Subgroups — #30764864

Order 17152.

290 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	#22562008	C₂	1	2	yes	maximal
2	#22558127	C₂	1	2	yes	maximal, complemented
3	#22558126	C₂	1	2	yes	maximal, complemented
4	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	—
6	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	—
7	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₈²×C₆₇	C₄	1	1	yes	characteristic
12	C₈²×C₆₇	C₄	1	1	yes	characteristic
13	C₈²×C₆₇	C₂²	1	1	yes	characteristic
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₆₇	C₈	1	2	yes	—
17	C₂²×C₈×C₆₇	C₈	1	2	yes	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₆₇	C₈	1	2	yes	—
21	C₄×C₈×C₆₇	C₈	1	2	yes	—
22	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₆₇⋊C₁₆	—	2	8	no	—
30	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	—	2	4	no	—
35	C₂×C₈×C₆₇	—	2	4	no	—
36	C₂×C₈×C₆₇	C₄²	1	2	yes	—
37	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
42	C₄²×C₆₇	C₄²	1	1	yes	characteristic
43	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂³×C₆₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
53	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₆₇	—	2	2	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	—	2	2	no	—
59	C₈×C₆₇	—	2	2	no	—
60	C₈×C₆₇	M₃₂	1	4	yes	—
61	C₈×C₆₇	#131	1	4	yes	—
62	C₂²×C₆₇	—	2	1	no	—
63	C₂²×C₆₇	—	2	1	no	—
64	C₂²×C₆₇	#346	1	2	yes	—
65	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	#346	1	2	yes	—
69	C₄×C₆₇	#405	1	2	yes	—
70	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
71	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
72	#11411	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	—	2	2	no	—
74	C₂×C₆₇	#1450	1	1	yes	characteristic
75	C₂×C₆₇	C₂×#346	1	1	yes	characteristic
76	C₂×C₆₇	C₄×M₃₂	1	1	yes	characteristic
77	#2162	—	67	2	no	—
78	C₂×C₈²	D₆₇	1	1	yes	characteristic
79	#1312	—	67	2	no	—
80	#1311	—	67	2	no	—
81	C₆₇	#11411	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
83	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
85	C₄⋊C₁₆	—	67	4	no	—
86	C₂²⋊C₁₆	—	67	4	no	—
87	C₄×C₁₆	—	67	4	no	—
88	C₈²	Dic₆₇	1	1	yes	characteristic
89	C₈²	Dic₆₇	1	1	yes	characteristic
90	C₈²	C₂×D₆₇	1	1	yes	characteristic
91	C₈²	C₂×D₆₇	1	1	yes	characteristic
92	C₂²×C₈	C₆₇⋊C₈	1	2	yes	—
93	C₂²×C₈	C₆₇⋊C₈	1	2	yes	—
94	C₂²×C₈	C₄×D₆₇	1	2	yes	—
95	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
96	C₂×C₁₆	—	67	8	no	—
97	C₄×C₈	C₆₇⋊C₈	1	2	yes	—
98	C₄×C₈	C₆₇⋊C₈	1	2	yes	—
99	C₄×C₈	C₄×D₆₇	1	2	yes	—
100	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
113	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
114	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
115	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
116	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
118	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
119	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
120	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
121	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
122	C₁₆	—	134	8	no	—
123	C₂³	C₄×C₆₇⋊C₈	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×C₆₇⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₄×C₆₇⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₄×C₆₇⋊C₈	1	1	yes	characteristic
129	C₂×C₄	#14523983	1	4	yes	—
130	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	#14523983	1	4	yes	—
138	C₈	#14524049	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#16863172	1	2	yes	—
142	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	#16863172	1	2	yes	—
146	C₄	#16863351	1	2	yes	—
147	C₄	C₂×#14523983	1	2	yes	—
148	C₄	C₂×#14523983	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#22558492	1	1	yes	characteristic
151	C₂	C₂×#16863172	1	1	yes	characteristic
152	C₂	C₄×#14523983	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₂×C₈²×C₆₇	C₂
prime radical	C₂³×C₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#11411
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #22558127	C₂	semidirect
3. #22558126	C₂	semidirect
81. C₆₇	#11411	semidirect

Maximal subgroups

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

subgroup	index
#22562008	2
#22558127	2
#22558126	2
C₂×C₈²×C₆₇	2
#11411	67