	—	2	2	no	—
101	#121×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	4	no	—
103	#121×C₆₇	—	2	8	no	—
104	#121×C₆₇	D₄	1	4	yes	—
105	#121×C₆₇	C₂³	1	8	yes	—
106	#120×C₆₇	—	2	8	no	—
107	#120×C₆₇	—	2	8	no	—
108	C₄×D₄×C₆₇	—	2	2	no	—
109	C₄×D₄×C₆₇	—	2	2	no	—
110	C₄×D₄×C₆₇	—	2	2	no	—
111	C₄×D₄×C₆₇	—	2	2	no	—
112	C₄×D₄×C₆₇	C₂³	1	4	yes	—
113	C₄×D₄×C₆₇	C₂³	1	4	yes	—
114	C₄×D₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	#106×C₆₇	C₂³	1	4	yes	—
121	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
122	#103×C₆₇	C₂³	1	8	yes	—
123	#102×C₆₇	C₂³	1	8	yes	—
124	#102×C₆₇	C₂³	1	8	yes	—
125	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
126	#97×C₆₇	C₂³	1	4	yes	—
127	C₂⁴×C₆₇	—	2	2	no	—
128	C₂⁴×C₆₇	—	2	8	no	—
129	#41×C₆₇	—	2	4	no	—
130	#41×C₆₇	—	2	4	no	—
131	#41×C₆₇	—	2	16	no	—
132	#41×C₆₇	—	2	8	no	—
133	#41×C₆₇	—	2	8	no	—
134	#41×C₆₇	—	4	8	no	—
135	#41×C₆₇	—	4	16	no	—
136	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
138	C₂×D₄×C₆₇	—	2	16	no	—
139	C₂×D₄×C₆₇	—	2	8	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	8	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	8	no	—
145	C₂×D₄×C₆₇	—	2	2	no	—
146	C₂×D₄×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	4	8	no	—
148	C₂×D₄×C₆₇	—	4	16	no	—
149	C₂×D₄×C₆₇	—	4	32	no	—
150	C₂×D₄×C₆₇	—	8	8	no	—
151	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	complemented
155	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
156	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	8	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	8	4	no	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₆₇	—	2	4	no	—
167	Q₁₆×C₆₇	—	4	2	no	—
168	SD₁₆×C₆₇	—	2	16	no	—
169	SD₁₆×C₆₇	—	2	8	no	—
170	SD₁₆×C₆₇	—	4	4	no	—
171	D₈×C₆₇	—	2	16	no	—
172	D₈×C₆₇	—	2	4	no	—
173	D₈×C₆₇	—	4	2	no	—
174	D₈×C₆₇	C₂×D₄	1	16	yes	complemented
175	M₁₆×C₆₇	—	2	4	no	—
176	M₁₆×C₆₇	—	2	8	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	2	8	no	—
179	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	complemented
180	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
181	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
182	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	8	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
188	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₆₇	—	2	8	no	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	—	2	4	no	—
192	C₂²⋊C₄×C₆₇	—	2	4	no	—
193	C₂²⋊C₄×C₆₇	—	2	8	no	—
194	C₂²⋊C₄×C₆₇	—	2	4	no	—
195	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
196	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₆₇	C₂⁴	1	2	yes	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	—	2	8	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	—	4	16	no	—
203	C₂³×C₆₇	—	8	2	no	—
204	C₂³×C₆₇	—	8	2	no	—
205	C₂³×C₆₇	—	8	8	no	—
206	C₂³×C₆₇	—	8	8	no	—
207	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
208	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₆₇	—	2	4	no	—
210	Q₈×C₆₇	—	2	4	no	—
211	Q₈×C₆₇	—	2	4	no	—
212	D₄×C₆₇	—	2	8	no	—
213	D₄×C₆₇	—	2	8	no	—
214	D₄×C₆₇	—	2	4	no	—
215	D₄×C₆₇	—	2	4	no	—
216	D₄×C₆₇	—	2	4	no	—
217	D₄×C₆₇	—	2	4	no	—
218	D₄×C₆₇	—	4	4	no	—
219	D₄×C₆₇	—	4	16	no	—
