Subgroups — #30776323

Order 17152.

198 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863230	C₂	1	1	yes	maximal, characteristic
2	#22561805	C₂	1	1	yes	maximal, characteristic
3	#22558406	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₄	1	1	yes	characteristic
6	#16863351	C₄	1	1	yes	characteristic
7	#16863230	—	2	1	no	—
8	#16863230	C₂²	1	2	yes	—
9	C₂×C₆₇⋊C₃₂	—	2	1	no	—
10	#16863171	C₄	1	2	yes	—
11	C₁₆×Dic₆₇	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	M₆₄×C₆₇	—	2	1	no	—
14	M₆₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₃₂×C₆₇	—	2	1	no	—
16	C₂×#11761612	C₈	1	1	yes	characteristic
17	#14524002	C₈	1	1	yes	characteristic
18	#14523983	—	2	1	no	—
19	#14523983	—	2	1	no	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14523953	C₈	1	2	yes	—
22	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
23	C₆₇⋊C₃₂	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
29	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
31	C₃₂×C₆₇	—	2	2	no	—
32	#11761612	C₁₆	1	2	yes	—
33	#11761612	C₁₆	1	2	yes	—
34	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
35	C₄⋊Dic₆₇	—	2	1	no	—
36	C₂²⋊Dic₆₇	—	2	1	no	—
37	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
38	C₆₇⋊C₁₆	—	2	2	no	—
39	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₆₇	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₆₇	—	2	2	no	—
58	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
62	C₆₇⋊C₈	C₂×C₁₆	1	4	yes	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
67	C₄×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
68	C₄×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
69	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
71	Dic₆₇	—	4	2	no	—
72	#13630	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	—	2	1	no	—
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
77	C₂×M₆₄	—	67	1	no	—
78	C₂×M₆₄	D₆₇	1	1	yes	characteristic
79	#2108	—	67	1	no	—
80	#1422	—	67	4	no	—
81	C₆₇	#13630	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₃₂	—	67	1	no	—
83	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
84	#405	—	67	1	no	—
85	M₆₄	—	2	1	no	—
86	M₆₄	—	67	2	no	—
87	M₆₄	—	134	1	no	—
88	M₆₄	C₂×D₆₇	1	2	yes	—
89	C₂×C₃₂	—	2	1	no	—
90	C₂×C₃₂	—	134	1	no	—
91	#346	—	67	2	no	—
92	C₄×C₁₆	—	67	2	no	—
93	C₂×M₁₆	—	67	1	no	—
94	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
95	#117	—	67	1	no	—
96	M₃₂	—	134	1	no	—
97	M₃₂	—	134	1	no	—
98	C₂×C₁₆	—	67	2	no	—
99	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
100	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
103	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
105	#97	—	67	2	no	—
106	C₄×C₈	—	67	2	no	—
107	C₃₂	—	2	2	no	—
108	C₃₂	—	134	2	no	—
109	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
110	M₁₆	—	67	2	no	—
111	M₁₆	—	67	2	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
114	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
115	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
116	C₂×C₈	#11761611	1	1	yes	characteristic
117	C₂×C₈	#11761611	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
119	C₄⋊C₄	—	134	1	no	—
120	C₂²⋊C₄	—	134	1	no	—
121	C₄²	—	67	2	no	—
122	C₁₆	—	134	2	no	—
123	C₁₆	Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₁₆	Dic₆₇⋊C₄	1	1	yes	characteristic
125	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
126	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
127	C₂³	#14523982	1	1	yes	characteristic
128	C₂×C₄	—	134	2	no	—
129	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
130	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
131	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
132	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
133	C₂×C₄	#14523982	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
135	C₈	—	67	4	no	—
136	C₈	Dic₆₇⋊C₈	1	1	yes	characteristic
137	C₈	Dic₆₇⋊C₈	1	1	yes	characteristic
138	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
139	C₈	C₂×#11761611	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	Dic₆₇⋊C₁₆	1	1	yes	characteristic
142	C₂²	Dic₆₇⋊C₁₆	1	1	yes	characteristic
143	C₂²	C₂×#14523982	1	1	yes	characteristic
144	C₄	—	268	2	no	—
145	C₄	Dic₆₇⋊C₁₆	1	1	yes	characteristic
146	C₄	Dic₆₇⋊C₁₆	1	1	yes	characteristic
147	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
148	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×Dic₆₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₂×M₆₄×C₆₇	C₂
socle	C₂×C₆₇	C₂×C₄⋊C₁₆
prime radical	C₂³×C₆₇	M₃₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	C₂×M₆₄	D₆₇
hypocenter	C₆₇	#13630
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #22558406	C₂	semidirect
81. C₆₇	#13630	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863230	2
#22561805	2
#22558406	2
C₂×M₆₄×C₆₇	2
#13630	67