Subgroups — C₆₇×#32938 #30712189

Order 17152.

1436 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#468×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2616	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2615	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1895	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1894	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1892	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1809	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	—	2	8	no	—
15	C₂×#131×C₆₇	—	2	4	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	#468×C₆₇	—	2	8	no	—
25	#465×C₆₇	—	2	4	no	—
26	C₂×#107×C₆₇	C₂²	1	2	yes	—
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	#395×C₆₇	—	2	1	no	—
33	#394×C₆₇	—	2	2	no	—
34	#394×C₆₇	—	2	1	no	—
35	#393×C₆₇	—	2	1	no	—
36	#392×C₆₇	—	2	1	no	—
37	#392×C₆₇	—	2	2	no	—
38	#380×C₆₇	C₂²	1	1	yes	characteristic
39	#380×C₆₇	C₂²	1	1	yes	characteristic
40	#380×C₆₇	C₂²	1	2	yes	—
41	#380×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	C₂×#41×C₆₇	—	2	4	no	—
46	C₂×#41×C₆₇	—	2	2	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	4	no	—
49	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	D₄	1	2	yes	—
54	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×SD₁₆×C₆₇	—	2	2	no	—
59	C₂×SD₁₆×C₆₇	—	2	4	no	—
60	#131×C₆₇	—	2	32	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	4	no	—
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
66	C₂²×C₈×C₆₇	—	2	4	no	—
67	C₂²×C₈×C₆₇	—	2	4	no	—
68	C₂²×C₈×C₆₇	—	2	4	no	—
69	C₂²×C₈×C₆₇	—	2	4	no	—
70	C₂²×C₈×C₆₇	D₄	1	4	yes	complemented
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	—	2	4	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	4	no	—
78	#121×C₆₇	—	2	2	no	—
79	#121×C₆₇	—	2	4	no	—
80	#121×C₆₇	—	2	2	no	—
81	#121×C₆₇	—	2	4	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	#107×C₆₇	—	2	4	no	—
97	#106×C₆₇	—	2	2	no	—
98	#103×C₆₇	—	2	2	no	—
99	#103×C₆₇	—	2	4	no	—
100	#102×C₆₇	—	2	2	no	—
101	#102×C₆₇	—	2	4	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	2	no	—
104	#95×C₆₇	—	2	1	no	—
105	#95×C₆₇	—	2	2	no	—
106	C₂⁴×C₆₇	—	2	1	no	—
107	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
109	#41×C₆₇	—	2	8	no	—
110	#41×C₆₇	—	2	16	no	—
111	#41×C₆₇	—	2	8	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	4	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	1	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	4	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	—	4	4	no	—
129	C₂×D₄×C₆₇	—	8	2	no	—
130	C₂×D₄×C₆₇	—	8	2	no	—
131	C₂×D₄×C₆₇	—	8	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	#41	1	2	yes	—
148	C₂²×C₄×C₆₇	#41	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	SD₁₆×C₆₇	—	4	4	no	—
151	M₁₆×C₆₇	—	2	8	no	—
152	M₁₆×C₆₇	—	2	16	no	—
153	C₂×C₈×C₆₇	—	2	16	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	8	no	—
158	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
160	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	1	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	4	2	no	—
171	C₄⋊C₄×C₆₇	—	4	4	no	—
172	C₂²⋊C₄×C₆₇	—	4	2	no	—
173	C₂²⋊C₄×C₆₇	—	4	4	no	—
174	C₂²⋊C₄×C₆₇	—	4	2	no	—
175	C₂²⋊C₄×C₆₇	—	4	4	no	—
176	C₂²⋊C₄×C₆₇	—	8	1	no	—
177	C₂²⋊C₄×C₆₇	—	8	2	no	—
178	C₂²⋊C₄×C₆₇	—	8	1	no	—
179	C₂²⋊C₄×C₆₇	—	8	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	2	1	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	2	2	no	—
186	C₂³×C₆₇	—	2	2	no	—
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	2	1	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	4	1	no	—
191	C₂³×C₆₇	—	4	2	no	—
192	C₂³×C₆₇	—	4	1	no	—
193	C₂³×C₆₇	—	4	2	no	—
194	C₂³×C₆₇	—	4	1	no	—
195	C₂³×C₆₇	#121	1	2	yes	—
196	C₂³×C₆₇	#121	1	4	yes	—
197	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₆₇	—	2	8	no	—
199	Q₈×C₆₇	—	4	1	no	—
200	Q₈×C₆₇	—	4	1	no	—
201	Q₈×C₆₇	—	4	2	no	—
202	D₄×C₆₇	—	2	16	no	—
203	D₄×C₆₇	—	2	8	no	—
204	D₄×C₆₇	—	4	1	no	—
205	D₄×C₆₇	—	4	1	no	—
206	D₄×C₆₇	—	4	2	no	—
207	D₄×C₆₇	—	8	4	no	—
208	D₄×C₆₇	—	8	8	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	4	1	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	1	no	—
