Subgroups — C₆₇×#32918 #30712169

Order 17152.

1372 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#464×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#451×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#448×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2620	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1875	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
18	#519×C₆₇	—	2	8	no	—
19	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#117×C₆₇	—	2	2	no	—
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	#464×C₆₇	—	2	8	no	—
26	#451×C₆₇	—	2	4	no	—
27	#448×C₆₇	—	2	4	no	—
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#396×C₆₇	—	2	2	no	—
36	#394×C₆₇	—	2	2	no	—
37	#391×C₆₇	C₂²	1	1	yes	characteristic
38	#390×C₆₇	C₂²	1	1	yes	characteristic
39	#384×C₆₇	C₂²	1	1	yes	characteristic
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	4	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	D₄	1	4	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×Q₁₆×C₆₇	—	2	4	no	—
55	C₂×Q₁₆×C₆₇	—	2	4	no	—
56	C₂×Q₁₆×C₆₇	—	2	4	no	—
57	C₂×SD₁₆×C₆₇	—	2	4	no	—
58	C₂×SD₁₆×C₆₇	—	2	4	no	—
59	C₂×SD₁₆×C₆₇	—	2	4	no	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	#124×C₆₇	—	2	4	no	—
63	C₂²⋊Q₈×C₆₇	—	2	2	no	—
64	C₂²⋊Q₈×C₆₇	—	2	2	no	—
65	#121×C₆₇	—	2	2	no	—
66	#121×C₆₇	—	2	2	no	—
67	C₄×Q₈×C₆₇	—	2	16	no	—
68	C₄×Q₈×C₆₇	—	2	4	no	—
69	C₄×Q₈×C₆₇	—	2	4	no	—
70	C₄×Q₈×C₆₇	—	2	4	no	—
71	C₄×Q₈×C₆₇	D₄	1	8	yes	complemented
72	#117×C₆₇	—	2	16	no	—
73	#117×C₆₇	—	2	4	no	—
74	#117×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
89	C₂×C₄²×C₆₇	—	2	2	no	—
90	C₂×C₄²×C₆₇	—	2	2	no	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	—	2	4	no	—
95	#103×C₆₇	—	2	4	no	—
96	#103×C₆₇	—	2	4	no	—
97	#103×C₆₇	—	2	4	no	—
98	#102×C₆₇	—	2	4	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	C₂²⋊C₈×C₆₇	—	2	2	no	—
101	#95×C₆₇	—	2	2	no	—
102	C₂⁴×C₆₇	—	2	1	no	—
103	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×Q₈×C₆₇	—	2	4	no	—
105	C₂×Q₈×C₆₇	—	2	16	no	—
106	C₂×Q₈×C₆₇	—	2	4	no	—
107	C₂×Q₈×C₆₇	—	2	16	no	—
108	C₂×Q₈×C₆₇	—	2	4	no	—
109	C₂×Q₈×C₆₇	—	2	2	no	—
110	C₂×Q₈×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
117	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
118	C₂×Q₈×C₆₇	#41	1	4	yes	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	1	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	4	2	no	—
124	C₂×D₄×C₆₇	—	8	2	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₆₇	#41	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₆₇	—	4	8	no	—
142	SD₁₆×C₆₇	—	4	8	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	8	no	—
150	C₄⋊C₄×C₆₇	—	2	8	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	4	2	no	—
158	C₄⋊C₄×C₆₇	—	8	2	no	—
159	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
160	C₂²⋊C₄×C₆₇	—	2	4	no	—
161	C₂²⋊C₄×C₆₇	—	2	4	no	—
162	C₂²⋊C₄×C₆₇	—	4	2	no	—
163	C₂²⋊C₄×C₆₇	—	4	4	no	—
164	C₂²⋊C₄×C₆₇	—	4	2	no	—
165	C₂²⋊C₄×C₆₇	—	8	2	no	—
166	C₄²×C₆₇	—	2	8	no	—
167	C₄²×C₆₇	—	2	4	no	—
168	C₄²×C₆₇	—	2	8	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	C₂×D₄	1	4	yes	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	4	1	no	—
176	C₂³×C₆₇	—	4	1	no	—
177	C₂³×C₆₇	—	4	1	no	—
178	C₂³×C₆₇	—	4	2	no	—
179	C₂³×C₆₇	—	4	2	no	—
180	C₂³×C₆₇	#121	1	2	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	16	no	—
183	Q₈×C₆₇	—	2	4	no	—
184	Q₈×C₆₇	—	2	4	no	—
185	Q₈×C₆₇	—	2	4	no	—
186	Q₈×C₆₇	—	4	2	no	—
187	Q₈×C₆₇	—	4	2	no	—
