Subgroups — #30767402

Order 17152.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561965	C₂	1	2	yes	maximal
2	#22558279	C₂	1	4	yes	maximal, complemented
3	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	2	no	—
5	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
6	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₈²×C₆₇	C₄	1	2	yes	—
11	C₈²×C₆₇	C₂²	1	2	yes	—
12	#14523983	—	4	4	no	—
13	C₂×C₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₈×C₆₇	—	2	1	no	—
15	C₂²×C₈×C₆₇	C₈	1	2	yes	—
16	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₆₇	—	2	1	no	—
20	C₄×C₈×C₆₇	C₈	1	1	yes	characteristic
21	C₄×C₈×C₆₇	C₈	1	1	yes	characteristic
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	2	yes	—
25	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₆₇⋊C₁₆	—	4	4	no	—
29	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₆₇	—	2	4	no	—
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	—	2	4	no	—
36	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₆₇	D₈	1	2	yes	—
41	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
47	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#107	1	2	yes	—
55	C₂×C₄×C₆₇	#108	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₆₇	—	2	4	no	—
60	C₈×C₆₇	—	2	1	no	—
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	—	2	2	no	—
63	C₈×C₆₇	#107	1	2	yes	—
64	C₈×C₆₇	M₃₂	1	4	yes	complemented
65	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
66	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	—	2	1	no	—
69	C₂²×C₆₇	#335	1	2	yes	—
70	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	#335	1	2	yes	—
74	C₄×C₆₇	#364	1	2	yes	—
75	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
76	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
77	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
78	#12004	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	—	2	2	no	—
80	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
81	C₂×C₆₇	#2150	1	1	yes	characteristic
82	C₂×C₆₇	C₂×#364	1	1	yes	characteristic
83	#2150	—	67	2	no	—
84	C₂×C₈²	D₆₇	1	1	yes	characteristic
85	#1370	—	67	4	no	—
86	C₆₇	#12004	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₃₂	—	134	2	no	—
88	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
89	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
91	C₄⋊C₁₆	—	67	4	no	—
92	C₄⋊C₁₆	—	134	2	no	—
93	C₈²	Dic₆₇	1	2	yes	—
94	C₈²	C₂×D₆₇	1	2	yes	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	C₆₇⋊C₈	1	2	yes	—
97	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
98	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
99	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
100	M₃₂	—	268	4	no	—
101	C₂×C₁₆	—	134	4	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₆₇⋊C₈	1	1	yes	characteristic
104	C₄×C₈	C₆₇⋊C₈	1	1	yes	characteristic
105	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
106	C₄×C₈	D₂₆₈	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
108	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
110	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
113	C₂²×C₄	#11761612	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
120	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
121	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
122	C₂×C₈	D₅₃₆	1	2	yes	—
123	C₂×C₈	Dic₂₆₈	1	2	yes	—
124	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
125	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
126	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
127	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
128	C₄²	#11761612	1	1	yes	characteristic
129	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
130	C₁₆	—	268	4	no	—
131	C₂³	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
135	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
136	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
137	C₂×C₄	#14523977	1	2	yes	—
138	C₂×C₄	#14523979	1	2	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
140	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
141	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	#14523977	1	2	yes	—
147	C₈	#14523983	1	4	yes	complemented
148	C₈	C₂×D₅₃₆	1	1	yes	characteristic
149	C₈	C₂×Dic₂₆₈	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16863149	1	2	yes	—
153	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#16863149	1	2	yes	—
157	C₄	#16863217	1	2	yes	—
158	C₄	C₂×#14523977	1	1	yes	characteristic
159	C₄	C₂×#14523979	1	1	yes	characteristic
160	C₄	C₂×#14523983	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₂×#16863149	1	1	yes	characteristic
163	C₂	#22561965	1	1	yes	characteristic
164	C₂	C₂×#16863217	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#12004
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #22558279	C₂	semidirect
64. C₈×C₆₇	M₃₂	semidirect
86. C₆₇	#12004	semidirect
147. C₈	#14523983	semidirect

Maximal subgroups

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

subgroup	index
#22561965	2
#22558279	2
C₂×C₈²×C₆₇	2
#12004	67