222	C₂×C₄×C₆₇	—	4	2	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	—	4	2	no	—
225	C₂×C₄×C₆₇	—	4	1	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	#121	1	2	yes	—
233	C₂×C₄×C₆₇	#121	1	4	yes	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
238	C₈×C₆₇	—	2	2	no	—
239	C₈×C₆₇	—	2	4	no	—
240	C₈×C₆₇	#120	1	4	yes	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	2	4	no	—
243	C₂²×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	2	2	no	—
245	C₂²×C₆₇	—	2	4	no	—
246	C₂²×C₆₇	—	2	4	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	4	2	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	8	1	no	—
254	C₂²×C₆₇	—	8	1	no	—
255	C₂²×C₆₇	—	8	1	no	—
256	C₂²×C₆₇	—	8	2	no	—
257	C₂²×C₆₇	—	8	2	no	—
258	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
259	C₂²×C₆₇	C₂×#121	1	2	yes	—
260	C₂²×C₆₇	#576	1	1	yes	characteristic
261	C₂²×C₆₇	#577	1	2	yes	—
262	C₂²×C₆₇	#578	1	1	yes	characteristic
263	C₄×C₆₇	—	2	4	no	—
264	C₄×C₆₇	—	2	2	no	—
265	C₄×C₆₇	—	8	1	no	—
266	C₄×C₆₇	—	8	1	no	—
267	C₄×C₆₇	—	8	1	no	—
268	C₄×C₆₇	—	8	2	no	—
269	C₄×C₆₇	—	8	2	no	—
270	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
271	C₄×C₆₇	#525	1	2	yes	—
272	C₄×C₆₇	#525	1	1	yes	characteristic
273	#32938	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂×C₆₇	—	4	2	no	—
275	C₂×C₆₇	—	4	4	no	—
276	C₂×C₆₇	—	16	1	no	—
277	C₂×C₆₇	#2847	1	1	yes	characteristic
278	C₂×C₆₇	#3057	1	1	yes	characteristic
279	C₂×C₆₇	#3061	1	1	yes	characteristic
280	C₂×C₆₇	#3062	1	2	yes	—
281	C₂×C₆₇	#3063	1	2	yes	—
282	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
283	C₂×#468	C₂×C₆₇	1	2	yes	complemented
284	C₂×#465	C₂×C₆₇	1	1	yes	characteristic, complemented
285	#2616	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#2615	C₂×C₆₇	1	1	yes	characteristic
287	#1895	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#1894	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#1893	C₂×C₆₇	1	2	yes	complemented
290	#1892	C₂×C₆₇	1	2	yes	complemented
291	#1809	C₂×C₆₇	1	2	yes	complemented
292	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
293	C₆₇	#32938	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×#131	—	2	8	no	—
297	C₂×#131	—	2	4	no	—
298	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
299	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
300	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
301	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
303	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#121	C₂²×C₆₇	1	2	yes	—
305	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
306	#468	—	2	8	no	—
307	#465	—	2	4	no	—
308	C₂×#107	C₂²×C₆₇	1	2	yes	—
309	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×#103	C₂²×C₆₇	1	2	yes	—
312	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#102	C₂²×C₆₇	1	2	yes	—
314	#395	—	2	1	no	—
315	#394	—	2	1	no	—
316	#394	—	2	2	no	—
317	#393	—	2	1	no	—
318	#392	—	2	1	no	—
319	#392	—	2	2	no	—
320	#380	C₂²×C₆₇	1	1	yes	characteristic
321	#380	C₂²×C₆₇	1	1	yes	characteristic
322	#380	C₂²×C₆₇	1	2	yes	—
323	#380	C₂²×C₆₇	1	2	yes	—
324	#340	C₂²×C₆₇	1	2	yes	—
325	#340	C₂²×C₆₇	1	2	yes	—
326	#340	C₂²×C₆₇	1	2	yes	—
327	C₂×#41	—	2	4	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	—	2	2	no	—
330	C₂×#41	—	2	4	no	—
331	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
332	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	D₄×C₆₇	1	2	yes	—
336	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
337	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
339	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	4	no	—
342	#131	—	2	32	no	—
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	—	2	4	no	—
345	C₂×M₁₆	—	2	4	no	—
346	C₂×M₁₆	—	2	4	no	—
347	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
348	C₂²×C₈	—	2	4	no	—
349	C₂²×C₈	—	2	4	no	—
350	C₂²×C₈	—	2	4	no	—
351	C₂²×C₈	—	2	4	no	—
352	C₂²×C₈	D₄×C₆₇	1	4	yes	complemented
353	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
354	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
355	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
356	C₂²⋊Q₈	—	2	2	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	2	no	—
360	#121	—	2	2	no	—
361	#121	—	2	4	no	—
362	#121	—	2	4	no	—
