Subgroups — #30764839

Order 17152.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
2	#22558492	C₂	1	1	yes	maximal, characteristic
3	#22558129	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
6	#16863449	C₄	1	2	yes	—
7	#16863172	C₄	1	2	yes	—
8	#16863172	C₂²	1	2	yes	—
9	#16863117	C₂²	1	2	yes	—
10	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
11	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#346×C₆₇	C₂²	1	2	yes	—
14	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
15	C₂×C₆₇⋊C₁₆	—	2	4	no	—
16	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₆₇	C₈	1	8	yes	complemented
24	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
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	C₂×C₆₇⋊C₈	—	2	4	no	—
30	C₆₇⋊C₁₆	—	2	4	no	—
31	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
33	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
37	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₆₇	C₄²	1	1	yes	characteristic
40	C₄²×C₆₇	M₁₆	1	2	yes	—
41	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₆₇	C₂×C₈	1	8	yes	—
43	C₂³×C₆₇	#97	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₆₇	#97	1	2	yes	—
47	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#97	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
51	C₈×C₆₇	C₄×C₈	1	2	yes	—
52	C₈×C₆₇	#97	1	2	yes	—
53	C₈×C₆₇	C₂²×C₈	1	2	yes	—
54	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
55	C₆₇⋊C₈	—	4	4	no	—
56	C₂²×C₆₇	—	2	2	no	—
57	C₂²×C₆₇	#322	1	2	yes	—
58	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
59	C₄×C₆₇	—	2	2	no	—
60	C₄×C₆₇	—	2	2	no	—
61	C₄×C₆₇	#322	1	2	yes	—
62	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
64	#11406	—	67	1	no	maximal, Sylow
65	C₂×C₆₇	—	2	2	no	—
66	C₂×C₆₇	C₂×#322	1	1	yes	characteristic
67	C₂×C₆₇	#2110	1	2	yes	—
68	C₄×M₃₂	D₆₇	1	1	yes	characteristic
69	C₂×#346	—	67	1	no	—
70	#1450	—	67	1	no	—
71	#1313	—	67	4	no	—
72	C₆₇	#11406	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	Dic₆₇	1	2	yes	—
74	C₂²×C₁₆	—	67	2	no	—
75	#431	—	67	2	no	—
76	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
77	#346	—	67	2	no	—
78	#346	—	67	2	no	—
79	#346	C₂×D₆₇	1	2	yes	—
80	C₄×C₁₆	Dic₆₇	1	2	yes	—
81	#322	—	67	2	no	—
82	C₈²	—	67	2	no	—
83	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
86	M₃₂	C₆₇⋊C₈	1	8	yes	complemented
87	C₂×C₁₆	—	67	4	no	—
88	C₂×C₁₆	—	134	4	no	—
89	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
90	C₄⋊C₈	—	134	2	no	—
91	C₂²⋊C₈	—	134	2	no	—
92	C₄×C₈	—	67	4	no	—
93	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
94	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
98	C₂²×C₄	#11761611	1	2	yes	—
99	C₂×C₈	—	134	4	no	—
100	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
102	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
103	C₂×C₈	#11761611	1	4	yes	—
104	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
106	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
107	C₄²	#11761611	1	2	yes	—
108	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
109	C₁₆	—	134	4	no	—
110	C₁₆	C₂×C₆₇⋊C₈	1	8	yes	—
111	C₂³	#14523953	1	1	yes	characteristic
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
114	C₂×C₄	#14523953	1	2	yes	—
115	C₂×C₄	#14523953	1	2	yes	—
116	C₂×C₄	#14523953	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
118	C₂×C₄	C₂×#11761611	1	2	yes	—
119	C₈	—	268	4	no	—
120	C₈	C₄×C₆₇⋊C₈	1	2	yes	—
121	C₈	#14523953	1	2	yes	—
122	C₈	C₂²×C₆₇⋊C₈	1	2	yes	—
123	C₈	C₂×#11761611	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#16863117	1	2	yes	—
126	C₂²	C₂×#14523953	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#16863117	1	2	yes	—
130	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
131	C₄	C₂×#14523953	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#16863117	1	1	yes	characteristic
134	C₂	#22561813	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×C₆₇	#97
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#11406
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558129	C₂	semidirect
23. M₃₂×C₆₇	C₈	semidirect
72. C₆₇	#11406	semidirect
86. M₃₂	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863172	2
#22558492	2
#22558129	2
C₄×M₃₂×C₆₇	2
#11406	67