Subgroups — #30859192

Order 17152.

982 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
3	#22562832	C₂	1	2	yes	maximal
4	#22560948	C₂	1	4	yes	maximal, complemented
5	#22560938	C₂	1	4	yes	maximal, complemented
6	#22559930	C₂	1	2	yes	maximal
7	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	C₂×#14523975	—	2	4	no	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	C₂×#14523975	C₂²	1	2	yes	—
14	#16863351	—	2	4	no	—
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	#16863295	C₂²	1	8	yes	—
18	#16863271	—	2	4	no	—
19	#16863266	—	2	2	no	—
20	#16863262	—	2	2	no	—
21	#16863158	C₂²	1	8	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	#427×C₆₇	C₄	1	8	yes	complemented
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	C₂³×Dic₆₇	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	Q₈	1	4	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	#14524002	—	2	2	no	—
33	#14524002	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	8	no	—
35	C₂×C₄⋊Dic₆₇	—	2	4	no	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	4	no	—
38	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₆₇	D₄	1	2	yes	—
41	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
42	#14523975	—	2	4	no	—
43	#14523975	—	2	4	no	—
44	#14523953	—	2	4	no	—
45	C₈×Dic₆₇	—	2	4	no	—
46	C₆₇⋊#95	—	2	8	no	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	D₄	1	4	yes	—
50	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	#117×C₆₇	—	2	4	no	—
54	#117×C₆₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	#107×C₆₇	C₂×C₄	1	8	yes	complemented
59	#106×C₆₇	C₂×C₄	1	8	yes	complemented
60	C₂²×Dic₆₇	—	2	4	no	—
61	C₂²×Dic₆₇	—	4	4	no	—
62	C₂²×Dic₆₇	—	8	4	no	—
63	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
64	#11761612	—	2	8	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	4	4	no	—
72	C₄⋊Dic₆₇	—	8	8	no	—
73	C₂²⋊Dic₆₇	—	2	4	no	—
74	C₄×Dic₆₇	—	2	2	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂²⋊C₄×C₆₇	—	2	8	no	—
96	C₄²×C₆₇	—	2	8	no	—
97	C₂³×C₆₇	—	2	4	no	—
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	#121	1	2	yes	—
100	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₆₇	#121	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₆₇	#125	1	4	yes	—
111	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₆₇	—	2	4	no	—
119	C₂×Dic₆₇	—	2	8	no	—
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	—	4	8	no	—
122	C₂×Dic₆₇	—	8	8	no	—
123	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
124	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
125	C₆₇⋊C₈	—	2	4	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	#389	1	4	yes	—
130	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₂²×C₆₇	#577	1	4	yes	—
134	C₄×C₆₇	—	4	8	no	—
135	C₄×C₆₇	#389	1	4	yes	—
136	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
140	Dic₆₇	—	4	4	no	—
141	Dic₆₇	—	8	4	no	—
142	#21155	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	2	4	no	—
144	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
145	C₂×C₆₇	#2952	1	2	yes	—
146	C₂×C₆₇	#3057	1	2	yes	—
147	C₂×C₆₇	#3163	1	2	yes	—
148	C₂×#427	D₆₇	1	1	yes	characteristic
149	C₂²×#106	—	67	1	no	—
150	C₂×#405	—	67	1	no	—
151	#2317	—	67	2	no	—
152	#1943	—	67	4	no	—
153	#1941	—	67	4	no	—
154	#1736	—	67	2	no	—
155	C₆₇	#21155	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	67	1	no	—
157	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂×#117	—	67	1	no	—
159	C₂×#117	C₂×D₆₇	1	2	yes	—
160	C₂²×C₄⋊C₄	—	67	2	no	—
161	#427	Dic₆₇	1	8	yes	complemented
162	C₂×#107	C₂×D₆₇	1	2	yes	—
163	C₂×#106	—	67	2	no	—
164	C₂×#106	—	67	2	no	—
165	C₂×#106	—	134	4	no	—
166	C₂×#106	C₂×D₆₇	1	2	yes	—
167	#405	—	134	4	no	—
168	C₂×#97	—	67	2	no	—
169	C₂×C₄×C₈	—	67	2	no	—
170	#389	—	67	8	no	—
171	#385	—	134	4	no	—
172	#384	—	134	2	no	—
173	#382	—	134	2	no	—
174	#340	—	67	8	no	—
175	C₂³×C₄	—	134	2	no	—
176	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×M₁₆	—	67	4	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂×M₁₆	—	134	2	no	—
