Subgroups — #30761626

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₆₇⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863151	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863150	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863150	C₂	1	1	yes	maximal, characteristic
7	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×Dic₆₇⋊C₄	C₄	1	2	yes	—
11	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	—	2	2	no	—
14	C₂×Dic₆₇⋊C₈	—	2	2	no	—
15	C₂×Dic₆₇⋊C₈	C₄	1	4	yes	complemented
16	C₂×#14523953	—	2	2	no	—
17	C₂×#14523953	—	2	2	no	—
18	C₂×#14523953	C₄	1	4	yes	complemented
19	C₄×Dic₆₇⋊C₄	—	2	2	no	—
20	C₄×Dic₆₇⋊C₄	—	2	2	no	—
21	C₂×C₆₇⋊#95	C₄	1	2	yes	—
22	#16863151	—	2	2	no	—
23	#16863150	—	2	2	no	—
24	#16863150	—	2	2	no	—
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#336×C₆₇	—	2	2	no	—
29	C₂³×Dic₆₇	C₂×C₄	1	2	yes	—
30	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₆₇⋊C₈	—	2	1	no	—
33	C₂²×C₆₇⋊C₈	—	2	1	no	—
34	C₂²×C₆₇⋊C₈	—	2	1	no	—
35	C₂²×C₆₇⋊C₈	—	2	1	no	—
36	C₂²×C₆₇⋊C₈	—	2	1	no	—
37	C₂²×C₆₇⋊C₈	—	2	1	no	—
38	C₂²×C₆₇⋊C₈	—	2	1	no	—
39	C₂²×C₆₇⋊C₈	—	2	1	no	—
40	C₂²×C₆₇⋊C₈	—	2	1	no	—
41	C₂²×C₆₇⋊C₈	—	2	1	no	—
42	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
45	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
46	C₂×Dic₆₇⋊C₄	—	2	4	no	—
47	C₂×Dic₆₇⋊C₄	—	2	4	no	—
48	C₂×Dic₆₇⋊C₄	—	2	4	no	—
49	C₂×C₄×Dic₆₇	—	2	1	no	—
50	C₂×C₄×Dic₆₇	—	2	1	no	—
51	C₂×C₄×Dic₆₇	—	2	1	no	—
52	C₂×C₄×Dic₆₇	—	2	1	no	—
53	C₂×C₄×Dic₆₇	—	2	1	no	—
54	C₂×C₄×Dic₆₇	—	2	1	no	—
55	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
58	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
59	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
60	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
61	Dic₆₇⋊C₈	—	2	8	no	—
62	#14523953	—	2	8	no	—
63	C₆₇⋊#95	—	2	4	no	—
64	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	—	2	1	no	—
67	C₂²×C₈×C₆₇	—	2	1	no	—
68	C₂²×C₈×C₆₇	—	2	1	no	—
69	C₂²×C₈×C₆₇	—	2	1	no	—
70	C₂²×C₈×C₆₇	—	2	1	no	—
71	C₂²×C₈×C₆₇	—	2	1	no	—
72	C₂²×C₈×C₆₇	—	2	1	no	—
73	C₂²×C₈×C₆₇	—	2	1	no	—
74	C₂²×C₈×C₆₇	—	2	1	no	—
75	C₂²×C₈×C₆₇	—	2	1	no	—
76	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
78	C₂²×C₈×C₆₇	D₄	1	2	yes	—
79	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
80	C₂×C₄²×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	—	2	1	no	—
82	C₂×C₄²×C₆₇	—	2	1	no	—
83	C₂×C₄²×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	—	2	1	no	—
86	C₂²×Dic₆₇	—	2	2	no	—
87	C₂²×Dic₆₇	—	2	2	no	—
88	C₂²×Dic₆₇	—	2	2	no	—
89	C₂²×Dic₆₇	—	2	2	no	—
90	C₂²×Dic₆₇	—	2	2	no	—
91	C₂²×Dic₆₇	—	2	2	no	—
92	C₂²×Dic₆₇	—	2	2	no	—
93	C₂²×Dic₆₇	—	2	2	no	—
94	C₂²×Dic₆₇	—	2	2	no	—