188	Q₈×C₆₇	#121	1	8	yes	—
189	D₄×C₆₇	—	4	2	no	—
190	D₄×C₆₇	—	4	2	no	—
191	D₄×C₆₇	—	8	4	no	—
192	C₂×C₄×C₆₇	—	2	16	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	4	1	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	#120	1	4	yes	—
216	C₂×C₄×C₆₇	#121	1	2	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
221	C₈×C₆₇	—	4	2	no	—
222	C₈×C₆₇	—	4	2	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	—	8	1	no	—
229	C₂²×C₆₇	—	8	2	no	—
230	C₂²×C₆₇	—	8	2	no	—
231	C₂²×C₆₇	—	8	1	no	—
232	C₂²×C₆₇	—	8	1	no	—
233	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
234	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
235	C₂²×C₆₇	#525	1	1	yes	characteristic
236	C₂²×C₆₇	#577	1	2	yes	—
237	C₂²×C₆₇	#578	1	2	yes	—
238	C₄×C₆₇	—	2	8	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	4	no	—
243	C₄×C₆₇	—	8	1	no	—
244	C₄×C₆₇	—	8	1	no	—
245	C₄×C₆₇	—	8	1	no	—
246	C₄×C₆₇	C₂×#121	1	2	yes	—
247	C₄×C₆₇	#525	1	2	yes	—
248	#32918	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂×C₆₇	—	4	2	no	—
250	C₂×C₆₇	—	16	1	no	—
251	C₂×C₆₇	#2847	1	1	yes	characteristic
252	C₂×C₆₇	#3013	1	2	yes	—
253	C₂×C₆₇	#3043	1	1	yes	characteristic
254	C₂×C₆₇	#3044	1	1	yes	characteristic
255	C₂×C₆₇	#3121	1	2	yes	—
256	C₂×#519	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₂×#464	C₂×C₆₇	1	2	yes	complemented
258	C₂×#451	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#448	C₂×C₆₇	1	1	yes	characteristic
260	#2620	C₂×C₆₇	1	1	yes	characteristic
261	#1880	C₂×C₆₇	1	2	yes	complemented
262	#1878	C₂×C₆₇	1	2	yes	complemented
263	#1875	C₂×C₆₇	1	1	yes	characteristic
264	#1873	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1803	C₂×C₆₇	1	2	yes	complemented
266	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₆₇	#32918	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
269	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
270	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
271	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
274	#519	—	2	8	no	—
275	C₂×C₄×Q₈	—	2	2	no	—
276	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
277	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
278	C₂×#117	—	2	2	no	—
279	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
281	#464	—	2	8	no	—
282	#451	—	2	4	no	—
283	#448	—	2	4	no	—
284	C₂×C₄⋊C₈	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×#102	C₂²×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
291	#396	—	2	2	no	—
292	#394	—	2	2	no	—
293	#391	C₂²×C₆₇	1	1	yes	characteristic
294	#390	C₂²×C₆₇	1	1	yes	characteristic
295	#384	C₂²×C₆₇	1	1	yes	characteristic
296	#340	C₂²×C₆₇	1	2	yes	—
297	#340	C₂²×C₆₇	1	2	yes	—
298	#336	C₂²×C₆₇	1	2	yes	—
299	C₂²×Q₈	—	2	4	no	—
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×Q₈	—	2	2	no	—
303	C₂²×Q₈	D₄×C₆₇	1	4	yes	—
304	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂³×C₄	—	2	1	no	—
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂×Q₁₆	—	2	4	no	—
311	C₂×Q₁₆	—	2	4	no	—
312	C₂×Q₁₆	—	2	4	no	—
313	C₂×SD₁₆	—	2	4	no	—
314	C₂×SD₁₆	—	2	4	no	—
315	C₂×SD₁₆	—	2	4	no	—
316	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
317	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
318	#124	—	2	4	no	—
319	C₂²⋊Q₈	—	2	2	no	—
320	C₂²⋊Q₈	—	2	2	no	—
321	#121	—	2	2	no	—
322	#121	—	2	2	no	—
323	C₄×Q₈	—	2	16	no	—
324	C₄×Q₈	—	2	4	no	—
325	C₄×Q₈	—	2	4	no	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	D₄×C₆₇	1	8	yes	complemented
328	#117	—	2	16	no	—
