Subgroups — #30856216

Order 17152.

1350 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	6	yes	maximal
2	#22560361	C₂	1	8	yes	maximal, complemented
3	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	C₂²	1	4	yes	—
5	#16863420	—	2	24	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	24	yes	—
7	C₂²×C₄²×C₆₇	C₄	1	4	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
9	C₄³×C₆₇	C₄	1	3	yes	—
10	C₄³×C₆₇	C₄	1	1	yes	characteristic
11	C₄³×C₆₇	C₂²	1	4	yes	—
12	C₂×#11761612	—	2	24	no	—
13	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
14	C₄⋊(C₆₇⋊C₈)	—	2	48	no	—
15	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
16	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	—	2	16	no	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₆₇	D₄	1	12	yes	—
23	C₂×C₄²×C₆₇	Q₈	1	12	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
25	#11761612	—	4	16	no	—
26	C₂×C₆₇⋊C₈	—	2	24	no	—
27	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	—	2	24	no	—
29	C₂²×C₄×C₆₇	C₄⋊C₄	1	24	yes	—
30	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
34	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
35	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
36	C₄²×C₆₇	—	2	24	no	—
37	C₄²×C₆₇	C₄⋊C₄	1	12	yes	—
38	C₄²×C₆₇	C₄⋊C₄	1	12	yes	—
39	C₄²×C₆₇	M₁₆	1	16	yes	complemented
40	C₄²×C₆₇	C₂×D₄	1	12	yes	—
41	C₄²×C₆₇	C₂×Q₈	1	12	yes	—
42	C₂³×C₆₇	—	2	4	no	—
43	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
44	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	12	no	—
46	C₂×C₄×C₆₇	—	2	48	no	—
47	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
48	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
49	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
50	C₂×C₄×C₆₇	#127	1	4	yes	—
51	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
52	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	—
53	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
54	C₂×C₄×C₆₇	C₂²×Q₈	1	3	yes	—
55	C₆₇⋊C₈	—	4	8	no	—
56	C₂²×C₆₇	—	2	12	no	—
57	C₂²×C₆₇	#384	1	4	yes	—
58	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
59	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
60	C₄×C₆₇	—	2	12	no	—
61	C₄×C₆₇	#384	1	4	yes	—
62	C₄×C₆₇	#423	1	24	yes	complemented
63	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
64	C₄×C₆₇	C₂×C₄⋊Q₈	1	3	yes	—
65	#20791	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	—	2	4	no	—
67	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
68	C₂×C₆₇	C₂×#423	1	6	yes	—
69	C₂×#423	—	67	6	no	—
70	C₂×C₄³	D₆₇	1	1	yes	characteristic
71	#1842	—	67	8	no	—
72	C₆₇	#20791	1	1	yes	characteristic, complemented, Sylow
73	C₂²×M₁₆	—	67	4	no	—
74	C₂²×C₄²	Dic₆₇	1	4	yes	—
75	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
76	#423	—	134	24	no	—
77	C₂×C₄⋊C₈	—	67	24	no	—
78	C₄³	Dic₆₇	1	3	yes	—
79	C₄³	Dic₆₇	1	1	yes	characteristic
80	C₄³	C₂×D₆₇	1	4	yes	—
81	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
82	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
83	C₂×M₁₆	—	134	24	no	—
84	C₂²×C₈	—	67	8	no	—
85	C₂×C₄²	—	2	16	no	—
86	C₂×C₄²	Dic₁₃₄	1	12	yes	—
87	C₂×C₄²	D₂₆₈	1	12	yes	—
88	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
89	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
90	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
91	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
92	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
93	C₄⋊C₈	—	134	48	no	—
94	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
95	C₂²×C₄	—	2	24	no	—
96	C₂²×C₄	C₄⋊Dic₆₇	1	24	yes	—
97	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
98	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
100	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
101	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
102	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
103	M₁₆	—	268	16	no	—
104	C₂×C₈	—	134	24	no	—
105	C₄²	—	2	24	no	—
106	C₄²	C₄⋊Dic₆₇	1	12	yes	—
107	C₄²	C₄⋊Dic₆₇	1	12	yes	—
108	C₄²	#11761612	1	16	yes	complemented
109	C₄²	C₂×D₂₆₈	1	12	yes	—
110	C₄²	C₂×Dic₁₃₄	1	12	yes	—
111	C₂³	—	2	4	no	—
112	C₂³	C₂×C₄⋊Dic₆₇	1	6	yes	—
113	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
119	C₂×C₄	#14524040	1	4	yes	—
120	C₂×C₄	#14524043	1	12	yes	—
121	C₂×C₄	C₂×#11761612	1	24	yes	—
122	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
123	C₂×C₄	C₂²×Dic₁₃₄	1	3	yes	—
124	C₈	—	268	8	no	—
125	C₂²	—	2	12	no	—
126	C₂²	#16863266	1	4	yes	—
127	C₂²	C₂²×C₄⋊Dic₆₇	1	3	yes	—
128	C₂²	C₂²×#11761612	1	4	yes	—
129	C₄	—	2	12	no	—
130	C₄	#16863266	1	4	yes	—
131	C₄	#16863420	1	24	yes	complemented
132	C₄	C₂×#14524040	1	1	yes	characteristic
133	C₄	C₂×#14524043	1	3	yes	—
134	C₂	—	2	4	no	—
135	C₂	C₂×#16863266	1	1	yes	characteristic
136	C₂	C₂×#16863420	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₂×C₄³×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	#20791
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22560361	C₂	semidirect
39. C₄²×C₆₇	M₁₆	semidirect
62. C₄×C₆₇	#423	semidirect
72. C₆₇	#20791	semidirect
108. C₄²	#11761612	semidirect
131. C₄	#16863420	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
#22560361	2
C₂×C₄³×C₆₇	2
#20791	67