Subgroups — C₆₇×#32980 #30712231

Order 17152.

1508 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#481×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#480×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2847	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1875	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1854	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1824	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1823	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1779	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₆₇	C₂²	1	2	yes	—
13	C₂×#120×C₆₇	—	2	1	no	—
14	C₂×#117×C₆₇	—	2	2	no	—
15	C₂×#117×C₆₇	—	2	4	no	—
16	C₂×#117×C₆₇	—	2	4	no	—
17	C₂×#117×C₆₇	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
21	C₂²×C₄²×C₆₇	—	2	1	no	—
22	#481×C₆₇	—	2	4	no	—
23	#480×C₆₇	—	2	4	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
31	#386×C₆₇	—	2	1	no	—
32	#386×C₆₇	—	2	2	no	—
33	#380×C₆₇	C₂²	1	1	yes	characteristic
34	#380×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	4	yes	complemented
36	#340×C₆₇	C₂²	1	4	yes	complemented
37	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	—	2	8	no	—
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	8	no	—
55	#120×C₆₇	—	4	4	no	—
56	#117×C₆₇	—	2	16	no	—
57	#117×C₆₇	—	2	8	no	—
58	#117×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
64	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	—	2	8	no	—
76	C₂×C₄²×C₆₇	—	2	4	no	—
77	#107×C₆₇	—	2	4	no	—
78	#106×C₆₇	—	2	4	no	—
79	#102×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	4	no	—
81	C₂²⋊C₈×C₆₇	—	2	2	no	—
82	C₂²⋊C₈×C₆₇	—	2	2	no	—
83	C₂²⋊C₈×C₆₇	—	2	8	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	2	no	—
86	C₂⁴×C₆₇	—	2	2	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	—	2	1	no	—
91	C₂⁴×C₆₇	—	4	2	no	—
92	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
93	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	4	no	—
97	C₂×D₄×C₆₇	—	4	4	no	—
98	C₂×D₄×C₆₇	—	4	4	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	—	2	16	no	—
104	C₂²×C₄×C₆₇	—	2	16	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	4	4	no	—
121	C₂²×C₄×C₆₇	—	4	2	no	—
122	C₂²×C₄×C₆₇	—	4	2	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	#41	1	8	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	4	no	—
132	C₂×C₈×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	4	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	4	no	—
140	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
141	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
142	C₂²⋊C₄×C₆₇	—	2	16	no	—
143	C₂²⋊C₄×C₆₇	—	4	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	2	no	—
145	C₂²⋊C₄×C₆₇	—	4	2	no	—
146	C₂²⋊C₄×C₆₇	—	4	4	no	—
147	C₄²×C₆₇	—	2	16	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	8	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	4	8	no	—
158	C₂³×C₆₇	—	4	8	no	—
159	C₂³×C₆₇	—	4	4	no	—
160	C₂³×C₆₇	—	4	4	no	—
161	C₂³×C₆₇	—	4	1	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	4	1	no	—
164	C₂³×C₆₇	—	4	1	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	#120	1	2	yes	—
167	C₂³×C₆₇	#123	1	4	yes	—
168	C₂³×C₆₇	#135	1	2	yes	—
169	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
170	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	D₄×C₆₇	—	4	2	no	—
172	D₄×C₆₇	—	4	2	no	—
173	D₄×C₆₇	—	8	8	no	—
174	D₄×C₆₇	—	8	4	no	—
175	C₂×C₄×C₆₇	—	2	16	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	8	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	4	no	—
190	C₂×C₄×C₆₇	—	4	8	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	2	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	#120	1	2	yes	—
198	C₂×C₄×C₆₇	#123	1	4	yes	—
199	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
202	C₈×C₆₇	—	4	2	no	—