180	C₂²×C₈	—	67	2	no	—
181	C₂²×C₈	Dic₁₃₄	1	4	yes	—
182	C₂²×C₈	D₂₆₈	1	4	yes	—
183	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
184	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	#117	—	2	4	no	—
187	#117	—	134	2	no	—
188	#117	—	134	2	no	—
189	#117	C₂×Dic₆₇	1	8	yes	—
190	C₂×C₄⋊C₄	—	67	2	no	—
191	C₂×C₄⋊C₄	—	67	4	no	—
192	C₂×C₄⋊C₄	—	134	2	no	—
193	C₂×C₄⋊C₄	—	134	4	no	—
194	C₂×C₄⋊C₄	—	134	4	no	—
195	C₂×C₄⋊C₄	—	134	8	no	—
196	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	67	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	67	2	no	—
201	#107	C₂×Dic₆₇	1	8	yes	complemented
202	#106	—	134	4	no	—
203	#106	—	134	4	no	—
204	#106	C₂×Dic₆₇	1	8	yes	complemented
205	#97	—	134	4	no	—
206	C₄×C₈	—	134	4	no	—
207	#95	—	134	8	no	—
208	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	268	4	no	—
211	C₂²×C₄	—	536	4	no	—
212	C₂²×C₄	—	67	4	no	—
213	C₂²×C₄	—	134	4	no	—
214	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
215	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
217	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
218	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
219	C₂²×C₄	#11761627	1	2	yes	—
220	C₂²×C₄	#11761629	1	2	yes	—
221	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
222	M₁₆	—	134	8	no	—
223	C₂×C₈	—	67	4	no	—
224	C₂×C₈	—	134	2	no	—
225	C₂×C₈	—	134	2	no	—
226	C₂×C₈	C₄⋊Dic₆₇	1	16	yes	complemented
227	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
228	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
229	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
230	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
231	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
232	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	268	4	no	—
235	C₄⋊C₄	—	536	8	no	—
236	C₄⋊C₄	—	134	4	no	—
237	C₄⋊C₄	—	134	4	no	—
238	C₄⋊C₄	—	134	4	no	—
239	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	134	4	no	—
242	C₄²	—	2	8	no	—
243	C₄²	—	134	2	no	—
244	C₄²	—	134	2	no	—
245	C₂³	—	2	4	no	—
246	C₂³	D₄×Dic₆₇	1	2	yes	—
247	C₂³	#14524018	1	2	yes	—
248	C₂³	#14524042	1	2	yes	—
249	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	268	4	no	—
254	C₂×C₄	—	268	8	no	—
255	C₂×C₄	—	536	8	no	—
256	C₂×C₄	—	134	4	no	—
257	C₂×C₄	—	134	8	no	—
258	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
259	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
260	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
261	C₂×C₄	#14524018	1	2	yes	—
262	C₂×C₄	#14524023	1	4	yes	—
263	C₂×C₄	#14524034	1	4	yes	—
264	C₂×C₄	#14524042	1	2	yes	—
265	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
269	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
271	C₈	—	134	4	no	—
272	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
273	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#16863295	1	4	yes	—
278	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
279	C₂²	C₂×#14524018	1	1	yes	characteristic
280	C₂²	C₂×#14524042	1	1	yes	characteristic
281	C₂²	#16864203	1	4	yes	—
282	C₄	—	4	8	no	—
283	C₄	—	268	4	no	—
284	C₄	—	536	4	no	—
285	C₄	#16863295	1	4	yes	—
286	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
288	C₄	C₂×#14524023	1	1	yes	characteristic
289	C₄	C₂×#14524034	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	C₂×#16863295	1	1	yes	characteristic
292	C₂	#22569093	1	2	yes	—
293	C₂	#22569849	1	2	yes	—
294	C₂	#22570741	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#21155
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #22560948	C₂	semidirect
5. #22560938	C₂	semidirect
24. #427×C₆₇	C₄	semidirect
58. #107×C₆₇	C₂×C₄	semidirect
59. #106×C₆₇	C₂×C₄	semidirect
86. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
155. C₆₇	#21155	semidirect
161. #427	Dic₆₇	semidirect
201. #107	C₂×Dic₆₇	semidirect
204. #106	C₂×Dic₆₇	semidirect
226. C₂×C₈	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523975	2
C₂×#16863351	2
#22562832	2
#22560948	2
#22560938	2
#22559930	2
C₂×#427×C₆₇	2
#21155	67