95	C₂²×Dic₆₇	—	2	2	no	—
96	C₂²×Dic₆₇	C₄²	1	2	yes	—
97	C₂²×Dic₆₇	C₄²	1	2	yes	—
98	C₂²×Dic₆₇	C₂²⋊C₄	1	2	yes	—
99	C₂²×Dic₆₇	C₂²⋊C₄	1	2	yes	—
100	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
101	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
102	C₂²×Dic₆₇	M₁₆	1	4	yes	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₂×C₆₇⋊C₈	—	2	2	no	—
110	C₂×C₆₇⋊C₈	—	2	2	no	—
111	C₂×C₆₇⋊C₈	—	2	2	no	—
112	C₂×C₆₇⋊C₈	—	4	1	no	—
113	C₂×C₆₇⋊C₈	—	4	1	no	—
114	C₂×C₆₇⋊C₈	—	4	1	no	—
115	C₂×C₆₇⋊C₈	—	4	1	no	—
116	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	complemented
117	Dic₆₇⋊C₄	—	2	16	no	—
118	C₄×Dic₆₇	—	4	1	no	—
119	C₄×Dic₆₇	—	4	1	no	—
120	C₄×Dic₆₇	—	4	1	no	—
121	C₄×Dic₆₇	—	4	1	no	—
122	C₄×Dic₆₇	C₂²⋊C₄	1	4	yes	—
123	C₄×Dic₆₇	C₂²⋊C₄	1	4	yes	—
124	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
125	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
126	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	—	2	2	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	2	no	—
133	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²×C₄	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	C₂×Q₈	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	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	—	4	1	no	—
158	C₂×C₈×C₆₇	—	4	1	no	—
159	C₂×C₈×C₆₇	—	4	1	no	—
160	C₂×C₈×C₆₇	—	4	1	no	—
161	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
162	C₄²×C₆₇	—	4	1	no	—
163	C₄²×C₆₇	—	4	1	no	—
164	C₄²×C₆₇	—	4	1	no	—
165	C₄²×C₆₇	—	4	1	no	—
166	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
167	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
170	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
171	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
172	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
173	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
174	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
175	C₂³×C₆₇	#131	1	2	yes	—
176	C₂³×C₆₇	#131	1	2	yes	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
191	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
197	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
198	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
199	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
204	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
205	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
206	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
207	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
208	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
212	C₂×Dic₆₇	—	2	2	no	—
213	C₂×Dic₆₇	—	2	2	no	—
214	C₂×Dic₆₇	—	2	2	no	—
215	C₂×Dic₆₇	—	2	2	no	—
216	C₂×Dic₆₇	—	2	4	no	—
