Subgroups — C₆₇×#29492 #30708743

Order 17152.

896 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2863	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2084	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2072	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2041	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2014	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	—	2	1	no	—
15	C₂×C₄×D₄×C₆₇	—	2	1	no	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×#103×C₆₇	C₂²	1	2	yes	—
19	C₂×#102×C₆₇	C₂²	1	2	yes	—
20	C₂×#102×C₆₇	C₂²	1	2	yes	—
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	#391×C₆₇	C₂²	1	1	yes	characteristic
24	#391×C₆₇	C₂²	1	1	yes	characteristic
25	#391×C₆₇	C₂²	1	1	yes	characteristic
26	#390×C₆₇	—	2	1	no	—
27	#390×C₆₇	—	2	1	no	—
28	#390×C₆₇	C₂²	1	2	yes	—
29	#389×C₆₇	C₂²	1	1	yes	characteristic
30	#389×C₆₇	C₂²	1	1	yes	characteristic
31	#389×C₆₇	C₂²	1	1	yes	characteristic
32	#387×C₆₇	—	2	1	no	—
33	#387×C₆₇	—	2	1	no	—
34	#387×C₆₇	—	2	1	no	—
35	#384×C₆₇	C₂²	1	1	yes	characteristic
36	#383×C₆₇	—	2	1	no	—
37	C₄³×C₆₇	C₂²	1	1	yes	characteristic
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	1	yes	characteristic
41	#336×C₆₇	C₂²	1	1	yes	characteristic
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	—	2	4	no	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	4	2	no	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	#127×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	—	4	2	no	—
55	C₄×D₄×C₆₇	—	4	2	no	—
56	C₄×D₄×C₆₇	—	4	4	no	—
57	C₄×D₄×C₆₇	—	4	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#106×C₆₇	—	4	2	no	—
78	#103×C₆₇	—	4	2	no	—
79	#102×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	4	2	no	—
82	C₄×C₈×C₆₇	—	2	4	no	—
83	#95×C₆₇	—	2	1	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	1	no	—
86	#95×C₆₇	—	2	1	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	1	no	—
89	#95×C₆₇	—	2	1	no	—
90	#95×C₆₇	—	2	1	no	—
91	#95×C₆₇	—	2	1	no	—
92	C₂⁴×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	4	2	no	—
106	C₂×D₄×C₆₇	—	4	2	no	—
107	C₂×D₄×C₆₇	—	4	2	no	—
108	C₂×D₄×C₆₇	#41	1	4	yes	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	4	4	no	—
115	C₂²×C₄×C₆₇	—	4	4	no	—
116	C₂²×C₄×C₆₇	—	4	4	no	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₆₇	—	4	8	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	#41	1	4	yes	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	—	4	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₂²⋊C₄×C₆₇	—	4	4	no	—
143	C₂²⋊C₄×C₆₇	—	4	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	4	1	no	—
152	C₄²×C₆₇	SD₁₆	1	4	yes	complemented
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	2	no	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	4	2	no	—
162	Q₈×C₆₇	—	4	2	no	—
163	D₄×C₆₇	—	2	4	no	—
164	D₄×C₆₇	—	2	4	no	—
165	D₄×C₆₇	—	4	8	no	—
166	D₄×C₆₇	—	4	8	no	—
167	D₄×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	1	no	—
194	C₂×C₄×C₆₇	—	4	1	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	8	8	no	—
198	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
200	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
201	C₂×C₄×C₆₇	#135	1	2	yes	—
202	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
204	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
205	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
