Finite Group Databasebeta

Subgroups — #28362491

Order 13952.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20295377	C₂	1	1	yes	maximal, characteristic
2	#20295322	C₂	1	1	yes	maximal, characteristic
3	#20295312	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
5	#20294774	C₂	1	1	yes	maximal, characteristic
6	#20294771	C₂	1	1	yes	maximal, characteristic
7	#335×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812699	C₂²	1	1	yes	characteristic
9	#16812699	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	#16812633	—	2	1	no	—
12	#16812631	—	2	1	no	—
13	C₄⋊#11724784	C₂²	1	1	yes	characteristic
14	#16812628	—	2	1	no	—
15	#16812624	—	2	1	no	—
16	#16812624	—	2	1	no	—
17	#16812624	—	2	1	no	—
18	#16812624	—	2	1	no	—
19	C₈×Dic₁₀₉	—	2	1	no	—
20	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×Dic₂₁₈	—	2	1	no	—
25	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
26	C₂×#11724784	—	2	1	no	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
32	#13182008	—	2	1	no	—
33	C₄×Dic₁₀₉	—	2	1	no	—
34	C₂×C₈×C₁₀₉	—	2	1	no	—
35	C₂×C₈×C₁₀₉	—	2	1	no	—
36	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
42	C₂×Dic₁₀₉	—	2	1	no	—
43	C₂×Dic₁₀₉	—	2	1	no	—
44	C₂×Dic₁₀₉	—	4	1	no	—
45	C₂×Dic₁₀₉	—	4	1	no	—
46	Dic₂₁₈	—	2	1	no	—
47	Dic₂₁₈	—	4	1	no	—
48	Dic₂₁₈	—	4	1	no	—
49	Dic₂₁₈	—	8	1	no	—
50	Dic₂₁₈	Q₁₆	1	2	yes	—
51	C₈×C₁₀₉	—	2	1	no	—
52	C₈×C₁₀₉	—	2	1	no	—
53	C₈×C₁₀₉	—	4	1	no	—
54	C₈×C₁₀₉	—	4	1	no	—
55	#11724784	—	4	1	no	—
56	C₂²×C₁₀₉	#121	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	1	no	—
58	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₁₀₉	#135	1	1	yes	characteristic
60	C₄×C₁₀₉	#135	1	1	yes	characteristic
61	C₄×C₁₀₉	#137	1	1	yes	characteristic
62	Dic₁₀₉	—	2	2	no	—
63	Dic₁₀₉	—	4	1	no	—
64	Dic₁₀₉	—	8	1	no	—
65	Dic₁₀₉	—	8	1	no	—
66	C₂×C₁₀₉	#464	1	1	yes	characteristic
67	C₂×C₁₀₉	#467	1	1	yes	characteristic
68	C₂×C₁₀₉	#471	1	1	yes	characteristic
69	#1684	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1684	1	1	yes	characteristic, complemented, Sylow
71	#487	—	109	1	no	—
72	#477	—	109	1	no	—
73	#475	—	109	1	no	—
74	C₈×Q₈	—	109	1	no	—
75	#335	D₁₀₉	1	1	yes	characteristic
76	#333	—	109	1	no	—
77	#332	—	109	1	no	—
78	C₄⋊Q₈	—	109	1	no	—
79	C₄⋊Q₈	—	109	1	no	—
80	C₄×Q₈	—	109	1	no	—
81	#107	—	218	1	no	—
82	#106	—	218	1	no	—
83	C₄⋊C₈	—	109	1	no	—
84	C₄⋊C₈	—	218	1	no	—
85	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
87	#103	—	218	1	no	—
88	#103	—	218	1	no	—
89	#103	—	218	1	no	—
90	#103	—	218	1	no	—
91	C₄×C₈	—	218	1	no	—
92	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
93	C₂×Q₈	—	109	1	no	—
94	C₂×Q₈	—	218	1	no	—
95	C₂×Q₈	—	218	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	218	1	no	—
99	C₂×C₈	D₄₃₆	1	1	yes	characteristic
100	C₂×C₈	D₄₃₆	1	1	yes	characteristic
101	C₄⋊C₄	—	109	1	no	—
102	C₄⋊C₄	—	218	1	no	—
103	C₄⋊C₄	—	218	1	no	—
104	C₄⋊C₄	—	218	1	no	—
105	C₄⋊C₄	—	218	1	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄²	—	218	1	no	—
108	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
109	Q₈	—	872	1	no	—
110	Q₈	—	109	2	no	—
111	Q₈	—	436	1	no	—
112	Q₈	—	436	1	no	—
113	Q₈	—	218	1	no	—
114	C₂×C₄	—	436	1	no	—
115	C₂×C₄	—	436	1	no	—
116	C₂×C₄	—	218	1	no	—
117	C₂×C₄	—	218	1	no	—
118	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
120	C₂×C₄	#13182030	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	436	1	no	—
126	C₂²	#16812674	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	872	1	no	—
129	C₄	—	872	1	no	—
130	C₄	—	436	1	no	—
131	C₄	—	218	2	no	—
132	C₄	Q₁₆×D₁₀₉	1	1	yes	characteristic
133	C₄	#16812718	1	1	yes	characteristic
134	C₄	#16812721	1	1	yes	characteristic
135	C₄	#16812732	1	1	yes	characteristic
136	C₂	#20295249	1	1	yes	characteristic
137	C₂	#20295268	1	1	yes	characteristic
138	C₂	#20295290	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812674
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#335×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#121
hypercenter	#335	D₁₀₉
hypocenter	C₁₀₉	#1684
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
70. C₁₀₉	#1684	semidirect

Maximal subgroups

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

subgroup	index
#20295377	2
#20295322	2
#20295312	2
C₈×Dic₂₁₈	2
#20294774	2
#20294771	2
#335×C₁₀₉	2
#1684	109