Subgroups — #30749577

Order 17152.

418 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560360	C₂	1	1	yes	maximal, characteristic
2	#22560325	C₂	1	1	yes	maximal, characteristic
3	C₄×#14523953	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863148	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863146	C₂	1	1	yes	maximal, characteristic
7	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₇⋊C₈	—	2	2	no	—
11	C₂×#14523953	C₄	1	1	yes	characteristic
12	C₂×#14523953	C₄	1	1	yes	characteristic
13	C₂×#14523953	C₄	1	1	yes	characteristic
14	C₂×#14523953	C₄	1	1	yes	characteristic
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863150	—	2	2	no	—
18	#16863149	—	2	2	no	—
19	#16863148	—	2	2	no	—
20	#16863146	—	2	2	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#334×C₆₇	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	1	no	—
26	C₂²×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
32	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	Dic₆₇⋊C₈	—	4	4	no	—
38	#14523953	—	2	2	no	—
39	#14523953	—	2	2	no	—
40	#14523953	—	2	2	no	—
41	#14523953	C₈	1	4	yes	complemented
42	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
43	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	—	2	1	no	—
50	C₄⋊C₈×C₆₇	—	2	1	no	—
51	C₄⋊C₈×C₆₇	—	2	1	no	—
52	C₄⋊C₈×C₆₇	—	2	1	no	—
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	Q₈	1	2	yes	—
55	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₆₇⋊C₈	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	1	no	—
59	C₂×C₆₇⋊C₈	—	4	1	no	—
60	C₂×C₆₇⋊C₈	—	4	1	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
66	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
67	C₄×Dic₆₇	—	2	1	no	—
68	C₄×Dic₆₇	—	2	1	no	—
69	C₄×Dic₆₇	—	2	2	no	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	—	4	2	no	—
72	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
73	C₄×Dic₆₇	M₁₆	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	1	no	—
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	4	1	no	—
80	C₂×C₈×C₆₇	—	4	1	no	—
81	C₂×C₈×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
87	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
91	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₆₇	#137	1	2	yes	—
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	—	2	2	no	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	—	8	1	no	—
113	C₈×C₆₇	—	8	1	no	—
114	C₈×C₆₇	C₄⋊C₈	1	4	yes	complemented
115	C₆₇⋊C₈	—	2	2	no	—
116	C₆₇⋊C₈	—	8	1	no	—
117	C₆₇⋊C₈	—	8	1	no	—
118	C₆₇⋊C₈	C₄⋊C₈	1	4	yes	complemented
119	C₂²×C₆₇	#384	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₆₇	#423	1	1	yes	characteristic
122	C₂²×C₆₇	#428	1	2	yes	—
123	C₂²×C₆₇	#430	1	2	yes	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
127	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₄×C₆₇	#423	1	1	yes	characteristic
129	C₄×C₆₇	#423	1	1	yes	characteristic
130	C₄×C₆₇	#496	1	1	yes	characteristic
131	C₄×C₆₇	#497	1	1	yes	characteristic
132	C₄×C₆₇	#501	1	2	yes	—
133	Dic₆₇	—	2	4	no	—
134	Dic₆₇	—	4	2	no	—
135	#8808	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	#1565	1	2	yes	—
137	C₂×C₆₇	#1569	1	2	yes	—
138	C₂×C₆₇	#1842	1	1	yes	characteristic
139	C₂×C₆₇	#1847	1	1	yes	characteristic
140	C₂×C₆₇	#1849	1	1	yes	characteristic
141	#1842	—	67	1	no	—
142	#1831	—	67	1	no	—
143	C₄×#97	—	67	1	no	—
144	C₂×#335	—	67	1	no	—
145	C₂×#335	—	67	1	no	—
146	C₂×#334	—	67	1	no	—
147	C₂×#334	D₆₇	1	1	yes	characteristic
148	C₆₇	#8808	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊C₈	—	67	1	no	—
150	C₂×C₄⋊C₈	—	67	1	no	—
151	C₂×C₄⋊C₈	—	134	2	no	—
152	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
153	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₂×#97	—	67	1	no	—
155	C₂×#97	—	67	1	no	—
156	C₂×#97	—	67	1	no	—