203	C₈×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	2	4	no	—
206	C₂²×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	4	no	—
209	C₂²×C₆₇	—	2	4	no	—
210	C₂²×C₆₇	—	2	4	no	—
211	C₂²×C₆₇	—	4	16	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	4	2	no	—
214	C₂²×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	4	2	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	—	8	2	no	—
218	C₂²×C₆₇	—	8	1	no	—
219	C₂²×C₆₇	—	8	1	no	—
220	C₂²×C₆₇	—	8	1	no	—
221	C₂²×C₆₇	#449	1	4	yes	—
222	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#123	1	2	yes	—
224	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
225	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₂²×C₆₇	#575	1	2	yes	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	4	no	—
229	C₄×C₆₇	—	4	4	no	—
230	C₄×C₆₇	—	4	4	no	—
231	C₄×C₆₇	—	8	2	no	—
232	C₄×C₆₇	—	8	1	no	—
233	C₄×C₆₇	#525	1	2	yes	—
234	C₄×C₆₇	#525	1	2	yes	—
235	#32980	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂×C₆₇	—	2	4	no	—
237	C₂×C₆₇	—	4	4	no	—
238	C₂×C₆₇	—	16	1	no	—
239	C₂×C₆₇	#2849	1	1	yes	characteristic
240	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
241	C₂×C₆₇	#3005	1	1	yes	characteristic
242	C₂×C₆₇	#3091	1	2	yes	—
243	C₂×C₆₇	#3094	1	2	yes	—
244	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₂×#481	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂×#480	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#2847	C₂×C₆₇	1	1	yes	characteristic
248	#1875	C₂×C₆₇	1	4	yes	complemented
249	#1854	C₂×C₆₇	1	4	yes	complemented
250	#1824	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#1823	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#1779	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	#32980	1	1	yes	characteristic, direct product factor, Sylow
254	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×#121	C₂²×C₆₇	1	2	yes	—
257	C₂×#120	—	2	1	no	—
258	C₂×#117	—	2	2	no	—
259	C₂×#117	—	2	4	no	—
260	C₂×#117	—	2	4	no	—
261	C₂×#117	C₂²×C₆₇	1	4	yes	—
262	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
263	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×C₂²⋊C₄	—	2	1	no	—
265	C₂²×C₄²	—	2	1	no	—
266	#481	—	2	4	no	—
267	#480	—	2	4	no	—
268	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
269	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
270	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
271	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
272	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
275	#386	—	2	2	no	—
276	#386	—	2	1	no	—
277	#380	C₂²×C₆₇	1	1	yes	characteristic
278	#380	C₂²×C₆₇	1	2	yes	—
279	#340	C₂²×C₆₇	1	4	yes	complemented
280	#340	C₂²×C₆₇	1	4	yes	complemented
281	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	—	2	2	no	—
287	C₂³×C₄	—	2	8	no	—
288	C₂³×C₄	—	2	1	no	—
289	C₂³×C₄	—	2	1	no	—
290	C₂³×C₄	D₄×C₆₇	1	2	yes	—
291	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
294	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
295	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
296	#121	—	2	2	no	—
297	#121	—	2	2	no	—
298	#121	—	2	8	no	—
299	#120	—	4	4	no	—
300	#117	—	2	16	no	—
301	#117	—	2	8	no	—
302	#117	—	2	8	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
308	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
313	C₂×C₂²⋊C₄	—	2	8	no	—
314	C₂×C₂²⋊C₄	—	2	4	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₄²	—	2	8	no	—
320	C₂×C₄²	—	2	4	no	—
321	#107	—	2	4	no	—
322	#106	—	2	4	no	—
323	#102	—	2	4	no	—
324	#102	—	2	4	no	—
325	C₂²⋊C₈	—	2	2	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	—	2	8	no	—
328	#95	—	2	1	no	—
329	#95	—	2	2	no	—
330	C₂⁴	—	2	2	no	—
331	C₂⁴	—	2	4	no	—
332	C₂⁴	—	2	2	no	—
333	C₂⁴	—	2	2	no	—
334	C₂⁴	—	2	1	no	—
335	C₂⁴	—	4	2	no	—
336	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
