Subgroups — C₆₇×#33154 #30712405

Order 17152.

804 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
2	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1921	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1920	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1854	C₂	1	2	yes	maximal
10	C₆₇×#1825	C₂	1	2	yes	maximal
11	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	#391×C₆₇	—	2	1	no	—
23	#389×C₆₇	—	2	1	no	—
24	#389×C₆₇	C₂²	1	2	yes	—
25	#384×C₆₇	C₂²	1	1	yes	characteristic
26	#380×C₆₇	—	2	1	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
33	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
34	#376×C₆₇	—	2	1	no	—
35	C₄³×C₆₇	—	2	1	no	—
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	4	4	no	—
52	#117×C₆₇	—	2	2	no	—
53	#117×C₆₇	—	2	2	no	—
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	Q₈	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	1	no	—
70	C₂×C₄²×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	#107×C₆₇	—	2	4	no	—
82	#106×C₆₇	—	2	4	no	—
83	C₄⋊C₈×C₆₇	—	2	2	no	—
84	C₄⋊C₈×C₆₇	—	2	2	no	—
85	C₂²⋊C₈×C₆₇	—	2	4	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	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×Q₈×C₆₇	—	4	2	no	—
94	C₂×Q₈×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
118	C₂²×C₄×C₆₇	#41	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	—	2	1	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
139	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
140	C₄⋊C₄×C₆₇	#41	1	2	yes	—
141	C₄⋊C₄×C₆₇	#41	1	2	yes	—
142	C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₄²×C₆₇	—	2	4	no	—
146	C₄²×C₆₇	—	2	4	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	2	no	—
152	C₄²×C₆₇	—	4	2	no	—
153	C₄²×C₆₇	—	4	2	no	—
154	C₄²×C₆₇	—	4	2	no	—
155	C₄²×C₆₇	#41	1	2	yes	—
156	C₄²×C₆₇	#41	1	2	yes	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	—	2	1	no	—
163	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
164	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₆₇	—	8	4	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	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	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	4	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
200	C₂×C₄×C₆₇	#123	1	2	yes	—
201	C₂×C₄×C₆₇	#123	1	2	yes	—
202	C₂×C₄×C₆₇	#135	1	2	yes	—
203	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
204	C₂×C₄×C₆₇	#137	1	2	yes	—
205	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₈×C₆₇	—	4	2	no	—
212	C₈×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	2	2	no	—
214	C₂²×C₆₇	—	2	2	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	4	1	no	—
217	C₂²×C₆₇	—	4	1	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
222	C₂²×C₆₇	#525	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
224	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
226	C₂²×C₆₇	#575	1	1	yes	characteristic
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	4	2	no	—
231	C₄×C₆₇	—	4	2	no	—
232	C₄×C₆₇	—	4	1	no	—
233	C₄×C₆₇	—	4	2	no	—
234	C₄×C₆₇	—	4	2	no	—
235	C₄×C₆₇	—	8	2	no	—
236	C₄×C₆₇	—	8	2	no	—
237	C₄×C₆₇	#482	1	4	yes	—
238	C₄×C₆₇	#529	1	1	yes	characteristic
239	C₄×C₆₇	#529	1	1	yes	characteristic
240	C₄×C₆₇	#533	1	1	yes	characteristic
241	C₄×C₆₇	#533	1	1	yes	characteristic
242	#33154	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₆₇	—	4	2	no	—
244	C₂×C₆₇	#2855	1	1	yes	characteristic
245	C₂×C₆₇	#3076	1	1	yes	characteristic
246	C₂×C₆₇	#3079	1	1	yes	characteristic
247	C₂×C₆₇	C₂×#482	1	1	yes	characteristic
248	C₂×C₆₇	#3095	1	1	yes	characteristic
249	C₂×C₆₇	#3107	1	1	yes	characteristic
250	C₂×C₆₇	#3115	1	1	yes	characteristic
251	#2857	C₂×C₆₇	1	1	yes	characteristic
252	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
253	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
254	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
255	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
257	#1921	C₂×C₆₇	1	2	yes	complemented
258	#1920	C₂×C₆₇	1	2	yes	complemented
259	#1854	C₂×C₆₇	1	2	yes	—
260	#1825	C₂×C₆₇	1	2	yes	—
261	#1797	C₂×C₆₇	1	1	yes	characteristic
262	C₆₇	#33154	1	1	yes	characteristic, direct product factor, Sylow
263	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×C₂²⋊Q₈	—	2	1	no	—
265	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
267	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×#107	C₂²×C₆₇	1	2	yes	—
269	C₂×#106	C₂²×C₆₇	1	2	yes	—
270	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
273	#391	—	2	1	no	—