329	#117	—	2	4	no	—
330	#117	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	—	2	2	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
350	C₄⋊C₈	—	2	4	no	—
351	#103	—	2	4	no	—
352	#103	—	2	4	no	—
353	#103	—	2	4	no	—
354	#102	—	2	4	no	—
355	C₂²⋊C₈	—	2	2	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	#95	—	2	2	no	—
358	C₂⁴	—	2	1	no	—
359	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	2	16	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	16	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
373	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
374	C₂×Q₈	#41×C₆₇	1	4	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	4	2	no	—
380	C₂×D₄	—	8	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
395	C₂²×C₄	#41×C₆₇	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
397	Q₁₆	—	4	8	no	—
398	SD₁₆	—	4	8	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	—	8	2	no	—
415	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	2	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₂²⋊C₄	—	8	2	no	—
422	C₄²	—	2	8	no	—
423	C₄²	—	2	4	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	2	no	—
426	C₄²	C₂×D₄×C₆₇	1	4	yes	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	#121×C₆₇	1	2	yes	—
437	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
438	Q₈	—	2	16	no	—
439	Q₈	—	2	4	no	—
440	Q₈	—	2	4	no	—
441	Q₈	—	2	4	no	—
442	Q₈	—	4	2	no	—
443	Q₈	—	4	2	no	—
444	Q₈	#121×C₆₇	1	8	yes	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	8	4	no	—
448	C₂×C₄	—	2	16	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	#120×C₆₇	1	4	yes	—
472	C₂×C₄	#121×C₆₇	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
475	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
476	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	8	2	no	—
485	C₂²	—	8	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	2	no	—
489	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
490	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
491	C₂²	#525×C₆₇	1	1	yes	characteristic
492	C₂²	#577×C₆₇	1	2	yes	—
493	C₂²	#578×C₆₇	1	2	yes	—
494	C₄	—	2	4	no	—
495	C₄	—	2	8	no	—
496	C₄	—	2	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	8	1	no	—
500	C₄	—	8	1	no	—
501	C₄	—	8	1	no	—
502	C₄	C₂×#121×C₆₇	1	2	yes	—
503	C₄	#525×C₆₇	1	2	yes	—
504	C₂	—	4	2	no	—
505	C₂	—	16	1	no	—
506	C₂	C₆₇×#2847	1	1	yes	characteristic
507	C₂	C₆₇×#3013	1	2	yes	—
508	C₂	C₆₇×#3043	1	1	yes	characteristic
509	C₂	C₆₇×#3044	1	1	yes	characteristic
510	C₂	C₆₇×#3121	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#519×C₆₇	C₂	semidirect
2. C₂×#464×C₆₇	C₂	semidirect
3. C₂×#451×C₆₇	C₂	semidirect
6. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1878	C₂	semidirect
9. C₆₇×#1873	C₂	semidirect
10. C₆₇×#1803	C₂	semidirect
11. C₆₇×#1798	C₂	semidirect
71. C₄×Q₈×C₆₇	D₄	semidirect
248. #32918	C₆₇	direct (N × H)
256. C₂×#519	C₂×C₆₇	semidirect
257. C₂×#464	C₂×C₆₇	semidirect
258. C₂×#451	C₂×C₆₇	semidirect
261. #1880	C₂×C₆₇	semidirect
262. #1878	C₂×C₆₇	semidirect
264. #1873	C₂×C₆₇	semidirect
265. #1803	C₂×C₆₇	semidirect
266. #1798	C₂×C₆₇	semidirect
267. C₆₇	#32918	direct (N × H)
327. C₄×Q₈	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×#519×C₆₇	2
C₂×#464×C₆₇	2
C₂×#451×C₆₇	2
C₂×#448×C₆₇	2
C₆₇×#2620	2
C₆₇×#1880	2
C₆₇×#1878	2
C₆₇×#1875	2
C₆₇×#1873	2
C₆₇×#1803	2
C₆₇×#1798	2
#32918	67

Subgroups — C67×#32918 #30712169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32918 #30712169