363	#121	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₂²⋊C₄	—	2	2	no	—
378	#107	—	2	4	no	—
379	#106	—	2	2	no	—
380	#103	—	2	2	no	—
381	#103	—	2	4	no	—
382	#102	—	2	2	no	—
383	#102	—	2	4	no	—
384	#95	—	2	1	no	—
385	#95	—	2	2	no	—
386	#95	—	2	1	no	—
387	#95	—	2	2	no	—
388	C₂⁴	—	2	1	no	—
389	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
390	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
391	#41	—	2	8	no	—
392	#41	—	2	16	no	—
393	#41	—	2	8	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	4	2	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	8	2	no	—
412	C₂×D₄	—	8	2	no	—
413	C₂×D₄	—	8	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
428	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
429	C₂²×C₄	#41×C₆₇	1	2	yes	—
430	C₂²×C₄	#41×C₆₇	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
432	SD₁₆	—	4	4	no	—
433	M₁₆	—	2	8	no	—
434	M₁₆	—	2	16	no	—
435	C₂×C₈	—	2	16	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	8	no	—
440	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
441	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
442	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
443	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	4	2	no	—
453	C₄⋊C₄	—	4	4	no	—
454	C₂²⋊C₄	—	4	2	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	—	4	2	no	—
457	C₂²⋊C₄	—	4	4	no	—
458	C₂²⋊C₄	—	8	1	no	—
459	C₂²⋊C₄	—	8	2	no	—
460	C₂²⋊C₄	—	8	1	no	—
461	C₂²⋊C₄	—	8	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	2	no	—
476	C₂³	—	4	1	no	—
477	C₂³	#121×C₆₇	1	2	yes	—
478	C₂³	#121×C₆₇	1	4	yes	—
479	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
480	Q₈	—	2	8	no	—
481	Q₈	—	4	1	no	—
482	Q₈	—	4	1	no	—
483	Q₈	—	4	2	no	—
484	D₄	—	2	16	no	—
485	D₄	—	2	8	no	—
486	D₄	—	4	1	no	—
487	D₄	—	4	1	no	—
488	D₄	—	4	2	no	—
489	D₄	—	8	4	no	—
490	D₄	—	8	8	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	#121×C₆₇	1	2	yes	—
515	C₂×C₄	#121×C₆₇	1	4	yes	—
516	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
518	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
519	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
520	C₈	—	2	2	no	—
521	C₈	—	2	4	no	—
522	C₈	#120×C₆₇	1	4	yes	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	1	no	—
537	C₂²	—	8	2	no	—
538	C₂²	—	8	2	no	—
539	C₂²	—	8	1	no	—
540	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
541	C₂²	C₂×#121×C₆₇	1	2	yes	—
542	C₂²	#576×C₆₇	1	1	yes	characteristic
543	C₂²	#577×C₆₇	1	2	yes	—
544	C₂²	#578×C₆₇	1	1	yes	characteristic
545	C₄	—	2	4	no	—
546	C₄	—	2	2	no	—
547	C₄	—	8	1	no	—
548	C₄	—	8	1	no	—
549	C₄	—	8	2	no	—
550	C₄	—	8	2	no	—
551	C₄	—	8	1	no	—
552	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
553	C₄	#525×C₆₇	1	2	yes	—
554	C₄	#525×C₆₇	1	1	yes	characteristic
555	C₂	—	4	2	no	—
556	C₂	—	4	4	no	—
557	C₂	—	16	1	no	—
558	C₂	C₆₇×#2847	1	1	yes	characteristic
559	C₂	C₆₇×#3057	1	1	yes	characteristic
560	C₂	C₆₇×#3061	1	1	yes	characteristic
561	C₂	C₆₇×#3062	1	2	yes	—
562	C₂	C₆₇×#3063	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2615	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#131×C₆₇	C₂	semidirect
2. C₂×#468×C₆₇	C₂	semidirect
3. C₂×#465×C₆₇	C₂	semidirect
4. C₆₇×#2616	C₂	semidirect
6. C₆₇×#1895	C₂	semidirect
7. C₆₇×#1894	C₂	semidirect
8. C₆₇×#1893	C₂	semidirect
9. C₆₇×#1892	C₂	semidirect
10. C₆₇×#1809	C₂	semidirect
11. C₆₇×#1808	C₂	semidirect
65. C₂×M₁₆×C₆₇	D₄	semidirect
70. C₂²×C₈×C₆₇	D₄	semidirect
273. #32938	C₆₇	direct (N × H)
282. C₂²×#131	C₂×C₆₇	semidirect
283. C₂×#468	C₂×C₆₇	semidirect
284. C₂×#465	C₂×C₆₇	semidirect
285. #2616	C₂×C₆₇	semidirect
287. #1895	C₂×C₆₇	semidirect
288. #1894	C₂×C₆₇	semidirect
289. #1893	C₂×C₆₇	semidirect
290. #1892	C₂×C₆₇	semidirect
291. #1809	C₂×C₆₇	semidirect
292. #1808	C₂×C₆₇	semidirect
293. C₆₇	#32938	direct (N × H)
347. C₂×M₁₆	D₄×C₆₇	semidirect
352. C₂²×C₈	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×#131×C₆₇	2
C₂×#468×C₆₇	2
C₂×#465×C₆₇	2
C₆₇×#2616	2
C₆₇×#2615	2
C₆₇×#1895	2
C₆₇×#1894	2
C₆₇×#1893	2
C₆₇×#1892	2
C₆₇×#1809	2
C₆₇×#1808	2
#32938	67

Subgroups — C67×#32938 #30712189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32938 #30712189