217	C₂×Dic₆₇	—	2	4	no	—
218	C₂×Dic₆₇	—	2	4	no	—
219	C₂×Dic₆₇	—	2	4	no	—
220	C₂×Dic₆₇	—	2	4	no	—
221	C₂×Dic₆₇	—	2	4	no	—
222	C₂×Dic₆₇	—	2	4	no	—
223	C₂×Dic₆₇	—	4	2	no	—
224	C₂×Dic₆₇	—	4	2	no	—
225	C₂×Dic₆₇	—	4	2	no	—
226	C₂×Dic₆₇	—	4	2	no	—
227	C₂×Dic₆₇	#95	1	4	yes	—
228	C₂×Dic₆₇	#95	1	4	yes	—
229	C₈×C₆₇	—	2	4	no	—
230	C₈×C₆₇	—	4	2	no	—
231	C₆₇⋊C₈	—	2	4	no	—
232	C₆₇⋊C₈	—	4	2	no	—
233	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
234	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
235	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
236	C₂²×C₆₇	#389	1	1	yes	characteristic
237	C₂²×C₆₇	#389	1	1	yes	characteristic
238	C₂²×C₆₇	#390	1	1	yes	characteristic
239	C₂²×C₆₇	#391	1	1	yes	characteristic
240	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
241	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
242	C₂²×C₆₇	#405	1	1	yes	characteristic
243	C₂²×C₆₇	#405	1	1	yes	characteristic
244	C₂²×C₆₇	#407	1	2	yes	—
245	C₂²×C₆₇	#408	1	2	yes	—
246	C₂²×C₆₇	#423	1	2	yes	—
247	C₂²×C₆₇	#424	1	2	yes	—
248	C₂²×C₆₇	#435	1	2	yes	—
249	C₂²×C₆₇	#435	1	2	yes	—
250	C₂²×C₆₇	#435	1	2	yes	—
251	C₂²×C₆₇	#435	1	2	yes	—
252	C₂²×C₆₇	#446	1	2	yes	—
253	C₂²×C₆₇	#446	1	2	yes	—
254	C₂²×C₆₇	#446	1	2	yes	—
255	C₂²×C₆₇	#446	1	2	yes	—
256	C₄×C₆₇	—	4	4	no	—
257	C₄×C₆₇	#380	1	1	yes	characteristic
258	C₄×C₆₇	#380	1	1	yes	characteristic
259	C₄×C₆₇	#382	1	1	yes	characteristic
260	C₄×C₆₇	#384	1	1	yes	characteristic
261	C₄×C₆₇	#389	1	1	yes	characteristic
262	C₄×C₆₇	#389	1	1	yes	characteristic
263	C₄×C₆₇	#390	1	1	yes	characteristic
264	C₄×C₆₇	#391	1	1	yes	characteristic
265	Dic₆₇	—	2	4	no	—
266	Dic₆₇	—	2	4	no	—
267	Dic₆₇	—	4	4	no	—
268	#10835	—	67	1	no	maximal, Sylow
269	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
270	C₂×C₆₇	#1728	1	1	yes	characteristic
271	C₂×C₆₇	#1729	1	1	yes	characteristic
272	C₂×C₆₇	#1771	1	1	yes	characteristic
273	C₂×C₆₇	#1771	1	1	yes	characteristic
274	C₂×C₆₇	#1771	1	1	yes	characteristic
275	C₂×C₆₇	#1771	1	1	yes	characteristic
276	C₂×C₆₇	#1940	1	2	yes	—
277	C₂×C₆₇	#1940	1	2	yes	—
278	C₂×C₆₇	#1955	1	2	yes	—
279	C₂×C₆₇	#1985	1	2	yes	—
280	C₂²×C₄⋊C₈	—	67	1	no	—
281	C₂²×#97	—	67	1	no	—
282	C₂×C₄×C₄⋊C₄	—	67	1	no	—
283	C₂×#336	—	67	1	no	—
284	C₂×#336	—	67	1	no	—
285	C₂×#336	—	67	1	no	—
286	C₂×#336	D₆₇	1	1	yes	characteristic
287	C₆₇	#10835	1	1	yes	characteristic, complemented, Sylow
288	C₂³×C₈	—	67	1	no	—
289	C₂³×C₈	—	67	1	no	—
290	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
291	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
292	C₂²×C₄⋊C₄	—	67	2	no	—
293	C₂²×C₄²	—	67	1	no	—
294	C₂²×C₄²	—	67	1	no	—
295	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