220	D₄×C₆₇	—	4	4	no	—
221	D₄×C₆₇	—	4	16	no	—
222	D₄×C₆₇	—	4	8	no	—
223	D₄×C₆₇	—	8	2	no	—
224	D₄×C₆₇	—	8	8	no	—
225	D₄×C₆₇	C₂²×D₄	1	8	yes	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	4	16	no	—
232	C₂×C₄×C₆₇	—	4	8	no	—
233	C₂×C₄×C₆₇	—	4	8	no	—
234	C₂×C₄×C₆₇	—	8	4	no	—
235	C₂×C₄×C₆₇	—	8	2	no	—
236	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
242	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
243	C₈×C₆₇	—	2	4	no	—
244	C₈×C₆₇	—	2	2	no	—
245	C₈×C₆₇	C₂²×D₄	1	4	yes	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	8	no	—
248	C₂²×C₆₇	—	2	4	no	—
249	C₂²×C₆₇	—	2	8	no	—
250	C₂²×C₆₇	—	4	4	no	—
251	C₂²×C₆₇	—	4	4	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	—	4	4	no	—
255	C₂²×C₆₇	—	8	8	no	—
256	C₂²×C₆₇	—	16	16	no	—
257	C₂²×C₆₇	D₄²	1	2	yes	—
258	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
259	C₄×C₆₇	—	2	2	no	—
260	C₄×C₆₇	—	2	2	no	—
261	C₄×C₆₇	—	2	4	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	—	4	4	no	—
264	C₄×C₆₇	D₄²	1	2	yes	—
265	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
267	#61079	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂×C₆₇	—	2	2	no	—
269	C₂×C₆₇	—	4	4	no	—
270	C₂×C₆₇	—	4	8	no	—
271	C₂×C₆₇	—	8	4	no	—
272	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
273	C₂×C₆₇	#3585	1	2	yes	—
274	C₂×#576	C₂×C₆₇	1	2	yes	complemented
275	#3496	C₂×C₆₇	1	2	yes	complemented
276	#3267	C₂×C₆₇	1	8	yes	complemented
277	#3264	C₂×C₆₇	1	8	yes	complemented
278	#3150	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#3060	C₂×C₆₇	1	2	yes	complemented
280	#3036	C₂×C₆₇	1	2	yes	complemented
281	#3012	C₂×C₆₇	1	4	yes	complemented
282	C₂×#442	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#2932	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₆₇	#61079	1	1	yes	characteristic, direct product factor, Sylow
285	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	complemented
286	#576	—	2	16	no	—
287	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
288	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
289	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
290	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
291	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
292	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
293	C₂²×D₈	C₂²×C₆₇	1	4	yes	complemented
294	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
295	#547	—	2	2	no	—
296	#547	C₂²×C₆₇	1	4	yes	complemented
297	#546	—	2	4	no	—
298	#546	C₂²×C₆₇	1	8	yes	complemented
299	D₄²	—	2	2	no	—
300	D₄²	C₂²×C₆₇	1	4	yes	complemented
301	#537	C₂²×C₆₇	1	4	yes	complemented
302	#534	—	2	4	no	—
303	#532	C₂²×C₆₇	1	1	yes	characteristic
304	#532	C₂²×C₆₇	1	2	yes	complemented
305	C₂×#121	—	2	4	no	—
306	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
307	#497	C₂²×C₆₇	1	4	yes	complemented
308	#496	C₂²×C₆₇	1	4	yes	complemented
309	#470	C₂²×C₆₇	1	4	yes	complemented
310	#469	C₂²×C₆₇	1	4	yes	complemented
311	#465	C₂²×C₆₇	1	8	yes	complemented
312	#463	C₂²×C₆₇	1	8	yes	complemented
313	#461	C₂²×C₆₇	1	4	yes	complemented
314	#459	C₂²×C₆₇	1	4	yes	complemented
315	#452	C₂²×C₆₇	1	8	yes	complemented
316	#450	C₂²×C₆₇	1	8	yes	complemented