207	C₈×C₆₇	—	2	4	no	—
208	C₈×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	4	no	—
210	C₂²×C₆₇	—	4	4	no	—
211	C₂²×C₆₇	—	4	4	no	—
212	C₂²×C₆₇	—	8	2	no	—
213	C₂²×C₆₇	—	8	2	no	—
214	C₂²×C₆₇	—	8	2	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	#525	1	1	yes	characteristic
217	C₂²×C₆₇	#532	1	1	yes	characteristic
218	C₂²×C₆₇	#535	1	1	yes	characteristic
219	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₆₇	#575	1	1	yes	characteristic
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	4	1	no	—
229	C₄×C₆₇	—	8	1	no	—
230	C₄×C₆₇	—	8	1	no	—
231	C₄×C₆₇	—	8	1	no	—
232	C₄×C₆₇	—	8	2	no	—
233	C₄×C₆₇	—	8	1	no	—
234	C₄×C₆₇	#551	1	2	yes	—
235	C₄×C₆₇	#553	1	2	yes	—
236	#29492	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂×C₆₇	—	8	4	no	—
238	C₂×C₆₇	#2603	1	1	yes	characteristic
239	C₂×C₆₇	#3101	1	1	yes	characteristic
240	C₂×C₆₇	#3105	1	1	yes	characteristic
241	C₂×C₆₇	#3167	1	1	yes	characteristic
242	C₂×C₆₇	#3180	1	1	yes	characteristic
243	C₂×C₆₇	#3208	1	1	yes	characteristic
244	C₂×C₆₇	#3212	1	1	yes	characteristic
245	#2863	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₄×#127	C₂×C₆₇	1	1	yes	characteristic
247	#2084	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#2072	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#2041	C₂×C₆₇	1	1	yes	characteristic
250	#2014	C₂×C₆₇	1	1	yes	characteristic
251	#1968	C₂×C₆₇	1	2	yes	complemented
252	#1934	C₂×C₆₇	1	2	yes	complemented
253	#1877	C₂×C₆₇	1	2	yes	complemented
254	#1864	C₂×C₆₇	1	2	yes	complemented
255	#1744	C₂×C₆₇	1	1	yes	characteristic, complemented
256	C₆₇	#29492	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
258	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	—	2	1	no	—
261	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
262	C₂×#106	C₂²×C₆₇	1	2	yes	—
263	C₂×#103	C₂²×C₆₇	1	2	yes	—
264	C₂×#102	C₂²×C₆₇	1	2	yes	—
265	C₂×#102	C₂²×C₆₇	1	2	yes	—
266	C₂×#102	C₂²×C₆₇	1	2	yes	—
267	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
268	#391	C₂²×C₆₇	1	1	yes	characteristic
269	#391	C₂²×C₆₇	1	1	yes	characteristic
270	#391	C₂²×C₆₇	1	1	yes	characteristic
271	#390	—	2	1	no	—
272	#390	—	2	1	no	—
273	#390	C₂²×C₆₇	1	2	yes	—
274	#389	C₂²×C₆₇	1	1	yes	characteristic
275	#389	C₂²×C₆₇	1	1	yes	characteristic
276	#389	C₂²×C₆₇	1	1	yes	characteristic
277	#387	—	2	1	no	—
278	#387	—	2	1	no	—
279	#387	—	2	1	no	—
280	#384	C₂²×C₆₇	1	1	yes	characteristic
281	#383	—	2	1	no	—
282	C₄³	C₂²×C₆₇	1	1	yes	characteristic
283	#340	C₂²×C₆₇	1	2	yes	—
284	#340	C₂²×C₆₇	1	2	yes	—
285	#336	C₂²×C₆₇	1	1	yes	characteristic
286	#336	C₂²×C₆₇	1	1	yes	characteristic
287	#336	C₂²×C₆₇	1	2	yes	—
288	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²×D₄	—	2	1	no	—
290	C₂²×D₄	—	2	1	no	—
291	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
292	C₂³×C₄	—	2	2	no	—
293	C₂×SD₁₆	—	2	4	no	—
294	C₂×SD₁₆	—	2	4	no	—
295	C₂×SD₁₆	—	4	2	no	—
296	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
297	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
298	#127	—	2	4	no	—
299	C₄×D₄	—	4	2	no	—
300	C₄×D₄	—	4	2	no	—
301	C₄×D₄	—	4	4	no	—
302	C₄×D₄	—	4	4	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
316	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
317	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
318	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
