Subgroups — #30754368

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560927	C₂	1	2	yes	maximal
2	C₄×#14523975	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
4	#22558049	C₂	1	2	yes	maximal
5	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523975	C₄	1	1	yes	characteristic
7	C₂×#14523975	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₆₇⋊C₈	—	2	2	no	—
10	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	#16863159	C₄	1	2	yes	—
12	#16863150	—	2	2	no	—
13	#16863147	—	2	2	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₈²×C₆₇	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₆₇	—	2	2	no	—
22	#14523975	—	2	2	no	—
23	#14523975	C₈	1	4	yes	complemented
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	Dic₆₇⋊C₈	—	4	4	no	—
27	C₆₇⋊#95	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	1	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₆₇	—	2	1	no	—
34	C₄×C₈×C₆₇	—	2	1	no	—
35	C₄×C₈×C₆₇	—	2	1	no	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	C₄×C₈×C₆₇	Q₈	1	2	yes	—
39	C₂²×Dic₆₇	—	2	2	no	—
40	C₂²×Dic₆₇	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	C₄⋊Dic₆₇	—	2	2	no	—
47	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
48	C₄×Dic₆₇	—	4	2	no	—
49	C₄×Dic₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
63	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₇	#41	1	1	yes	characteristic
68	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₆₇	#135	1	2	yes	—
82	C₂×Dic₆₇	—	2	4	no	—
83	C₂×Dic₆₇	—	4	2	no	—
84	C₂×Dic₆₇	—	4	2	no	—
85	C₂×Dic₆₇	—	4	2	no	—
86	C₂×Dic₆₇	—	4	2	no	—
87	C₂×Dic₆₇	—	4	2	no	—
88	C₈×C₆₇	—	2	4	no	—
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	—	2	4	no	—
91	C₈×C₆₇	C₄⋊C₈	1	4	yes	complemented
92	C₆₇⋊C₈	—	8	2	no	—
93	C₂²×C₆₇	#389	1	1	yes	characteristic
94	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
95	C₂²×C₆₇	#436	1	1	yes	characteristic
96	C₂²×C₆₇	C₄×SD₁₆	1	2	yes	—
97	C₂²×C₆₇	#443	1	2	yes	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₆₇	#424	1	1	yes	characteristic
102	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
103	C₄×C₆₇	#446	1	1	yes	characteristic
104	C₄×C₆₇	#465	1	2	yes	—
105	C₄×C₆₇	#498	1	1	yes	characteristic
106	C₄×C₆₇	#499	1	1	yes	characteristic
107	Dic₆₇	—	4	4	no	—
108	Dic₆₇	—	8	2	no	—
109	#9593	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	C₈×SD₁₆	1	2	yes	—
111	C₂×C₆₇	#1590	1	2	yes	—
112	C₂×C₆₇	#1939	1	1	yes	characteristic
113	C₂×C₆₇	#1941	1	1	yes	characteristic
114	C₂×C₆₇	#1945	1	1	yes	characteristic
115	#1939	—	67	2	no	—
116	C₄×#106	—	67	1	no	—
117	C₂×#334	—	67	1	no	—
118	C₂×C₈²	D₆₇	1	1	yes	characteristic
119	#1279	—	67	2	no	—
120	C₆₇	#9593	1	1	yes	characteristic, complemented, Sylow
121	C₂×#106	—	67	1	no	—
122	C₂×#106	—	67	1	no	—
123	C₂×C₄⋊C₈	—	67	2	no	—
124	C₂×C₄⋊C₈	—	134	2	no	—
125	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	67	2	no	—
129	#340	—	67	2	no	—
130	#336	—	134	2	no	—
131	#334	—	134	2	no	—
132	C₈²	—	2	2	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	134	2	no	—
135	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
137	C₂²×C₈	D₂₆₈	1	2	yes	—
138	C₂×C₄⋊C₄	—	67	2	no	—
139	C₂×C₄⋊C₄	—	67	2	no	—
140	C₂×C₄²	—	134	2	no	—
141	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
142	#106	—	67	4	no	—
143	#106	—	134	2	no	—
144	C₄⋊C₈	—	268	4	no	—
145	C₄⋊C₈	—	134	2	no	—
146	C₄⋊C₈	—	134	2	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	Dic₁₃₄	1	2	yes	—
151	C₄×C₈	Dic₁₃₄	1	2	yes	—
152	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
153	#95	—	134	2	no	—
154	C₂²×C₄	—	134	2	no	—
155	C₂²×C₄	—	134	2	no	—
156	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
158	C₂²×C₄	#11761628	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	268	2	no	—
166	C₂×C₈	—	268	2	no	—
167	C₂×C₈	—	268	2	no	—
168	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
169	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
170	C₂×C₈	C₈×D₆₇	1	2	yes	—
171	C₂×C₈	#11761611	1	2	yes	—
172	C₂×C₈	#11761617	1	4	yes	—
173	C₄⋊C₄	—	67	4	no	—
174	C₄⋊C₄	—	134	2	no	—
175	C₄⋊C₄	—	134	2	no	—
176	C₄⋊C₄	—	134	2	no	—
177	C₄²	—	268	2	no	—
178	C₄²	—	268	2	no	—
179	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
181	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
182	C₄²	#11761628	1	1	yes	characteristic
183	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	268	2	no	—
186	C₂×C₄	—	268	2	no	—
187	C₂×C₄	—	268	2	no	—
188	C₂×C₄	—	268	2	no	—
189	C₂×C₄	—	268	2	no	—
190	C₂×C₄	—	134	4	no	—
191	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
193	C₂×C₄	#14524017	1	1	yes	characteristic
194	C₂×C₄	#14524022	1	1	yes	characteristic
195	C₂×C₄	#14524035	1	1	yes	characteristic
196	C₂×C₄	#14524043	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
199	C₂×C₄	#14524049	1	1	yes	characteristic
200	C₂×C₄	#14524049	1	1	yes	characteristic
201	C₂×C₄	C₂×#11761617	1	2	yes	—
202	C₂×C₄	#14524065	1	2	yes	—
203	C₈	—	2	4	no	—
204	C₈	—	2	4	no	—
205	C₈	—	2	2	no	—
206	C₈	—	536	2	no	—
207	C₈	Dic₆₇⋊C₈	1	4	yes	complemented
208	C₂²	#16863294	1	1	yes	characteristic
209	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
210	C₂²	#16863475	1	1	yes	characteristic
211	C₂²	C₄×#11761617	1	2	yes	—
212	C₂²	#16863514	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	268	4	no	—
216	C₄	—	536	2	no	—
217	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
218	C₄	#16863422	1	1	yes	characteristic
219	C₄	C₈×Dic₁₃₄	1	1	yes	characteristic
220	C₄	#16863526	1	1	yes	characteristic
221	C₄	#16863625	1	2	yes	—
222	C₄	#16863794	1	1	yes	characteristic
223	C₄	#16863798	1	1	yes	characteristic
224	C₂	C₈×#11761617	1	2	yes	—
225	C₂	#22559101	1	2	yes	—
226	C₂	#22560927	1	1	yes	characteristic
227	C₂	#22560937	1	1	yes	characteristic
228	C₂	#22560957	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#9593
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
23. #14523975	C₈	semidirect
91. C₈×C₆₇	C₄⋊C₈	semidirect
120. C₆₇	#9593	semidirect
207. 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
#22560927	2
C₄×#14523975	2
C₂×#16863147	2
#22558049	2
C₂×C₈²×C₆₇	2
#9593	67