317	#449	C₂²×C₆₇	1	8	yes	complemented
318	#447	C₂²×C₆₇	1	8	yes	complemented
319	#442	C₂²×C₆₇	1	8	yes	complemented
320	#435	C₂²×C₆₇	1	8	yes	complemented
321	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
322	#423	C₂²×C₆₇	1	1	yes	characteristic, complemented
323	#417	C₂²×C₆₇	1	4	yes	complemented
324	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
325	#408	C₂²×C₆₇	1	2	yes	complemented
326	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic, complemented
327	H(5,𝔽₂)	—	2	16	no	—
328	H(5,𝔽₂)	—	4	8	no	—
329	C₂×#41	—	2	4	no	—
330	C₂×#41	C₂³×C₆₇	1	4	yes	—
331	C₂×#41	C₂³×C₆₇	1	8	yes	—
332	C₂×#41	C₂³×C₆₇	1	2	yes	—
333	C₂²×D₄	—	2	8	no	—
334	C₂²×D₄	—	2	4	no	—
335	C₂²×D₄	—	2	4	no	—
336	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
337	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
338	C₂³×C₄	—	2	2	no	—
339	AGL(1,ℤ/8)	—	2	16	no	—
340	AGL(1,ℤ/8)	—	2	32	no	—
341	#135	—	2	16	no	—
342	#135	—	2	8	no	—
343	#135	—	4	4	no	—
344	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
345	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
346	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
347	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
348	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
349	C₂×D₈	—	2	16	no	—
350	C₂×D₈	D₄×C₆₇	1	8	yes	complemented
351	C₂×D₈	C₂³×C₆₇	1	2	yes	—
352	C₂×D₈	C₂³×C₆₇	1	8	yes	—
353	C₂×D₈	C₂³×C₆₇	1	2	yes	—
354	C₂×D₈	C₂³×C₆₇	1	4	yes	—
355	#131	—	2	8	no	—
356	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
357	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
358	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
359	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
361	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
362	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
363	#127	D₄×C₆₇	1	1	yes	characteristic
364	#127	C₂³×C₆₇	1	2	yes	—
365	#124	D₄×C₆₇	1	2	yes	—
366	#124	C₂³×C₆₇	1	4	yes	—
367	#123	—	2	8	no	—
368	#123	—	2	8	no	—
369	C₂²⋊Q₈	—	2	4	no	—
370	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
371	#121	—	2	8	no	—
372	#121	—	2	4	no	—
373	#121	—	2	4	no	—
374	#121	—	2	2	no	—
375	#121	—	2	2	no	—
376	#121	—	2	4	no	—
377	#121	—	2	8	no	—
378	#121	D₄×C₆₇	1	4	yes	—
379	#121	C₂³×C₆₇	1	8	yes	—
380	#120	—	2	8	no	—
381	#120	—	2	8	no	—
382	C₄×D₄	—	2	2	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	C₂³×C₆₇	1	4	yes	—
387	C₄×D₄	C₂³×C₆₇	1	4	yes	—
388	C₄×D₄	C₂³×C₆₇	1	4	yes	—
389	C₂×C₄⋊C₄	—	2	4	no	—
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	8	no	—
393	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
394	#106	C₂³×C₆₇	1	4	yes	—
395	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
396	#103	C₂³×C₆₇	1	8	yes	—
397	#102	C₂³×C₆₇	1	8	yes	—
398	#102	C₂³×C₆₇	1	8	yes	—
399	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
400	#97	C₂³×C₆₇	1	4	yes	—
401	C₂⁴	—	2	2	no	—
402	C₂⁴	—	2	8	no	—
403	#41	—	2	4	no	—
404	#41	—	2	4	no	—
405	#41	—	2	16	no	—
406	#41	—	2	8	no	—
407	#41	—	2	8	no	—
408	#41	—	4	8	no	—
409	#41	—	4	16	no	—
410	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
411	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
