Subgroups — #30859122

Order 17152.

1010 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863423	C₂	1	2	yes	maximal
2	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
4	#22560933	C₂	1	4	yes	maximal, complemented
5	#22560928	C₂	1	4	yes	maximal, complemented
6	#22559906	C₂	1	2	yes	maximal
7	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂²×#11761612	C₂²	1	1	yes	characteristic
10	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
11	#16863423	—	2	8	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
16	C₄×#11761612	—	2	4	no	—
17	C₂×#14523954	C₂²	1	2	yes	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
19	#16863151	C₂²	1	8	yes	—
20	C₂×#117×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#384×C₆₇	C₄	1	2	yes	—
23	#382×C₆₇	C₄	1	2	yes	—
24	#380×C₆₇	C₄	1	2	yes	—
25	#380×C₆₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	—	2	2	no	—
32	C₂×#11761612	D₄	1	4	yes	—
33	C₂×#11761612	Q₈	1	4	yes	—
34	C₂²×C₆₇⋊C₈	—	2	4	no	—
35	C₂²×C₆₇⋊C₈	—	2	4	no	—
36	C₂²×C₆₇⋊C₈	—	2	4	no	—
37	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
38	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
43	#14523954	—	2	4	no	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	#117×C₆₇	—	2	4	no	—
49	#117×C₆₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	4	yes	—
57	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#95×C₆₇	C₂×C₄	1	4	yes	—
62	#95×C₆₇	C₂×C₄	1	4	yes	—
63	#11761612	—	2	8	no	—
64	#11761612	—	2	8	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	4	8	no	—
73	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
74	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
75	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	4	2	no	—
78	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₆₇	#41	1	2	yes	—
91	C₂²×C₄×C₆₇	#41	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₇	—	2	8	no	—
94	C₄⋊C₄×C₆₇	—	2	8	no	—
95	C₄⋊C₄×C₆₇	—	4	4	no	—
96	C₂²⋊C₄×C₆₇	—	2	8	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
100	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₄²×C₆₇	C₂×D₄	1	2	yes	—
103	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₆₇	C₄×D₄	1	2	yes	—
109	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
110	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	8	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	2	8	no	—
117	C₂×C₄×C₆₇	—	2	8	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	#121	1	4	yes	—
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₆₇	#125	1	4	yes	—
129	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
130	C₂×C₄×C₆₇	#131	1	4	yes	—
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₆₇⋊C₈	—	2	4	no	—
138	C₆₇⋊C₈	—	2	4	no	—
139	C₆₇⋊C₈	—	4	4	no	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	4	no	—
144	C₂²×C₆₇	#389	1	4	yes	—
145	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×#131	1	2	yes	—
149	C₂²×C₆₇	#568	1	2	yes	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	4	8	no	—
154	C₄×C₆₇	#389	1	4	yes	—
155	C₄×C₆₇	#424	1	4	yes	complemented
156	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
157	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
159	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	#21145	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	4	no	—
162	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
164	C₂×C₆₇	#2905	1	1	yes	characteristic
165	C₂×C₆₇	#2935	1	2	yes	—
166	C₂×C₆₇	#2962	1	2	yes	—
167	C₂×#424	—	67	2	no	—
168	C₂²×C₄⋊C₈	—	67	1	no	—
169	C₂×C₄×M₁₆	—	67	1	no	—
170	#2285	D₆₇	1	1	yes	characteristic
171	#1940	—	67	4	no	—
172	#1939	—	67	4	no	—
173	#1729	—	67	2	no	—
174	C₆₇	#21145	1	1	yes	characteristic, complemented, Sylow
175	C₂²×M₁₆	—	67	1	no	—
176	C₂²×M₁₆	—	67	1	no	—
177	C₂³×C₈	—	67	2	no	—
178	C₂×#117	C₂×D₆₇	1	2	yes	—
179	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
180	#424	—	134	8	no	—
181	C₂×C₄⋊C₈	—	67	2	no	—
182	C₂×C₄⋊C₈	—	67	2	no	—
183	C₂×C₄⋊C₈	—	67	8	no	—
184	C₂×C₄⋊C₈	—	134	4	no	—
185	C₄×M₁₆	—	134	4	no	—
186	C₂×#97	—	67	2	no	—
187	C₂×C₄×C₈	—	67	2	no	—
188	#384	Dic₆₇	1	2	yes	—
189	#382	Dic₆₇	1	2	yes	—
190	#380	Dic₆₇	1	2	yes	—
191	#380	Dic₆₇	1	2	yes	—
