Subgroups — C₆₇×#25851 #30705102

Order 17152.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#501×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2059	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2058	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1864	C₂	1	2	yes	maximal
10	C₆₇×#1862	C₂	1	2	yes	maximal
11	C₆₇×#1762	C₂	1	1	yes	maximal, characteristic
12	#554×C₆₇	—	2	4	no	—
13	#554×C₆₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
21	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
27	#501×C₆₇	—	2	4	no	—
28	#476×C₆₇	—	2	4	no	—
29	C₂×#107×C₆₇	C₂²	1	2	yes	—
30	C₂×#106×C₆₇	C₂²	1	2	yes	—
31	C₂×#106×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#102×C₆₇	C₂²	1	2	yes	—
34	C₂×#102×C₆₇	C₂²	1	2	yes	—
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	#398×C₆₇	—	2	1	no	—
37	#398×C₆₇	—	2	1	no	—
38	#391×C₆₇	—	2	1	no	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#336×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×Q₈×C₆₇	—	2	1	no	—
49	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	—	2	1	no	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	—	2	2	no	—
59	C₄⋊Q₈×C₆₇	—	2	2	no	—
60	C₄⋊Q₈×C₆₇	—	4	4	no	—
61	#125×C₆₇	—	2	2	no	—
62	#125×C₆₇	—	2	2	no	—
63	#125×C₆₇	—	2	2	no	—
64	#125×C₆₇	D₄	1	4	yes	—
65	C₂²⋊Q₈×C₆₇	—	2	8	no	—
66	C₂²⋊Q₈×C₆₇	—	2	8	no	—
67	C₂²⋊Q₈×C₆₇	—	2	8	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	—	2	2	no	—
74	C₄×D₄×C₆₇	—	2	2	no	—
75	C₄×D₄×C₆₇	D₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	#107×C₆₇	—	2	4	no	—
101	#106×C₆₇	—	2	4	no	—
102	#106×C₆₇	—	2	4	no	—
103	C₄⋊C₈×C₆₇	—	2	2	no	—
104	#102×C₆₇	—	2	4	no	—
105	#102×C₆₇	Q₈	1	8	yes	complemented
106	#97×C₆₇	—	2	2	no	—
107	#95×C₆₇	—	2	1	no	—
108	#95×C₆₇	—	2	2	no	—
109	#95×C₆₇	—	2	2	no	—
110	C₂⁴×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	4	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	4	1	no	—
115	C₂×Q₈×C₆₇	—	4	2	no	—
116	C₂×Q₈×C₆₇	—	4	2	no	—
117	C₂×Q₈×C₆₇	—	4	2	no	—
118	C₂×Q₈×C₆₇	—	8	2	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	—	8	2	no	—
134	C₂²×C₄×C₆₇	—	8	2	no	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	4	no	—
146	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
147	C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₄⋊C₄×C₆₇	—	2	4	no	—
149	C₄⋊C₄×C₆₇	—	2	4	no	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	4	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	4	4	no	—
162	C₄⋊C₄×C₆₇	—	4	2	no	—
163	C₄⋊C₄×C₆₇	—	4	4	no	—
164	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	4	no	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	—	2	1	no	—
174	C₄²×C₆₇	—	2	1	no	—
175	C₄²×C₆₇	—	4	2	no	—
176	C₄²×C₆₇	C₂×D₄	1	2	yes	—
177	C₄²×C₆₇	C₂×D₄	1	2	yes	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	4	1	no	—
181	C₂³×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	—	4	2	no	—
183	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	Q₈×C₆₇	—	4	4	no	—
185	Q₈×C₆₇	—	8	4	no	—
186	Q₈×C₆₇	—	8	4	no	—
187	D₄×C₆₇	—	2	2	no	—
188	D₄×C₆₇	—	2	4	no	—
189	D₄×C₆₇	C₄⋊Q₈	1	4	yes	complemented
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	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	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	4	8	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	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	8	4	no	—
217	C₂×C₄×C₆₇	—	8	4	no	—
218	C₂×C₄×C₆₇	#120	1	2	yes	—