296	C₂×C₄⋊C₈	—	67	4	no	—
297	C₂×C₄⋊C₈	—	134	2	no	—
298	C₂×C₄⋊C₈	—	134	2	no	—
299	C₂×#97	—	67	4	no	—
300	C₂×#97	—	134	2	no	—
301	C₂×#97	—	134	2	no	—
302	C₄×C₄⋊C₄	—	134	2	no	—
303	C₄×C₄⋊C₄	—	134	2	no	—
304	C₂×#95	—	67	2	no	—
305	#336	—	2	2	no	—
306	#336	—	134	2	no	—
307	#336	—	134	2	no	—
308	#336	—	134	2	no	—
309	C₂³×C₄	—	67	1	no	—
310	C₂³×C₄	—	67	1	no	—
311	C₂³×C₄	—	67	2	no	—
312	C₂³×C₄	C₄×D₆₇	1	2	yes	—
313	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
314	C₂²×C₈	—	2	1	no	—
315	C₂²×C₈	—	2	1	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	1	no	—
319	C₂²×C₈	—	2	1	no	—
320	C₂²×C₈	—	2	1	no	—
321	C₂²×C₈	—	2	1	no	—
322	C₂²×C₈	—	2	1	no	—
323	C₂²×C₈	—	2	1	no	—
324	C₂²×C₈	—	67	2	no	—
325	C₂²×C₈	—	67	2	no	—
326	C₂²×C₈	—	67	2	no	—
327	C₂²×C₈	—	67	2	no	—
328	C₂²×C₈	—	134	1	no	—
329	C₂²×C₈	—	134	1	no	—
330	C₂²×C₈	—	134	1	no	—
331	C₂²×C₈	—	134	1	no	—
332	C₂²×C₈	—	134	1	no	—
333	C₂²×C₈	—	134	1	no	—
334	C₂²×C₈	—	134	1	no	—
335	C₂²×C₈	—	134	1	no	—
336	C₂²×C₈	—	134	1	no	—
337	C₂²×C₈	—	134	1	no	—
338	C₂²×C₈	Dic₁₃₄	1	2	yes	—
339	C₂²×C₈	C₄×D₆₇	1	2	yes	—
340	C₂²×C₈	C₄×D₆₇	1	2	yes	—
341	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
342	C₂×C₄⋊C₄	—	134	4	no	—
343	C₂×C₄⋊C₄	—	134	4	no	—
344	C₂×C₄⋊C₄	—	134	4	no	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	—	67	2	no	—
352	C₂×C₄²	—	67	2	no	—
353	C₂×C₄²	—	67	2	no	—
354	C₂×C₄²	—	67	2	no	—
355	C₂×C₄²	—	67	2	no	—
356	C₂×C₄²	—	67	2	no	—
357	C₂×C₄²	—	134	1	no	—
358	C₂×C₄²	—	134	1	no	—
359	C₂×C₄²	—	134	1	no	—
360	C₂×C₄²	—	134	1	no	—
361	C₂×C₄²	—	134	1	no	—
362	C₂×C₄²	—	134	1	no	—
363	C₄⋊C₈	—	134	8	no	—
364	#97	—	134	8	no	—
365	#95	—	134	4	no	—
366	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	67	2	no	—
374	C₂²×C₄	—	67	2	no	—
375	C₂²×C₄	—	67	2	no	—
376	C₂²×C₄	—	67	2	no	—
377	C₂²×C₄	—	67	2	no	—
378	C₂²×C₄	—	67	2	no	—
379	C₂²×C₄	—	67	4	no	—
380	C₂²×C₄	—	134	2	no	—
381	C₂²×C₄	—	134	2	no	—
382	C₂²×C₄	—	134	2	no	—
383	C₂²×C₄	—	134	2	no	—
384	C₂²×C₄	—	134	2	no	—
385	C₂²×C₄	—	134	2	no	—
386	C₂²×C₄	—	134	2	no	—
387	C₂²×C₄	—	134	2	no	—
388	C₂²×C₄	—	134	2	no	—
389	C₂²×C₄	—	134	2	no	—
390	C₂²×C₄	#11761611	1	4	yes	—
391	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
393	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
394	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
395	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
396	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
398	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
399	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
401	C₂²×C₄	#11761627	1	1	yes	characteristic
402	C₂²×C₄	#11761627	1	1	yes	characteristic
403	C₂²×C₄	#11761628	1	1	yes	characteristic
