Subgroups — #30755206

Order 17152.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560373	C₂	1	1	yes	maximal, characteristic
2	#22558773	C₂	1	4	yes	maximal
3	#22558492	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
5	#16863449	C₂²	1	2	yes	—
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523975	—	2	1	no	—
8	#16863262	C₂²	1	2	yes	—
9	#16863133	—	2	8	no	—
10	#16863117	—	2	1	no	—
11	C₈×C₆₇⋊C₈	—	2	1	no	—
12	#533×C₆₇	C₂²	1	2	yes	—
13	#500×C₆₇	C₄	1	2	yes	complemented
14	#499×C₆₇	—	2	1	no	—
15	#499×C₆₇	—	2	1	no	—
16	#498×C₆₇	C₄	1	2	yes	complemented
17	#427×C₆₇	—	2	1	no	—
18	#427×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	#14523977	—	4	2	no	—
22	#14523975	—	4	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₆₇⋊#95	—	2	4	no	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
30	#125×C₆₇	—	2	2	no	—
31	C₂²⋊Q₈×C₆₇	—	2	4	no	—
32	C₄×Q₈×C₆₇	—	2	4	no	—
33	#117×C₆₇	—	2	2	no	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#107×C₆₇	—	2	2	no	—
36	#107×C₆₇	—	2	2	no	—
37	#106×C₆₇	—	2	2	no	—
38	#106×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×Dic₆₇	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	2	4	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₂×Q₈×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
55	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	4	4	no	—
61	C₂²⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	—	4	2	no	—
63	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	#41	1	2	yes	—
66	C₂³×C₆₇	#127	1	1	yes	characteristic
67	Q₈×C₆₇	—	8	4	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₆₇	#121	1	2	yes	—
73	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₆₇	—	8	4	no	—
79	C₂×Dic₆₇	—	8	2	no	—
80	C₈×C₆₇	#103	1	2	yes	—
81	C₈×C₆₇	#103	1	2	yes	—
82	C₈×C₆₇	#103	1	2	yes	—
83	C₈×C₆₇	#103	1	2	yes	—
84	C₆₇⋊C₈	—	4	4	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	#390	1	1	yes	characteristic
87	C₂²×C₆₇	#492	1	1	yes	characteristic
88	C₂²×C₆₇	#494	1	1	yes	characteristic
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	—	8	4	no	—
92	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
93	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
94	C₄×C₆₇	#487	1	1	yes	characteristic
95	C₄×C₆₇	#487	1	1	yes	characteristic
96	Dic₆₇	—	16	2	no	—
97	#9755	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	—	2	1	no	—
100	C₂×C₆₇	#1956	1	1	yes	characteristic
101	C₂×C₆₇	#1976	1	2	yes	—
102	#3160	D₆₇	1	1	yes	characteristic
103	#1846	—	67	1	no	—
104	#1518	—	67	4	no	—
105	#1450	—	67	1	no	—
106	C₆₇	#9755	1	1	yes	characteristic, complemented, Sylow
107	#533	C₂×D₆₇	1	2	yes	—
108	#500	Dic₆₇	1	2	yes	complemented
109	#499	—	2	1	no	—
110	#499	—	2	1	no	—
111	#498	Dic₆₇	1	2	yes	complemented
112	#431	—	67	2	no	—
113	#427	—	2	1	no	—
114	#427	—	2	1	no	—
115	C₂×#107	—	134	1	no	—
116	C₂×#106	—	134	1	no	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	#382	—	67	2	no	—
119	#328	—	134	8	no	—
120	#322	—	134	1	no	—
121	C₈²	—	134	1	no	—
122	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	4	no	—
127	C₄×Q₈	—	2	4	no	—
128	#117	—	2	2	no	—
129	C₂×C₄⋊C₄	—	134	2	no	—
130	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	—	268	2	no	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	268	2	no	—
137	C₄⋊C₈	—	134	2	no	—
138	C₂²⋊C₈	—	67	4	no	—
139	C₄×C₈	—	134	2	no	—
140	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
144	#95	—	134	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂²×C₄	—	268	2	no	—
147	C₂²×C₄	D₄×D₆₇	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂×C₈	—	134	4	no	—
150	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
154	C₂×C₈	#11761615	1	2	yes	—
155	C₂×C₈	#11761615	1	2	yes	—
156	C₂×C₈	#11761618	1	2	yes	—
157	C₂×C₈	#11761618	1	2	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	268	2	no	—
162	C₄⋊C₄	—	268	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₄²	#11761627	1	2	yes	—
168	C₂³	#14524039	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	536	4	no	—
173	C₂×C₄	—	536	2	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
176	C₂×C₄	#14524014	1	2	yes	—
177	C₂×C₄	#14524030	1	1	yes	characteristic
178	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
179	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
180	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
181	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
182	C₈	—	268	4	no	—
183	C₈	#14523967	1	2	yes	—
184	C₈	#14523967	1	2	yes	—
185	C₈	#14523967	1	2	yes	—
186	C₈	#14523967	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#16863299	1	1	yes	characteristic
189	C₂²	#16863770	1	1	yes	characteristic
190	C₂²	#16863775	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	1072	2	no	—
195	C₄	C₂×#14523967	1	1	yes	characteristic
196	C₄	C₂×#14523967	1	1	yes	characteristic
197	C₄	#16863746	1	1	yes	characteristic
198	C₄	#16863746	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#22561012	1	1	yes	characteristic
202	C₂	#22561108	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#9755
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
13. #500×C₆₇	C₄	semidirect
16. #498×C₆₇	C₄	semidirect
106. C₆₇	#9755	semidirect
108. #500	Dic₆₇	semidirect
111. #498	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560373	2
#22558773	2
#22558492	2
C₆₇×#3160	2
#9755	67