192	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
193	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
194	#336	—	67	8	no	—
195	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
198	C₂×M₁₆	—	67	4	no	—
199	C₂×M₁₆	—	67	4	no	—
200	C₂×M₁₆	—	134	2	no	—
201	C₂×M₁₆	—	134	2	no	—
202	C₂×M₁₆	—	134	2	no	—
203	C₂×M₁₆	—	134	2	no	—
204	C₂²×C₈	—	67	2	no	—
205	C₂²×C₈	—	67	2	no	—
206	C₂²×C₈	—	67	4	no	—
207	C₂²×C₈	—	134	4	no	—
208	C₂²×C₈	—	134	4	no	—
209	C₂²×C₈	—	134	4	no	—
210	#117	—	2	4	no	—
211	#117	—	2	4	no	—
212	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
214	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
215	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
216	C₂×C₄²	Dic₁₃₄	1	4	yes	—
217	C₂×C₄²	D₂₆₈	1	4	yes	—
218	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
221	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	C₄⋊C₈	—	134	16	no	—
224	C₄⋊C₈	—	134	4	no	—
225	C₄⋊C₈	—	134	4	no	—
226	#97	—	134	4	no	—
227	C₄×C₈	—	134	4	no	—
228	#95	C₂×Dic₆₇	1	4	yes	—
229	#95	C₂×Dic₆₇	1	4	yes	—
230	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
234	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
235	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
236	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₆₇	1	2	yes	—
242	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
243	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
244	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
245	C₂²×C₄	#11761627	1	2	yes	—
246	C₂²×C₄	#11761629	1	2	yes	—
247	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
248	M₁₆	—	134	8	no	—
249	M₁₆	—	134	8	no	—
250	C₂×C₈	—	268	8	no	—
251	C₂×C₈	—	67	4	no	—
252	C₂×C₈	—	67	4	no	—
253	C₂×C₈	—	134	4	no	—
254	C₂×C₈	—	134	4	no	—
255	C₂×C₈	—	134	4	no	—
256	C₂×C₈	—	134	2	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	2	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	2	8	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	C₄⋊Dic₆₇	1	4	yes	—
267	C₄²	C₄⋊Dic₆₇	1	4	yes	—
268	C₄²	C₂×D₂₆₈	1	2	yes	—
269	C₄²	C₂×D₂₆₈	1	2	yes	—
270	C₄²	C₂×Dic₁₃₄	1	2	yes	—
271	C₄²	C₂×Dic₁₃₄	1	2	yes	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
275	C₂³	D₄×Dic₆₇	1	2	yes	—
276	C₂³	Q₈×Dic₆₇	1	2	yes	—
277	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
289	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
291	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
292	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
293	C₂×C₄	#14524018	1	4	yes	—
294	C₂×C₄	#14524023	1	4	yes	—
295	C₂×C₄	#14524034	1	4	yes	—
296	C₂×C₄	#14524042	1	4	yes	—
297	C₂×C₄	#14524050	1	4	yes	—
298	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
302	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
303	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
304	C₈	—	268	4	no	—
305	C₈	—	134	4	no	—
306	C₈	—	134	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	#16863295	1	4	yes	—
312	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
313	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
314	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
315	C₂²	C₂×#14524050	1	2	yes	—
316	C₂²	#16864163	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	8	no	—
321	C₄	#16863295	1	4	yes	—
322	C₄	#16863423	1	4	yes	complemented
323	C₄	C₂×#14524018	1	1	yes	characteristic
324	C₄	C₂×#14524023	1	1	yes	characteristic
325	C₄	C₂×#14524034	1	1	yes	characteristic
326	C₄	C₂×#14524042	1	1	yes	characteristic
327	C₂	—	2	4	no	—
328	C₂	C₂×#16863295	1	1	yes	characteristic
329	C₂	C₂×#16863423	1	1	yes	characteristic
330	C₂	#22568773	1	1	yes	characteristic
331	C₂	#22568953	1	2	yes	—
332	C₂	#22569158	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2285	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2285	D₆₇
hypocenter	C₆₇	#21145
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
4. #22560933	C₂	semidirect
5. #22560928	C₂	semidirect
155. C₄×C₆₇	#424	semidirect
174. C₆₇	#21145	semidirect
322. C₄	#16863423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863423	2
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂×C₄×#11761612	2
#22560933	2
#22560928	2
#22559906	2
C₆₇×#2285	2
#21145	67