412	C₂×D₄	—	2	16	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	4	8	no	—
422	C₂×D₄	—	4	16	no	—
423	C₂×D₄	—	4	32	no	—
424	C₂×D₄	—	8	8	no	—
425	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
426	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
427	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
428	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	complemented
429	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
430	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	8	4	no	—
437	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
440	Q₁₆	—	2	4	no	—
441	Q₁₆	—	4	2	no	—
442	SD₁₆	—	2	16	no	—
443	SD₁₆	—	2	8	no	—
444	SD₁₆	—	4	4	no	—
445	D₈	—	2	16	no	—
446	D₈	—	2	4	no	—
447	D₈	—	4	2	no	—
448	D₈	C₂×D₄×C₆₇	1	16	yes	complemented
449	M₁₆	—	2	4	no	—
450	M₁₆	—	2	8	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	8	no	—
453	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	complemented
454	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
455	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
456	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
470	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
472	C₄²	C₂⁴×C₆₇	1	2	yes	—
473	C₂³	—	2	4	no	—
474	C₂³	—	2	8	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	4	16	no	—
477	C₂³	—	8	2	no	—
478	C₂³	—	8	2	no	—
479	C₂³	—	8	8	no	—
480	C₂³	—	8	8	no	—
481	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
482	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
483	Q₈	—	2	4	no	—
484	Q₈	—	2	4	no	—
485	Q₈	—	2	4	no	—
486	D₄	—	2	8	no	—
487	D₄	—	2	8	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	2	4	no	—
492	D₄	—	4	4	no	—
493	D₄	—	4	16	no	—
494	D₄	—	4	4	no	—
495	D₄	—	4	16	no	—
496	D₄	—	4	8	no	—
497	D₄	—	8	2	no	—
498	D₄	—	8	8	no	—
499	D₄	C₂²×D₄×C₆₇	1	8	yes	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	16	no	—
506	C₂×C₄	—	4	8	no	—
507	C₂×C₄	—	4	8	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	—	8	2	no	—
510	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
511	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
513	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
515	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
516	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
517	C₈	—	2	4	no	—
518	C₈	—	2	2	no	—
519	C₈	C₂²×D₄×C₆₇	1	4	yes	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	8	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	8	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	8	8	no	—
530	C₂²	—	16	16	no	—
531	C₂²	D₄²×C₆₇	1	2	yes	—
532	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
533	C₄	—	2	2	no	—
534	C₄	—	2	2	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	4	no	—
538	C₄	D₄²×C₆₇	1	2	yes	—
539	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
540	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
541	C₂	—	2	2	no	—
542	C₂	—	4	4	no	—
543	C₂	—	4	8	no	—
544	C₂	—	8	4	no	—
545	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
546	C₂	C₆₇×#3585	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. C₂×#576×C₆₇	C₂	semidirect
2. C₆₇×#3496	C₂	semidirect
3. C₆₇×#3267	C₂	semidirect
4. C₆₇×#3264	C₂	semidirect
5. C₆₇×#3150	C₂	semidirect