274	#389	—	2	1	no	—
275	#389	C₂²×C₆₇	1	2	yes	—
276	#384	C₂²×C₆₇	1	1	yes	characteristic
277	#380	—	2	1	no	—
278	#380	C₂²×C₆₇	1	2	yes	—
279	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
280	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
282	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
283	C₄×C₂²⋊C₄	—	2	1	no	—
284	C₄×C₂²⋊C₄	—	2	1	no	—
285	#376	—	2	1	no	—
286	C₄³	—	2	1	no	—
287	#340	C₂²×C₆₇	1	2	yes	—
288	#340	C₂²×C₆₇	1	2	yes	—
289	#340	C₂²×C₆₇	1	2	yes	—
290	#340	C₂²×C₆₇	1	2	yes	—
291	#340	C₂²×C₆₇	1	2	yes	—
292	#340	C₂²×C₆₇	1	2	yes	—
293	#336	C₂²×C₆₇	1	2	yes	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
296	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
299	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
300	C₄⋊Q₈	—	2	2	no	—
301	C₄⋊Q₈	—	2	2	no	—
302	C₂²⋊Q₈	—	4	4	no	—
303	#117	—	2	2	no	—
304	#117	—	2	2	no	—
305	#117	—	2	2	no	—
306	#117	Q₈×C₆₇	1	4	yes	complemented
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	—	2	1	no	—
322	C₂×C₄²	—	2	2	no	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
332	#107	—	2	4	no	—
333	#106	—	2	4	no	—
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	2	no	—
336	C₂²⋊C₈	—	2	4	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	#95	—	2	1	no	—
341	#95	—	2	2	no	—
342	#95	—	2	2	no	—
343	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂×Q₈	—	4	2	no	—
345	C₂×Q₈	—	4	2	no	—
346	C₂×Q₈	—	4	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	4	1	no	—
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	C₂×Q₈×C₆₇	1	2	yes	—
369	C₂²×C₄	#41×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
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	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	4	4	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
390	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
391	C₄⋊C₄	#41×C₆₇	1	2	yes	—
392	C₄⋊C₄	#41×C₆₇	1	2	yes	—
393	C₂²⋊C₄	—	2	4	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₄²	—	2	4	no	—
397	C₄²	—	2	4	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	—	4	2	no	—
403	C₄²	—	4	2	no	—
404	C₄²	—	4	2	no	—
405	C₄²	—	4	2	no	—
406	C₄²	#41×C₆₇	1	2	yes	—
407	C₄²	#41×C₆₇	1	2	yes	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
415	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
416	Q₈	—	8	4	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	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	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	—	4	1	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
451	C₂×C₄	#123×C₆₇	1	2	yes	—
452	C₂×C₄	#123×C₆₇	1	2	yes	—
453	C₂×C₄	#135×C₆₇	1	2	yes	—
454	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
455	C₂×C₄	#137×C₆₇	1	2	yes	—
456	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
457	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
458	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
472	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
473	C₂²	#525×C₆₇	1	1	yes	characteristic
474	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
475	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
476	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
477	C₂²	#575×C₆₇	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	2	no	—
487	C₄	—	8	2	no	—
488	C₄	#482×C₆₇	1	4	yes	—
489	C₄	#529×C₆₇	1	1	yes	characteristic
490	C₄	#529×C₆₇	1	1	yes	characteristic
491	C₄	#533×C₆₇	1	1	yes	characteristic
492	C₄	#533×C₆₇	1	1	yes	characteristic
493	C₂	—	4	2	no	—
494	C₂	C₆₇×#2855	1	1	yes	characteristic
495	C₂	C₆₇×#3076	1	1	yes	characteristic
496	C₂	C₆₇×#3079	1	1	yes	characteristic
497	C₂	C₂×#482×C₆₇	1	1	yes	characteristic
498	C₂	C₆₇×#3095	1	1	yes	characteristic
499	C₂	C₆₇×#3107	1	1	yes	characteristic
500	C₂	C₆₇×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1988	C₂	semidirect
4. C₆₇×#1988	C₂	semidirect
5. C₆₇×#1948	C₂	semidirect
6. C₆₇×#1947	C₂	semidirect
7. C₆₇×#1921	C₂	semidirect
8. C₆₇×#1920	C₂	semidirect
55. #117×C₆₇	Q₈	semidirect
242. #33154	C₆₇	direct (N × H)
253. #1988	C₂×C₆₇	semidirect
254. #1988	C₂×C₆₇	semidirect
255. #1948	C₂×C₆₇	semidirect
256. #1947	C₂×C₆₇	semidirect
257. #1921	C₂×C₆₇	semidirect
258. #1920	C₂×C₆₇	semidirect
262. C₆₇	#33154	direct (N × H)
306. #117	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2857	2
C₄×#117×C₆₇	2
C₆₇×#1988	2
C₆₇×#1948	2
C₆₇×#1947	2
C₆₇×#1921	2
C₆₇×#1920	2
C₆₇×#1854	2
C₆₇×#1825	2
C₆₇×#1797	2
#33154	67

Subgroups — C67×#33154 #30712405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33154 #30712405