Finite Group Databasebeta

Subgroups — #28362362

Order 13952.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₀₉⋊#564	C₂	1	1	yes	maximal, characteristic
2	#20295436	C₂	1	1	yes	maximal, characteristic
3	#20295435	C₂	1	1	yes	maximal, characteristic
4	#20295395	C₂	1	1	yes	maximal, characteristic
5	#20294767	C₂	1	1	yes	maximal, characteristic
6	#20294767	C₂	1	1	yes	maximal, characteristic
7	#330×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812699	C₂²	1	1	yes	characteristic
9	#16812691	C₂²	1	1	yes	characteristic
10	#16812691	C₂²	1	1	yes	characteristic
11	#16812690	—	2	1	no	—
12	#16812690	—	2	1	no	—
13	#16812689	—	2	1	no	—
14	#16812689	—	2	1	no	—
15	#16812689	—	2	1	no	—
16	#16812689	—	2	1	no	—
17	#16812633	—	2	1	no	—
18	#16812632	—	2	1	no	—
19	#16812631	—	2	1	no	—
20	#16812630	—	2	1	no	—
21	#16812624	—	2	1	no	—
22	#16812624	—	2	1	no	—
23	#16812610	C₂²	1	1	yes	characteristic
24	#125×C₁₀₉	C₂²	1	1	yes	characteristic
25	#97×C₁₀₉	C₂²	1	1	yes	characteristic
26	#97×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×Dic₂₁₈	—	2	1	no	—
28	C₂×#11724784	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	—	2	1	no	—
33	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
34	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
37	#13182008	—	2	1	no	—
38	#13182008	—	2	1	no	—
39	#13182008	—	2	1	no	—
40	#13182008	—	2	1	no	—
41	#13182008	—	2	1	no	—
42	#13182008	—	2	1	no	—
43	#13182008	—	2	1	no	—
44	#13182008	—	2	1	no	—
45	C₄×Dic₁₀₉	—	2	1	no	—
46	C₄×Dic₁₀₉	—	2	1	no	—
47	C₂×C₈×C₁₀₉	—	2	1	no	—
48	C₂×C₈×C₁₀₉	—	2	1	no	—
49	C₄⋊C₄×C₁₀₉	—	2	1	no	—
50	C₄⋊C₄×C₁₀₉	—	2	1	no	—
51	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
53	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	—	2	1	no	—
55	C₂×C₄×C₁₀₉	—	2	1	no	—
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₁₀₉	—	2	1	no	—
60	C₂×Dic₁₀₉	—	2	1	no	—
61	C₂×Dic₁₀₉	—	2	1	no	—
62	C₂×Dic₁₀₉	—	2	1	no	—
63	C₂×Dic₁₀₉	—	4	1	no	—
64	Dic₂₁₈	—	4	2	no	—
65	C₈×C₁₀₉	—	2	2	no	—
66	C₈×C₁₀₉	—	4	1	no	—
67	#11724784	—	2	2	no	—
68	C₂²×C₁₀₉	#120	1	1	yes	characteristic
69	C₄×C₁₀₉	—	2	1	no	—
70	C₄×C₁₀₉	—	2	1	no	—
71	C₄×C₁₀₉	—	2	1	no	—
72	C₄×C₁₀₉	—	4	1	no	—
73	C₄×C₁₀₉	—	4	1	no	—
74	Dic₁₀₉	—	4	1	no	—
75	Dic₁₀₉	—	4	1	no	—
76	Dic₁₀₉	—	4	1	no	—
77	Dic₁₀₉	—	4	1	no	—
78	Dic₁₀₉	—	8	1	no	—
79	C₂×C₁₀₉	#454	1	1	yes	characteristic
80	C₂×C₁₀₉	#456	1	1	yes	characteristic
81	C₂×C₁₀₉	#456	1	1	yes	characteristic
82	#1663	—	109	1	no	maximal, Sylow
83	C₁₀₉	#1663	1	1	yes	characteristic, complemented, Sylow
84	#564	—	109	1	no	—
85	#501	—	109	1	no	—
86	#501	—	109	1	no	—
87	#491	—	109	1	no	—
88	#330	—	109	1	no	—
89	#330	—	109	1	no	—
90	#330	D₁₀₉	1	1	yes	characteristic
91	C₄⋊Q₈	—	109	1	no	—
92	#125	—	109	1	no	—
93	#125	—	109	1	no	—
94	#125	—	218	1	no	—
95	#125	—	218	1	no	—
96	#125	—	218	1	no	—
97	#125	—	218	1	no	—
98	#125	—	218	1	no	—
99	#125	—	218	1	no	—
100	#125	C₂×D₁₀₉	1	1	yes	characteristic
101	#107	—	218	1	no	—
102	#107	—	218	1	no	—
103	#106	—	218	1	no	—
104	#106	—	218	1	no	—
105	#103	—	218	1	no	—
106	#103	—	218	1	no	—
107	#97	—	109	1	no	—
108	#97	C₂×D₁₀₉	1	1	yes	characteristic
109	#97	C₂×D₁₀₉	1	1	yes	characteristic
110	C₂×Q₈	—	218	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	218	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	109	1	no	—
117	C₄⋊C₄	—	109	1	no	—
118	C₄⋊C₄	—	109	1	no	—
119	C₄⋊C₄	—	109	1	no	—
120	C₄⋊C₄	—	218	1	no	—
121	C₄⋊C₄	—	218	1	no	—
122	C₄⋊C₄	—	218	1	no	—
123	C₄⋊C₄	—	218	1	no	—
124	C₄⋊C₄	—	218	1	no	—
125	C₄⋊C₄	—	218	1	no	—
126	C₄⋊C₄	—	218	1	no	—
127	C₄⋊C₄	—	218	1	no	—
128	C₄⋊C₄	—	218	1	no	—
129	C₄⋊C₄	—	218	1	no	—
130	C₄⋊C₄	—	218	1	no	—
131	C₄⋊C₄	—	218	1	no	—
132	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
133	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
134	C₄²	—	218	1	no	—
135	C₄²	—	218	1	no	—
136	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
137	Q₈	—	436	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	436	1	no	—
141	C₂×C₄	—	218	1	no	—
142	C₂×C₄	—	218	1	no	—
143	C₂×C₄	—	218	1	no	—
144	C₂×C₄	—	218	1	no	—
145	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
146	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	218	2	no	—
151	C₂²	#16812669	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	4	1	no	—
157	C₄	—	872	1	no	—
158	C₄	—	436	1	no	—
159	C₄	—	436	1	no	—
160	C₄	—	436	1	no	—
161	C₄	—	436	1	no	—
162	C₂	#20295200	1	1	yes	characteristic
163	C₂	#20295210	1	1	yes	characteristic
164	C₂	#20295211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812669
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#330×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#120
hypercenter	#330	D₁₀₉
hypocenter	C₁₀₉	#1663
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

Complements

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

N	H ≅ G/N	type
83. C₁₀₉	#1663	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉⋊#564	2
#20295436	2
#20295435	2
#20295395	2
#20294767	2
#330×C₁₀₉	2
#1663	109