404	C₂²×C₄	#11761628	1	1	yes	characteristic
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	4	1	no	—
415	C₂×C₈	—	4	1	no	—
416	C₂×C₈	—	4	1	no	—
417	C₂×C₈	—	4	1	no	—
418	C₂×C₈	—	268	1	no	—
419	C₂×C₈	—	268	1	no	—
420	C₂×C₈	—	268	1	no	—
421	C₂×C₈	—	268	1	no	—
422	C₂×C₈	—	67	4	no	—
423	C₂×C₈	—	134	2	no	—
424	C₂×C₈	—	134	2	no	—
425	C₂×C₈	—	134	2	no	—
426	C₂×C₈	—	134	2	no	—
427	C₂×C₈	—	134	2	no	—
428	C₂×C₈	—	134	2	no	—
429	C₂×C₈	—	134	2	no	—
430	C₂×C₈	—	134	2	no	—
431	C₂×C₈	—	134	2	no	—
432	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	complemented
433	C₄⋊C₄	—	134	16	no	—
434	C₄²	—	4	1	no	—
435	C₄²	—	4	1	no	—
436	C₄²	—	4	1	no	—
437	C₄²	—	4	1	no	—
438	C₄²	—	268	1	no	—
439	C₄²	—	268	1	no	—
440	C₄²	—	268	1	no	—
441	C₄²	—	268	1	no	—
442	C₄²	—	67	4	no	—
443	C₄²	—	67	4	no	—
444	C₄²	—	67	4	no	—
445	C₄²	—	67	4	no	—
446	C₂³	C₄²×D₆₇	1	1	yes	characteristic
447	C₂³	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
448	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
449	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
450	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
451	C₂³	#14524008	1	1	yes	characteristic
452	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
453	C₂³	M₁₆×D₆₇	1	2	yes	—
454	C₂³	C₂×#11761611	1	2	yes	—
455	C₂³	#14524049	1	2	yes	—
456	C₂³	#14524051	1	2	yes	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	268	2	no	—
465	C₂×C₄	—	268	2	no	—
466	C₂×C₄	—	268	2	no	—
467	C₂×C₄	—	268	2	no	—
468	C₂×C₄	—	67	4	no	—
469	C₂×C₄	—	67	4	no	—
470	C₂×C₄	—	134	4	no	—
471	C₂×C₄	—	134	4	no	—
472	C₂×C₄	—	134	4	no	—
473	C₂×C₄	—	134	2	no	—
474	C₂×C₄	—	134	2	no	—
475	C₂×C₄	—	134	2	no	—
476	C₂×C₄	—	134	2	no	—
477	C₂×C₄	—	134	4	no	—
478	C₂×C₄	—	134	4	no	—
479	C₂×C₄	—	134	4	no	—
480	C₂×C₄	—	134	4	no	—
481	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
482	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
483	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
484	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
485	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
487	C₂×C₄	#14523999	1	1	yes	characteristic
488	C₂×C₄	#14524000	1	1	yes	characteristic
489	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
490	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
491	C₂×C₄	#14524008	1	1	yes	characteristic
492	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
493	C₂×C₄	#14524015	1	1	yes	characteristic
494	C₂×C₄	#14524015	1	1	yes	characteristic
495	C₂×C₄	#14524016	1	1	yes	characteristic
496	C₂×C₄	#14524016	1	1	yes	characteristic
497	C₂×C₄	#14524020	1	1	yes	characteristic
498	C₂×C₄	#14524020	1	1	yes	characteristic
499	C₂×C₄	#14524021	1	1	yes	characteristic
