Subgroups — C₆₇×#33114 #30712365

Order 17152.

876 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#485×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1928	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1927	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1873	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	C₂²	1	2	yes	—
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	#485×C₆₇	—	2	8	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#106×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₆₇	C₂²	1	2	yes	—
23	C₂×#103×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	#391×C₆₇	C₂²	1	2	yes	—
28	#389×C₆₇	—	2	1	no	—
29	#389×C₆₇	—	2	1	no	—
30	#384×C₆₇	C₂²	1	1	yes	characteristic
31	#380×C₆₇	—	2	1	no	—
32	#380×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
38	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
39	#376×C₆₇	—	2	1	no	—
40	C₄³×C₆₇	—	2	1	no	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	—	2	2	no	—
52	C₂²⋊Q₈×C₆₇	—	2	4	no	—
53	C₂²⋊Q₈×C₆₇	—	2	4	no	—
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	D₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	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×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	—	2	1	no	—
79	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	#107×C₆₇	—	2	4	no	—
84	#106×C₆₇	—	2	4	no	—
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	C₄⋊C₈×C₆₇	—	2	2	no	—
87	#103×C₆₇	—	2	4	no	—
88	#103×C₆₇	—	2	4	no	—
89	#103×C₆₇	—	2	4	no	—
90	#103×C₆₇	—	2	4	no	—
91	C₂²⋊C₈×C₆₇	—	2	4	no	—
92	#95×C₆₇	—	2	1	no	—
93	#95×C₆₇	—	2	1	no	—
94	#95×C₆₇	—	2	1	no	—
95	#95×C₆₇	—	2	1	no	—
96	#95×C₆₇	—	2	2	no	—
97	#95×C₆₇	—	2	2	no	—
98	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	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	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	4	1	no	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	#41	1	2	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	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₄⋊C₄×C₆₇	—	4	4	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	4	2	no	—
145	C₄⋊C₄×C₆₇	—	4	2	no	—
146	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₄⋊C₄×C₆₇	#41	1	2	yes	—
149	C₄⋊C₄×C₆₇	#41	1	2	yes	—
150	C₂²⋊C₄×C₆₇	—	2	4	no	—
151	C₂²⋊C₄×C₆₇	—	2	4	no	—
152	C₂²⋊C₄×C₆₇	—	4	4	no	—
153	C₄²×C₆₇	—	2	4	no	—
154	C₄²×C₆₇	—	2	4	no	—
155	C₄²×C₆₇	—	2	2	no	—
156	C₄²×C₆₇	—	2	2	no	—
157	C₄²×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	4	2	no	—
160	C₄²×C₆₇	—	4	2	no	—
161	C₄²×C₆₇	—	4	2	no	—
162	C₄²×C₆₇	—	4	2	no	—
163	C₄²×C₆₇	#41	1	2	yes	—
164	C₄²×C₆₇	#41	1	2	yes	—
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	1	no	—
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	#121	1	2	yes	—
172	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₆₇	—	4	2	no	—
174	Q₈×C₆₇	—	4	2	no	—
175	Q₈×C₆₇	—	4	2	no	—
176	Q₈×C₆₇	—	4	2	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	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	4	1	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	—	4	1	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	4	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	#121	1	2	yes	—
211	C₂×C₄×C₆₇	#123	1	2	yes	—
212	C₂×C₄×C₆₇	#123	1	2	yes	—
213	C₂×C₄×C₆₇	#135	1	2	yes	—
214	C₂×C₄×C₆₇	#137	1	2	yes	—
215	C₂×C₄×C₆₇	#137	1	2	yes	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
222	C₈×C₆₇	—	4	2	no	—
223	C₈×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	4	1	no	—
228	C₂²×C₆₇	—	4	1	no	—
229	C₂²×C₆₇	—	4	1	no	—
230	C₂²×C₆₇	—	4	1	no	—
231	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
232	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
233	C₂²×C₆₇	#525	1	1	yes	characteristic
234	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
235	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
236	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
237	C₂²×C₆₇	#575	1	1	yes	characteristic
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	2	2	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	—	4	2	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	—	8	2	no	—
247	C₄×C₆₇	—	8	2	no	—
248	C₄×C₆₇	#485	1	4	yes	complemented
249	C₄×C₆₇	#529	1	1	yes	characteristic
250	C₄×C₆₇	#529	1	1	yes	characteristic
251	C₄×C₆₇	#532	1	1	yes	characteristic
252	C₄×C₆₇	#532	1	1	yes	characteristic
253	#33114	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂×C₆₇	—	4	2	no	—
255	C₂×C₆₇	#2854	1	1	yes	characteristic
256	C₂×C₆₇	#3035	1	1	yes	characteristic
257	C₂×C₆₇	#3039	1	1	yes	characteristic
258	C₂×C₆₇	C₂×#485	1	1	yes	characteristic
259	C₂×C₆₇	#3095	1	1	yes	characteristic
260	C₂×C₆₇	#3107	1	1	yes	characteristic
261	C₂×C₆₇	#3116	1	1	yes	characteristic
262	C₂×#485	C₂×C₆₇	1	2	yes	—
263	#2857	C₂×C₆₇	1	1	yes	characteristic
264	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
265	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
267	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
269	#1928	C₂×C₆₇	1	2	yes	complemented