337	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	4	4	no	—
342	C₂×D₄	—	4	4	no	—
343	C₂×D₄	—	4	2	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	—	4	2	no	—
347	C₂²×C₄	—	2	16	no	—
348	C₂²×C₄	—	2	16	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
371	C₂²×C₄	#41×C₆₇	1	8	yes	—
372	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	4	no	—
376	C₂×C₈	—	2	4	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	4	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
385	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
386	C₂²⋊C₄	—	2	16	no	—
387	C₂²⋊C₄	—	4	4	no	—
388	C₂²⋊C₄	—	4	2	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₄²	—	2	16	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	8	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	4	8	no	—
402	C₂³	—	4	8	no	—
403	C₂³	—	4	4	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	1	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	2	no	—
410	C₂³	#120×C₆₇	1	2	yes	—
411	C₂³	#123×C₆₇	1	4	yes	—
412	C₂³	#135×C₆₇	1	2	yes	—
413	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
414	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	8	8	no	—
418	D₄	—	8	4	no	—
419	C₂×C₄	—	2	16	no	—
420	C₂×C₄	—	2	8	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	4	8	no	—
431	C₂×C₄	—	4	8	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	#120×C₆₇	1	2	yes	—
442	C₂×C₄	#123×C₆₇	1	4	yes	—
443	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
445	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
446	C₈	—	4	2	no	—
447	C₈	—	4	2	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	4	16	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	1	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	1	no	—
465	C₂²	#449×C₆₇	1	4	yes	—
466	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
467	C₂²	C₂×#123×C₆₇	1	2	yes	—
468	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
469	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
470	C₂²	#575×C₆₇	1	2	yes	—
471	C₄	—	2	2	no	—
472	C₄	—	2	4	no	—
473	C₄	—	4	4	no	—
474	C₄	—	4	4	no	—
475	C₄	—	8	2	no	—
476	C₄	—	8	1	no	—
477	C₄	#525×C₆₇	1	2	yes	—
478	C₄	#525×C₆₇	1	2	yes	—
479	C₂	—	2	4	no	—
480	C₂	—	4	4	no	—
481	C₂	—	16	1	no	—
482	C₂	C₆₇×#2849	1	1	yes	characteristic
483	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
484	C₂	C₆₇×#3005	1	1	yes	characteristic
485	C₂	C₆₇×#3091	1	2	yes	—
486	C₂	C₆₇×#3094	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#117×C₆₇	C₂	semidirect
2. C₂×#481×C₆₇	C₂	semidirect
3. C₂×#480×C₆₇	C₂	semidirect
5. C₆₇×#1875	C₂	semidirect
6. C₆₇×#1854	C₂	semidirect
7. C₆₇×#1824	C₂	semidirect
8. C₆₇×#1823	C₂	semidirect
9. C₆₇×#1779	C₂	semidirect
24. C₂×#107×C₆₇	C₂²	semidirect
25. C₂×#106×C₆₇	C₂²	semidirect
26. C₂×#102×C₆₇	C₂²	semidirect
27. C₂×#102×C₆₇	C₂²	semidirect
35. #340×C₆₇	C₂²	semidirect
36. #340×C₆₇	C₂²	semidirect
63. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
64. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
235. #32980	C₆₇	direct (N × H)
244. C₂²×#117	C₂×C₆₇	semidirect
245. C₂×#481	C₂×C₆₇	semidirect
246. C₂×#480	C₂×C₆₇	semidirect
248. #1875	C₂×C₆₇	semidirect
249. #1854	C₂×C₆₇	semidirect
250. #1824	C₂×C₆₇	semidirect
251. #1823	C₂×C₆₇	semidirect
252. #1779	C₂×C₆₇	semidirect
253. C₆₇	#32980	direct (N × H)
268. C₂×#107	C₂²×C₆₇	semidirect
269. C₂×#106	C₂²×C₆₇	semidirect
270. C₂×#102	C₂²×C₆₇	semidirect
271. C₂×#102	C₂²×C₆₇	semidirect
279. #340	C₂²×C₆₇	semidirect
280. #340	C₂²×C₆₇	semidirect
307. C₂×C₄⋊C₄	D₄×C₆₇	semidirect
308. C₂×C₄⋊C₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₆₇	2
C₂×#481×C₆₇	2
C₂×#480×C₆₇	2
C₆₇×#2847	2
C₆₇×#1875	2
C₆₇×#1854	2
C₆₇×#1824	2
C₆₇×#1823	2
C₆₇×#1779	2
#32980	67

Subgroups — C67×#32980 #30712231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32980 #30712231