Finite Group Databasebeta

Subgroups — #28378643

Order 13952.

152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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	#20296076	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295889	C₂	1	1	yes	maximal, characteristic
4	#20295876	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#11724786	C₂	1	1	yes	maximal, characteristic
6	#11724786⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#20295118	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#11724786	—	2	1	no	—
9	C₄⋊#11724786	—	2	1	no	—
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₄⋊#11724786	C₂²	1	1	yes	characteristic
12	#16812754	—	2	1	no	—
13	C₄×#11724786	—	2	1	no	—
14	C₄×#11724786	C₂²	1	1	yes	characteristic
15	C₄×#11724786	C₂²	1	1	yes	characteristic
16	C₂×#13182019	C₄	1	1	yes	characteristic
17	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	#16812664	C₂²	1	1	yes	characteristic
19	#16812632	C₄	1	1	yes	characteristic
20	#16812624	C₄	1	1	yes	characteristic
21	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
22	#103×C₁₀₉	C₄	1	1	yes	characteristic
23	C₂×#11724786	—	2	1	no	—
24	C₂×#11724786	—	2	1	no	—
25	C₂×#11724786	—	2	1	no	—
26	C₂×#11724786	D₄	1	1	yes	characteristic
27	C₂×#11724786	D₄	1	1	yes	characteristic
28	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
29	#13182019	—	2	2	no	—
30	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
33	#13182008	—	2	1	no	—
34	C₄×Dic₁₀₉	—	2	1	no	—
35	C₄×Dic₁₀₉	—	2	1	no	—
36	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	2	1	no	—
41	Q₈×C₁₀₉	M₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₉	—	2	1	no	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
49	Dic₂₁₈	—	2	1	no	—
50	Dic₂₁₈	—	2	1	no	—
51	#11724786	—	2	1	no	—
52	#11724786	—	4	1	no	—
53	#11724786	—	4	1	no	—
54	#11724786	—	4	1	no	—
55	#11724786	Q₁₆	1	2	yes	complemented
56	C₈×C₁₀₉	—	2	1	no	—
57	#11724784	—	2	1	no	—
58	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	4	1	no	—
61	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
62	C₄×C₁₀₉	#131	1	1	yes	characteristic
63	Dic₁₀₉	—	2	1	no	—
64	Dic₁₀₉	—	4	1	no	—
65	Dic₁₀₉	—	4	1	no	—
66	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
67	Dic₁₀₉	#135	1	1	yes	characteristic
68	C₂×C₁₀₉	#435	1	1	yes	characteristic
69	C₂×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₀₉	#444	1	1	yes	characteristic
71	#1584	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1584	1	1	yes	characteristic, complemented, Sylow
73	#446	—	109	1	no	—
74	C₈×Q₈	—	109	1	no	—
75	C₄×Q₁₆	—	109	1	no	—
76	#335	—	109	1	no	—
77	Q₈⋊C₈	—	109	1	no	—
78	Q₈⋊C₈	—	109	1	no	—
79	#322	—	109	1	no	—
80	C₂×Q₁₆	—	109	1	no	—
81	C₄×Q₈	—	109	1	no	—
82	C₄×Q₈	—	109	1	no	—
83	#107	—	109	1	no	—
84	C₄⋊C₈	—	109	1	no	—
85	C₄⋊C₈	—	109	1	no	—
86	C₄⋊C₈	—	218	1	no	—
87	C₄⋊C₈	—	218	1	no	—
88	#103	—	109	1	no	—
89	#103	#89137	1	1	yes	characteristic
90	#97	—	218	1	no	—
91	C₄×C₈	—	109	1	no	—
92	C₄×C₈	—	109	1	no	—
93	C₄×C₈	—	109	1	no	—
94	C₄×C₈	—	218	1	no	—
95	C₂×Q₈	—	109	1	no	—
96	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
97	Q₁₆	—	218	2	no	—
98	C₂×C₈	—	109	1	no	—
99	C₂×C₈	—	109	1	no	—
100	C₂×C₈	—	109	1	no	—
101	C₂×C₈	—	218	1	no	—
102	C₂×C₈	—	218	1	no	—
103	C₂×C₈	—	218	1	no	—
104	C₂×C₈	C₂×#89137	1	1	yes	characteristic
105	C₄⋊C₄	—	109	1	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
109	C₄²	—	109	1	no	—
110	C₄²	—	218	1	no	—
111	C₄²	—	218	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	218	1	no	—
114	Q₈	—	218	1	no	—
115	Q₈	#13182039	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	218	1	no	—
121	C₂×C₄	—	218	1	no	—
122	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	109	2	no	—
125	C₈	—	436	1	no	—
126	C₈	—	436	1	no	—
127	C₈	—	436	1	no	—
128	C₈	—	218	1	no	—
129	C₈	—	218	1	no	—
130	C₂²	D₄×#89137	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	109	1	no	—
134	C₄	—	109	1	no	—
135	C₄	—	436	1	no	—
136	C₄	—	436	1	no	—
137	C₄	—	218	1	no	—
138	C₄	C₂×#13182039	1	1	yes	characteristic
139	C₄	#16812786	1	1	yes	characteristic
140	C₂	#20296055	1	1	yes	characteristic
141	C₂	Q₁₆×#89137	1	1	yes	characteristic
142	C₂	#20296071	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#103×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	#103	#89137
hypocenter	C₁₀₉	#1584
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
55. #11724786	Q₁₆	semidirect
72. C₁₀₉	#1584	semidirect

Maximal subgroups

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

subgroup	index
#20296076	2
Q₈×#11724786	2
#20295889	2
#20295876	2
Q₈⋊#11724786	2
#11724786⋊C₈	2
#20295118	2
#1584	109