219	C₂×C₄×C₆₇	#120	1	2	yes	—
220	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
224	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	4	4	no	—
229	C₂²×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	—	4	2	no	—
232	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
233	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
234	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂²×C₆₇	#575	1	1	yes	characteristic
236	C₂²×C₆₇	#577	1	2	yes	—
237	C₄×C₆₇	—	2	2	no	—
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	4	1	no	—
244	C₄×C₆₇	—	4	1	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	—	8	2	no	—
248	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
249	C₄×C₆₇	#536	1	1	yes	characteristic
250	C₄×C₆₇	D₄×Q₈	1	2	yes	—
251	#25851	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	#2434	1	1	yes	characteristic
254	C₂×C₆₇	#3006	1	1	yes	characteristic
255	C₂×C₆₇	#3009	1	1	yes	characteristic
256	C₂×C₆₇	#3384	1	2	yes	—
257	C₂×C₆₇	#3388	1	2	yes	—
258	C₂×#554	C₂×C₆₇	1	1	yes	characteristic
259	C₂×#501	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#476	C₂×C₆₇	1	1	yes	characteristic
261	#2546	C₂×C₆₇	1	1	yes	characteristic, complemented
262	#2059	C₂×C₆₇	1	2	yes	complemented
263	#2058	C₂×C₆₇	1	2	yes	complemented
264	#1989	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1985	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#1864	C₂×C₆₇	1	2	yes	—
267	#1862	C₂×C₆₇	1	2	yes	—
268	#1762	C₂×C₆₇	1	1	yes	characteristic
269	C₆₇	#25851	1	1	yes	characteristic, direct product factor, Sylow
270	#554	—	2	4	no	—
271	#554	—	2	4	no	—
272	C₂×C₄⋊Q₈	—	2	1	no	—
273	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₂²⋊Q₈	—	2	1	no	—
278	C₂×C₂²⋊Q₈	—	2	1	no	—
279	C₂×C₂²⋊Q₈	—	2	1	no	—
280	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
282	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
284	C₂²×C₄⋊C₄	—	2	1	no	—
285	#501	—	2	4	no	—
286	#476	—	2	4	no	—
287	C₂×#107	C₂²×C₆₇	1	2	yes	—
288	C₂×#106	C₂²×C₆₇	1	2	yes	—
289	C₂×#106	C₂²×C₆₇	1	2	yes	—
290	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×#102	C₂²×C₆₇	1	2	yes	—
292	C₂×#102	C₂²×C₆₇	1	2	yes	—
293	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
294	#398	—	2	1	no	—
295	#398	—	2	1	no	—
296	#391	—	2	1	no	—
297	#389	C₂²×C₆₇	1	2	yes	—
298	#389	C₂²×C₆₇	1	2	yes	—
299	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
300	#340	C₂²×C₆₇	1	2	yes	—
301	#340	C₂²×C₆₇	1	2	yes	—
302	#340	C₂²×C₆₇	1	2	yes	—
303	#336	C₂²×C₆₇	1	2	yes	—
304	#336	C₂²×C₆₇	1	2	yes	—
305	C₂²×Q₈	—	2	1	no	—
306	C₂²×Q₈	—	2	1	no	—
307	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
308	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
309	C₂³×C₄	—	2	1	no	—
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	1	no	—
312	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
313	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	—	4	4	no	—
319	#125	—	2	2	no	—
320	#125	—	2	2	no	—
321	#125	—	2	2	no	—
322	#125	D₄×C₆₇	1	4	yes	—
323	C₂²⋊Q₈	—	2	8	no	—
324	C₂²⋊Q₈	—	2	8	no	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×D₄	—	2	2	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	2	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	D₄×C₆₇	1	4	yes	—
334	C₂×C₄⋊C₄	—	2	8	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
351	C₂×C₂²⋊C₄	—	2	1	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₂²⋊C₄	—	2	1	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
