Finite Group Databasebeta

Subgroups — #30770579

Order 17152.

246 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562491	C₂	1	2	yes	maximal
2	#22562484	C₂	1	2	yes	maximal
3	#22559827	C₂	1	1	yes	maximal, characteristic
4	#22559819	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1311	C₂	1	1	yes	maximal, characteristic
6	#16863801	—	2	1	no	—
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863801	C₂²	1	1	yes	characteristic
9	#16863801	C₂²	1	2	yes	—
10	#16863776	—	2	1	no	—
11	#16863747	—	2	1	no	—
12	#16863747	—	2	1	no	—
13	#16863224	—	2	4	no	—
14	#16863222	—	2	4	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₈	—	4	2	no	—
18	#14524043	—	2	1	no	—
19	#14524043	—	2	1	no	—
20	#14524043	—	2	2	no	—
21	#14523977	—	2	2	no	—
22	#14523977	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523977	—	2	2	no	—
25	#14523977	—	4	2	no	—
26	#14523968	—	4	2	no	—
27	#14523968	—	4	2	no	—
28	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
29	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₃₄	—	4	1	no	—
33	C₂×Dic₁₃₄	—	4	1	no	—
34	C₂×Dic₁₃₄	—	4	2	no	—
35	Dic₂₆₈	—	8	4	no	—
36	C₄⋊Dic₆₇	—	4	1	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	1	no	—
39	C₄⋊Dic₆₇	—	4	1	no	—
40	C₄⋊Dic₆₇	—	4	1	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₂×C₈×C₆₇	D₈	1	2	yes	—
43	C₂×C₈×C₆₇	D₈	1	2	yes	—
44	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
45	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₆₇	Q₁₆	1	8	yes	complemented
47	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₆₇	—	8	1	no	—
51	C₂×Dic₆₇	—	8	2	no	—
52	C₂×Dic₆₇	—	8	1	no	—
53	Dic₁₃₄	—	8	4	no	—
54	Dic₁₃₄	—	8	2	no	—
55	Dic₁₃₄	—	8	2	no	—
56	C₈×C₆₇	—	2	2	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₆₇	#493	1	1	yes	characteristic
60	C₄×C₆₇	#494	1	2	yes	—
61	C₄×C₆₇	#509	1	2	yes	—
62	C₄×C₆₇	#510	1	2	yes	—
63	Dic₆₇	—	16	2	no	—
64	Dic₆₇	—	16	1	no	—
65	Dic₆₇	—	16	1	no	—
66	#12524	—	67	1	no	maximal, Sylow
67	C₂×C₆₇	#1713	1	1	yes	characteristic
68	C₂×C₆₇	#2250	1	1	yes	characteristic
69	C₂×C₆₇	#2251	1	1	yes	characteristic
70	#2254	—	67	2	no	—
71	#2252	—	67	2	no	—
72	#1713	—	67	1	no	—
73	#1710	—	67	1	no	—
74	#1311	D₆₇	1	1	yes	characteristic
75	C₆₇	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	67	1	no	—
77	#500	—	67	1	no	—
78	#500	—	67	2	no	—
79	#500	—	134	1	no	—
80	#494	—	134	1	no	—
81	#487	—	134	1	no	—
82	#487	—	134	1	no	—
83	#367	—	134	4	no	—
84	#366	—	134	4	no	—
85	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
86	C₈²	C₂×D₆₇	1	1	yes	characteristic
87	C₂×Q₁₆	—	268	2	no	—
88	C₄⋊Q₈	—	134	1	no	—
89	C₄⋊Q₈	—	134	1	no	—
90	C₄⋊Q₈	—	134	2	no	—
91	C₂×C₁₆	D₂₆₈	1	4	yes	—
92	#107	—	268	2	no	—
93	#107	—	134	2	no	—
94	#107	—	134	2	no	—
95	#107	—	134	2	no	—
96	#107	—	134	2	no	—
97	#103	—	268	2	no	—
98	#103	—	268	2	no	—
99	C₄×C₈	D₂₆₈	1	1	yes	characteristic
100	C₄×C₈	D₂₆₈	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	C₂×Q₈	—	268	1	no	—
103	C₂×Q₈	—	268	1	no	—
104	C₂×Q₈	—	268	2	no	—
105	Q₁₆	—	536	4	no	—
106	C₂×C₈	D₅₃₆	1	2	yes	—
107	C₂×C₈	D₅₃₆	1	2	yes	—
108	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
109	C₄⋊C₄	—	268	1	no	—
110	C₄⋊C₄	—	268	2	no	—
111	C₄⋊C₄	—	268	1	no	—
112	C₄⋊C₄	—	268	1	no	—
113	C₄⋊C₄	—	268	1	no	—
114	C₄⋊C₄	—	268	2	no	—
115	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
116	C₁₆	Dic₂₆₈	1	8	yes	complemented
117	Q₈	—	536	4	no	—
118	Q₈	—	536	2	no	—
119	Q₈	—	536	2	no	—
120	C₂×C₄	—	536	1	no	—
121	C₂×C₄	—	536	2	no	—
122	C₂×C₄	—	536	1	no	—
123	C₂×C₄	#14524040	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
125	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Dic₂₆₈	1	4	yes	—
129	C₂²	#16863773	1	1	yes	characteristic
130	C₄	—	1072	2	no	—
131	C₄	—	1072	1	no	—
132	C₄	—	1072	1	no	—
133	C₄	#16863776	1	2	yes	—
134	C₄	#16863834	1	2	yes	—
135	C₄	#16863839	1	2	yes	—
136	C₂	#22559827	1	1	yes	characteristic
137	C₂	#22562475	1	1	yes	characteristic
138	C₂	#22562480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1311	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1311	D₆₇
hypocenter	C₆₇	#12524
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
46. C₁₆×C₆₇	Q₁₆	semidirect
75. C₆₇	#12524	semidirect
116. C₁₆	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22562491	2
#22562484	2
#22559827	2
#22559819	2
C₆₇×#1311	2
#12524	67