157	C₂×#97	—	67	1	no	—
158	C₂×C₄×C₈	—	67	1	no	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₄³	—	67	1	no	—
161	#336	—	134	2	no	—
162	#335	—	134	2	no	—
163	#335	—	134	2	no	—
164	#334	—	2	2	no	—
165	#334	—	134	2	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	67	1	no	—
169	C₂²×C₈	—	67	1	no	—
170	C₂²×C₈	—	134	1	no	—
171	C₂²×C₈	—	134	1	no	—
172	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
174	C₂×C₄²	—	67	2	no	—
175	C₂×C₄²	—	67	1	no	—
176	C₂×C₄²	—	67	1	no	—
177	C₂×C₄²	—	67	2	no	—
178	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	268	4	no	—
184	C₄⋊C₈	—	134	1	no	—
185	C₄⋊C₈	—	134	1	no	—
186	C₄⋊C₈	—	134	1	no	—
187	C₄⋊C₈	—	134	1	no	—
188	#97	—	67	4	no	—
189	#97	—	134	2	no	—
190	#97	—	134	2	no	—
191	#97	—	134	2	no	—
192	C₄×C₈	—	67	2	no	—
193	C₄×C₈	—	67	2	no	—
194	C₄×C₈	Dic₁₃₄	1	2	yes	—
195	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
196	C₂²×C₄	—	67	2	no	—
197	C₂²×C₄	—	67	2	no	—
198	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
200	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	268	1	no	—
208	C₂×C₈	—	268	1	no	—
209	C₂×C₈	—	268	2	no	—
210	C₂×C₈	—	268	2	no	—
211	C₂×C₈	—	67	2	no	—
212	C₂×C₈	—	67	2	no	—
213	C₂×C₈	—	67	2	no	—
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	—	134	1	no	—
216	C₂×C₈	—	134	1	no	—
217	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
218	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
219	C₂×C₈	C₈×D₆₇	1	2	yes	—
220	C₂×C₈	#11761611	1	2	yes	—
221	C₄²	—	268	2	no	—
222	C₄²	—	67	2	no	—
223	C₄²	—	67	2	no	—
224	C₄²	—	134	1	no	—
225	C₄²	—	134	1	no	—
226	C₄²	—	134	2	no	—
227	C₄²	—	134	2	no	—
228	C₄²	D₄×D₆₇	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₄²	Q₈×D₆₇	1	1	yes	characteristic
231	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
232	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	67	4	no	—
235	C₂×C₄	—	134	2	no	—
236	C₂×C₄	—	134	2	no	—
237	C₂×C₄	—	134	2	no	—
238	C₂×C₄	—	134	2	no	—
239	C₂×C₄	—	134	2	no	—
240	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
242	C₂×C₄	#14524039	1	1	yes	characteristic
243	C₂×C₄	#14524041	1	1	yes	characteristic
244	C₂×C₄	#14524042	1	1	yes	characteristic
245	C₂×C₄	#14524042	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
248	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
249	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
250	C₂×C₄	#14524070	1	2	yes	—
251	C₂×C₄	#14524072	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	8	1	no	—
254	C₈	—	8	1	no	—
255	C₈	—	536	1	no	—
256	C₈	—	536	1	no	—
257	C₈	—	67	4	no	—
258	C₈	—	134	2	no	—
259	C₈	Dic₆₇⋊C₈	1	4	yes	complemented
260	C₂²	#16863265	1	1	yes	characteristic
261	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
262	C₂²	#16863418	1	1	yes	characteristic
263	C₂²	#16863435	1	2	yes	—
264	C₂²	#16863443	1	2	yes	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	268	2	no	—
268	C₄	—	134	4	no	—
269	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
270	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
271	C₄	#16863418	1	1	yes	characteristic
272	C₄	#16863419	1	1	yes	characteristic
273	C₄	#16863782	1	1	yes	characteristic
274	C₄	#16863787	1	1	yes	characteristic
275	C₄	#16863803	1	2	yes	—
276	C₂	#22558958	1	2	yes	—
277	C₂	#22558973	1	2	yes	—
278	C₂	#22560360	1	1	yes	characteristic
279	C₂	#22560374	1	1	yes	characteristic
280	C₂	#22560382	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#8808
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
41. #14523953	C₈	semidirect
114. C₈×C₆₇	C₄⋊C₈	semidirect
118. C₆₇⋊C₈	C₄⋊C₈	semidirect
148. C₆₇	#8808	semidirect
259. C₈	Dic₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22560360	2
#22560325	2
C₄×#14523953	2
C₂×#16863149	2
C₂×#16863148	2
C₂×#16863146	2
C₂×#334×C₆₇	2
#8808	67