6. C₆₇×#3060	C₂	semidirect
7. C₆₇×#3036	C₂	semidirect
8. C₆₇×#3012	C₂	semidirect
9. C₂×#442×C₆₇	C₂	semidirect
10. C₆₇×#2932	C₂	semidirect
11. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
14. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
15. C₂×#135×C₆₇	C₂²	semidirect
16. C₂×#135×C₆₇	C₂²	semidirect
17. C₂×#135×C₆₇	C₂²	semidirect
19. C₂²×D₈×C₆₇	C₂²	semidirect
20. C₂×#131×C₆₇	C₂²	semidirect
22. #547×C₆₇	C₂²	semidirect
24. #546×C₆₇	C₂²	semidirect
26. D₄²×C₆₇	C₂²	semidirect
27. #537×C₆₇	C₂²	semidirect
30. #532×C₆₇	C₂²	semidirect
32. C₂×C₄×D₄×C₆₇	C₂²	semidirect
33. #497×C₆₇	C₂²	semidirect
34. #496×C₆₇	C₂²	semidirect
35. #470×C₆₇	C₂²	semidirect
36. #469×C₆₇	C₂²	semidirect
37. #465×C₆₇	C₂²	semidirect
38. #463×C₆₇	C₂²	semidirect
39. #461×C₆₇	C₂²	semidirect
40. #459×C₆₇	C₂²	semidirect
41. #452×C₆₇	C₂²	semidirect
42. #450×C₆₇	C₂²	semidirect
43. #449×C₆₇	C₂²	semidirect
44. #447×C₆₇	C₂²	semidirect
45. #442×C₆₇	C₂²	semidirect
46. #435×C₆₇	C₂²	semidirect
47. C₂×#106×C₆₇	C₂²	semidirect
48. #423×C₆₇	C₂²	semidirect
49. #417×C₆₇	C₂²	semidirect
50. C₂×#102×C₆₇	C₂²	semidirect
51. #408×C₆₇	C₂²	semidirect
52. C₂×#97×C₆₇	C₂²	semidirect
76. C₂×D₈×C₆₇	D₄	semidirect
154. C₂×D₄×C₆₇	C₂×D₄	semidirect
174. D₈×C₆₇	C₂×D₄	semidirect
179. C₂×C₈×C₆₇	C₂×D₄	semidirect
267. #61079	C₆₇	direct (N × H)
274. C₂×#576	C₂×C₆₇	semidirect
275. #3496	C₂×C₆₇	semidirect
276. #3267	C₂×C₆₇	semidirect
277. #3264	C₂×C₆₇	semidirect
278. #3150	C₂×C₆₇	semidirect
279. #3060	C₂×C₆₇	semidirect
280. #3036	C₂×C₆₇	semidirect
281. #3012	C₂×C₆₇	semidirect
282. C₂×#442	C₂×C₆₇	semidirect
283. #2932	C₂×C₆₇	semidirect
284. C₆₇	#61079	direct (N × H)
285. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
287. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
288. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
289. C₂×#135	C₂²×C₆₇	semidirect
290. C₂×#135	C₂²×C₆₇	semidirect
291. C₂×#135	C₂²×C₆₇	semidirect
293. C₂²×D₈	C₂²×C₆₇	semidirect
294. C₂×#131	C₂²×C₆₇	semidirect
296. #547	C₂²×C₆₇	semidirect
298. #546	C₂²×C₆₇	semidirect
300. D₄²	C₂²×C₆₇	semidirect
301. #537	C₂²×C₆₇	semidirect
304. #532	C₂²×C₆₇	semidirect
306. C₂×C₄×D₄	C₂²×C₆₇	semidirect
307. #497	C₂²×C₆₇	semidirect
308. #496	C₂²×C₆₇	semidirect
309. #470	C₂²×C₆₇	semidirect
310. #469	C₂²×C₆₇	semidirect
311. #465	C₂²×C₆₇	semidirect
312. #463	C₂²×C₆₇	semidirect
313. #461	C₂²×C₆₇	semidirect
314. #459	C₂²×C₆₇	semidirect
315. #452	C₂²×C₆₇	semidirect
316. #450	C₂²×C₆₇	semidirect
317. #449	C₂²×C₆₇	semidirect
318. #447	C₂²×C₆₇	semidirect
319. #442	C₂²×C₆₇	semidirect
320. #435	C₂²×C₆₇	semidirect
321. C₂×#106	C₂²×C₆₇	semidirect
322. #423	C₂²×C₆₇	semidirect
323. #417	C₂²×C₆₇	semidirect
324. C₂×#102	C₂²×C₆₇	semidirect
325. #408	C₂²×C₆₇	semidirect
326. C₂×#97	C₂²×C₆₇	semidirect
350. C₂×D₈	D₄×C₆₇	semidirect
428. C₂×D₄	C₂×D₄×C₆₇	semidirect
448. D₈	C₂×D₄×C₆₇	semidirect
453. C₂×C₈	C₂×D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₆₇	2
C₆₇×#3496	2
C₆₇×#3267	2
C₆₇×#3264	2
C₆₇×#3150	2
C₆₇×#3060	2
C₆₇×#3036	2
C₆₇×#3012	2
C₂×#442×C₆₇	2
C₆₇×#2932	2
#61079	67

Subgroups — C67×#61079 #30740330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61079 #30740330