358	#107	—	2	4	no	—
359	#106	—	2	4	no	—
360	#106	—	2	4	no	—
361	C₄⋊C₈	—	2	2	no	—
362	#102	—	2	4	no	—
363	#102	Q₈×C₆₇	1	8	yes	complemented
364	#97	—	2	2	no	—
365	#95	—	2	1	no	—
366	#95	—	2	2	no	—
367	#95	—	2	2	no	—
368	C₂⁴	—	2	1	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	4	1	no	—
373	C₂×Q₈	—	4	2	no	—
374	C₂×Q₈	—	4	2	no	—
375	C₂×Q₈	—	4	2	no	—
376	C₂×Q₈	—	8	2	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
380	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	8	2	no	—
392	C₂²×C₄	—	8	2	no	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	4	no	—
404	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	4	4	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	4	4	no	—
422	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
423	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
424	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
425	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	2	1	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	4	2	no	—
434	C₄²	C₂×D₄×C₆₇	1	2	yes	—
435	C₄²	C₂×D₄×C₆₇	1	2	yes	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	2	no	—
441	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
442	Q₈	—	4	4	no	—
443	Q₈	—	8	4	no	—
444	Q₈	—	8	4	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	4	no	—
447	D₄	C₄⋊Q₈×C₆₇	1	4	yes	complemented
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	8	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	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	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	8	4	no	—
475	C₂×C₄	—	8	4	no	—
476	C₂×C₄	#120×C₆₇	1	2	yes	—
477	C₂×C₄	#120×C₆₇	1	2	yes	—
478	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
481	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
482	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
483	C₈	—	2	4	no	—
484	C₈	—	4	2	no	—
485	C₂²	—	2	4	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
491	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
492	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
493	C₂²	#575×C₆₇	1	1	yes	characteristic
494	C₂²	#577×C₆₇	1	2	yes	—
495	C₄	—	2	2	no	—
496	C₄	—	2	2	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	1	no	—
504	C₄	—	4	2	no	—
505	C₄	—	8	2	no	—
506	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
507	C₄	#536×C₆₇	1	1	yes	characteristic
508	C₄	D₄×Q₈×C₆₇	1	2	yes	—
509	C₂	—	4	4	no	—
510	C₂	C₆₇×#2434	1	1	yes	characteristic
511	C₂	C₆₇×#3006	1	1	yes	characteristic
512	C₂	C₆₇×#3009	1	1	yes	characteristic
513	C₂	C₆₇×#3384	1	2	yes	—
514	C₂	C₆₇×#3388	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#501×C₆₇	C₂	semidirect
4. C₆₇×#2546	C₂	semidirect
5. C₆₇×#2059	C₂	semidirect
6. C₆₇×#2058	C₂	semidirect
7. C₆₇×#1989	C₂	semidirect
8. C₆₇×#1985	C₂	semidirect
105. #102×C₆₇	Q₈	semidirect
189. D₄×C₆₇	C₄⋊Q₈	semidirect
251. #25851	C₆₇	direct (N × H)
259. C₂×#501	C₂×C₆₇	semidirect
261. #2546	C₂×C₆₇	semidirect
262. #2059	C₂×C₆₇	semidirect
263. #2058	C₂×C₆₇	semidirect
264. #1989	C₂×C₆₇	semidirect
265. #1985	C₂×C₆₇	semidirect
269. C₆₇	#25851	direct (N × H)
363. #102	Q₈×C₆₇	semidirect
447. D₄	C₄⋊Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
C₂×#501×C₆₇	2
C₂×#476×C₆₇	2
C₆₇×#2546	2
C₆₇×#2059	2
C₆₇×#2058	2
C₆₇×#1989	2
C₆₇×#1985	2
C₆₇×#1864	2
C₆₇×#1862	2
C₆₇×#1762	2
#25851	67

Subgroups — C67×#25851 #30705102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25851 #30705102