270	#1927	C₂×C₆₇	1	2	yes	complemented
271	#1873	C₂×C₆₇	1	2	yes	—
272	#1797	C₂×C₆₇	1	1	yes	characteristic
273	C₆₇	#33114	1	1	yes	characteristic, direct product factor, Sylow
274	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
276	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
278	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
279	#485	—	2	8	no	—
280	C₂×#107	C₂²×C₆₇	1	2	yes	—
281	C₂×#106	C₂²×C₆₇	1	2	yes	—
282	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×#103	C₂²×C₆₇	1	2	yes	—
285	C₂×#103	C₂²×C₆₇	1	2	yes	—
286	C₂×#103	C₂²×C₆₇	1	2	yes	—
287	C₂×#103	C₂²×C₆₇	1	2	yes	—
288	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
289	#391	C₂²×C₆₇	1	2	yes	—
290	#389	—	2	1	no	—
291	#389	—	2	1	no	—
292	#384	C₂²×C₆₇	1	1	yes	characteristic
293	#380	—	2	1	no	—
294	#380	—	2	1	no	—
295	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
296	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
297	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
298	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
299	C₄×C₂²⋊C₄	—	2	1	no	—
300	C₄×C₂²⋊C₄	—	2	1	no	—
301	#376	—	2	1	no	—
302	C₄³	—	2	1	no	—
303	#340	C₂²×C₆₇	1	2	yes	—
304	#340	C₂²×C₆₇	1	2	yes	—
305	#336	C₂²×C₆₇	1	2	yes	—
306	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
307	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
308	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
311	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₂²⋊Q₈	—	2	4	no	—
315	C₂²⋊Q₈	—	2	4	no	—
316	#117	—	2	2	no	—
317	#117	—	2	2	no	—
318	#117	—	2	2	no	—
319	#117	D₄×C₆₇	1	4	yes	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	2	2	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
345	#107	—	2	4	no	—
346	#106	—	2	4	no	—
347	C₄⋊C₈	—	2	2	no	—
348	C₄⋊C₈	—	2	2	no	—
349	#103	—	2	4	no	—
350	#103	—	2	4	no	—
351	#103	—	2	4	no	—
352	#103	—	2	4	no	—
353	C₂²⋊C₈	—	2	4	no	—
354	#95	—	2	1	no	—
355	#95	—	2	1	no	—
356	#95	—	2	1	no	—
357	#95	—	2	1	no	—
358	#95	—	2	2	no	—
359	#95	—	2	2	no	—
360	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	4	1	no	—
383	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
389	C₂²×C₄	#41×C₆₇	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
409	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
410	C₄⋊C₄	#41×C₆₇	1	2	yes	—
411	C₄⋊C₄	#41×C₆₇	1	2	yes	—
412	C₂²⋊C₄	—	2	4	no	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₄²	—	2	4	no	—
416	C₄²	—	2	4	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	4	2	no	—
422	C₄²	—	4	2	no	—
423	C₄²	—	4	2	no	—
424	C₄²	—	4	2	no	—
425	C₄²	#41×C₆₇	1	2	yes	—
426	C₄²	#41×C₆₇	1	2	yes	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	1	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	#121×C₆₇	1	2	yes	—
434	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	4	2	no	—
438	Q₈	—	4	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	#121×C₆₇	1	2	yes	—
473	C₂×C₄	#123×C₆₇	1	2	yes	—
474	C₂×C₄	#123×C₆₇	1	2	yes	—
475	C₂×C₄	#135×C₆₇	1	2	yes	—
476	C₂×C₄	#137×C₆₇	1	2	yes	—
477	C₂×C₄	#137×C₆₇	1	2	yes	—
478	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
484	C₈	—	4	2	no	—
485	C₈	—	4	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	4	1	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	4	1	no	—
493	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
494	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
495	C₂²	#525×C₆₇	1	1	yes	characteristic
496	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
497	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
498	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
499	C₂²	#575×C₆₇	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	1	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	2	no	—
509	C₄	—	8	2	no	—
510	C₄	#485×C₆₇	1	4	yes	complemented
511	C₄	#529×C₆₇	1	1	yes	characteristic
512	C₄	#529×C₆₇	1	1	yes	characteristic
513	C₄	#532×C₆₇	1	1	yes	characteristic
514	C₄	#532×C₆₇	1	1	yes	characteristic
515	C₂	—	4	2	no	—
516	C₂	C₆₇×#2854	1	1	yes	characteristic
517	C₂	C₆₇×#3035	1	1	yes	characteristic
518	C₂	C₆₇×#3039	1	1	yes	characteristic
519	C₂	C₂×#485×C₆₇	1	1	yes	characteristic
520	C₂	C₆₇×#3095	1	1	yes	characteristic
521	C₂	C₆₇×#3107	1	1	yes	characteristic
522	C₂	C₆₇×#3116	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
4. C₆₇×#1981	C₂	semidirect
5. C₆₇×#1981	C₂	semidirect
6. C₆₇×#1948	C₂	semidirect
7. C₆₇×#1947	C₂	semidirect
8. C₆₇×#1928	C₂	semidirect
9. C₆₇×#1927	C₂	semidirect
248. C₄×C₆₇	#485	semidirect
253. #33114	C₆₇	direct (N × H)
265. #1981	C₂×C₆₇	semidirect
266. #1981	C₂×C₆₇	semidirect
267. #1948	C₂×C₆₇	semidirect
268. #1947	C₂×C₆₇	semidirect
269. #1928	C₂×C₆₇	semidirect
270. #1927	C₂×C₆₇	semidirect
273. C₆₇	#33114	direct (N × H)
510. C₄	#485×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#485×C₆₇	2
C₆₇×#2857	2
C₄×#117×C₆₇	2
C₆₇×#1981	2
C₆₇×#1948	2
C₆₇×#1947	2
C₆₇×#1928	2
C₆₇×#1927	2
C₆₇×#1873	2
C₆₇×#1797	2
#33114	67

Subgroups — C67×#33114 #30712365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33114 #30712365