500	C₂×C₄	#14524021	1	1	yes	characteristic
501	C₂×C₄	#14524029	1	1	yes	characteristic
502	C₂×C₄	#14524029	1	1	yes	characteristic
503	C₂×C₄	#14524033	1	1	yes	characteristic
504	C₂×C₄	#14524033	1	1	yes	characteristic
505	C₂×C₄	#14524039	1	1	yes	characteristic
506	C₂×C₄	#14524041	1	1	yes	characteristic
507	C₂×C₄	#14524042	1	1	yes	characteristic
508	C₂×C₄	#14524042	1	1	yes	characteristic
509	C₈	—	2	4	no	—
510	C₈	—	4	2	no	—
511	C₈	—	268	2	no	—
512	C₈	—	134	4	no	—
513	C₂²	#95×D₆₇	1	1	yes	characteristic
514	C₂²	#16863247	1	1	yes	characteristic
515	C₂²	C₄×Dic₆₇⋊C₄	1	1	yes	characteristic
516	C₂²	#16863291	1	1	yes	characteristic
517	C₂²	#16863293	1	1	yes	characteristic
518	C₂²	#16863296	1	1	yes	characteristic
519	C₂²	#16863300	1	1	yes	characteristic
520	C₂²	C₄×#11761611	1	1	yes	characteristic
521	C₂²	#16863348	1	1	yes	characteristic
522	C₂²	#16863350	1	1	yes	characteristic
523	C₂²	#16863353	1	1	yes	characteristic
524	C₂²	#16863358	1	2	yes	—
525	C₂²	#16863361	1	2	yes	—
526	C₂²	#16863418	1	2	yes	—
527	C₂²	#16863421	1	2	yes	—
528	C₂²	#16863464	1	2	yes	—
529	C₂²	#16863465	1	2	yes	—
530	C₂²	#16863467	1	2	yes	—
531	C₂²	#16863470	1	2	yes	—
532	C₂²	#16863523	1	2	yes	—
533	C₂²	#16863524	1	2	yes	—
534	C₂²	#16863525	1	2	yes	—
535	C₂²	#16863526	1	2	yes	—
536	C₄	—	4	4	no	—
537	C₄	—	268	4	no	—
538	C₄	—	134	4	no	—
539	C₄	—	134	4	no	—
540	C₄	#16863254	1	1	yes	characteristic
541	C₄	#16863255	1	1	yes	characteristic
542	C₄	#16863260	1	1	yes	characteristic
543	C₄	#16863265	1	1	yes	characteristic
544	C₄	#16863291	1	1	yes	characteristic
545	C₄	#16863293	1	1	yes	characteristic
546	C₄	#16863296	1	1	yes	characteristic
547	C₄	#16863300	1	1	yes	characteristic
548	C₂	Dic₆₇⋊#95	1	1	yes	characteristic
549	C₂	#22559901	1	1	yes	characteristic
550	C₂	#22559904	1	1	yes	characteristic
551	C₂	#22560080	1	1	yes	characteristic
552	C₂	#22560082	1	1	yes	characteristic
553	C₂	#22560083	1	1	yes	characteristic
554	C₂	#22560085	1	1	yes	characteristic
555	C₂	#22560929	1	2	yes	—
556	C₂	#22560931	1	2	yes	—
557	C₂	#22561005	1	2	yes	—
558	C₂	#22561147	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₆₇
commutator	C₂³×C₆₇	C₂×C₄²
Fitting	C₂×#336×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#336	D₆₇
hypocenter	C₆₇	#10835
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
15. C₂×Dic₆₇⋊C₈	C₄	semidirect
18. C₂×#14523953	C₄	semidirect
116. C₂×C₆₇⋊C₈	C₄⋊C₄	semidirect
161. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
287. C₆₇	#10835	semidirect
432. C₂×C₈	Dic₆₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₇⋊C₈	2
C₂²×#14523953	2
C₂×C₄×Dic₆₇⋊C₄	2
C₂×#16863151	2
C₂×#16863150	2
C₂×#336×C₆₇	2
#10835	67