322	#106	—	4	2	no	—
323	#103	—	4	2	no	—
324	#102	—	2	4	no	—
325	#102	—	2	4	no	—
326	#102	—	4	2	no	—
327	C₄×C₈	—	2	4	no	—
328	#95	—	2	1	no	—
329	#95	—	2	1	no	—
330	#95	—	2	1	no	—
331	#95	—	2	1	no	—
332	#95	—	2	1	no	—
333	#95	—	2	1	no	—
334	#95	—	2	1	no	—
335	#95	—	2	1	no	—
336	#95	—	2	1	no	—
337	C₂⁴	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	1	no	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	#41×C₆₇	1	4	yes	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
369	SD₁₆	—	4	8	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	#41×C₆₇	1	4	yes	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₂²⋊C₄	—	4	4	no	—
388	C₂²⋊C₄	—	4	4	no	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	4	1	no	—
397	C₄²	SD₁₆×C₆₇	1	4	yes	complemented
398	C₂³	—	2	4	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	4	no	—
410	D₄	—	4	8	no	—
411	D₄	—	4	8	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	8	8	no	—
443	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
444	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
445	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
446	C₂×C₄	#135×C₆₇	1	2	yes	—
447	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
448	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
449	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
450	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
451	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
452	C₈	—	2	4	no	—
453	C₈	—	4	2	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	4	4	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	#525×C₆₇	1	1	yes	characteristic
462	C₂²	#532×C₆₇	1	1	yes	characteristic
463	C₂²	#535×C₆₇	1	1	yes	characteristic
464	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
465	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
466	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
467	C₂²	#575×C₆₇	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	4	1	no	—
474	C₄	—	8	1	no	—
475	C₄	—	8	1	no	—
476	C₄	—	8	1	no	—
477	C₄	—	8	2	no	—
478	C₄	—	8	1	no	—
479	C₄	#551×C₆₇	1	2	yes	—
480	C₄	#553×C₆₇	1	2	yes	—
481	C₂	—	8	4	no	—
482	C₂	C₆₇×#2603	1	1	yes	characteristic
483	C₂	C₆₇×#3101	1	1	yes	characteristic
484	C₂	C₆₇×#3105	1	1	yes	characteristic
485	C₂	C₆₇×#3167	1	1	yes	characteristic
486	C₂	C₆₇×#3180	1	1	yes	characteristic
487	C₂	C₆₇×#3208	1	1	yes	characteristic
488	C₂	C₆₇×#3212	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2863	C₂	semidirect
3. C₆₇×#2084	C₂	semidirect
4. C₆₇×#2072	C₂	semidirect
7. C₆₇×#1968	C₂	semidirect
8. C₆₇×#1934	C₂	semidirect
9. C₆₇×#1877	C₂	semidirect
10. C₆₇×#1864	C₂	semidirect
11. C₆₇×#1744	C₂	semidirect
152. C₄²×C₆₇	SD₁₆	semidirect
236. #29492	C₆₇	direct (N × H)
245. #2863	C₂×C₆₇	semidirect
247. #2084	C₂×C₆₇	semidirect
248. #2072	C₂×C₆₇	semidirect
251. #1968	C₂×C₆₇	semidirect
252. #1934	C₂×C₆₇	semidirect
253. #1877	C₂×C₆₇	semidirect
254. #1864	C₂×C₆₇	semidirect
255. #1744	C₂×C₆₇	semidirect
256. C₆₇	#29492	direct (N × H)
397. C₄²	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2863	2
C₄×#127×C₆₇	2
C₆₇×#2084	2
C₆₇×#2072	2
C₆₇×#2041	2
C₆₇×#2014	2
C₆₇×#1968	2
C₆₇×#1934	2
C₆₇×#1877	2
C₆₇×#1864	2
C₆₇×#1744	2
#29492	67

Subgroups — C67×#29492 #30708743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#29492 #30708743