Subgroups — #28371981

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295157	C₂	1	2	yes	maximal, complemented
2	#20295157	C₂	1	4	yes	maximal, complemented
3	Q₈×#11724784	C₂	1	2	yes	maximal, complemented
4	#20295053	C₂	1	2	yes	maximal
5	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
6	#20294982	C₂	1	2	yes	maximal
7	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#13182013	C₂²	1	1	yes	characteristic
10	C₂×#13182013	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	2	yes	—
12	C₄⋊#11724784	C₂²	1	1	yes	characteristic
13	C₄⋊#11724784	C₂²	1	1	yes	characteristic
14	C₄⋊#11724784	C₂²	1	2	yes	—
15	C₄⋊#11724784	C₂²	1	4	yes	—
16	C₄⋊#11724784	C₂²	1	4	yes	—
17	#16812610	C₂²	1	2	yes	—
18	#16812610	C₂²	1	4	yes	—
19	C₄×#11724784	C₂²	1	2	yes	—
20	C₄×#11724784	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
23	#125×C₁₀₉	C₄	1	2	yes	—
24	C₂²⋊Q₈×C₁₀₉	C₄	1	2	yes	—
25	C₂²⋊Q₈×C₁₀₉	C₄	1	2	yes	—
26	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
27	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
28	#117×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
30	#13182013	—	2	2	no	—
31	#13182013	Q₈	1	4	yes	complemented
32	C₂×#11724784	—	2	4	no	—
33	C₂×#11724784	C₂³	1	2	yes	—
34	C₂×#11724784	C₂³	1	2	yes	—
35	C₂×#11724784	C₂³	1	2	yes	—
36	C₂×#11724784	C₂³	1	2	yes	—
37	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
45	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
49	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₄²×C₁₀₉	C₂³	1	4	yes	—
54	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	Q₈×C₁₀₉	—	2	4	no	—
56	Q₈×C₁₀₉	—	4	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	4	1	no	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₀₉	#41	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	#11724784	—	2	2	no	—
70	#11724784	—	2	2	no	—
71	#11724784	—	2	2	no	—
72	#11724784	C₂×Q₈	1	4	yes	—
73	C₂²×C₁₀₉	—	2	1	no	—
74	C₂²×C₁₀₉	—	2	1	no	—
75	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₁₀₉	—	2	1	no	—
78	C₄×C₁₀₉	—	2	4	no	—
79	C₄×C₁₀₉	—	2	1	no	—
80	C₄×C₁₀₉	—	2	1	no	—
81	C₄×C₁₀₉	—	2	4	no	—
82	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
83	C₄×C₁₀₉	#131	1	2	yes	—
84	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
85	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
86	C₂×C₁₀₉	—	2	2	no	—
87	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
89	C₂×C₁₀₉	#568	1	1	yes	characteristic
90	#2962	—	109	1	no	maximal, Sylow
91	C₁₀₉	#2962	1	1	yes	characteristic, complemented, Sylow
92	#533	D₁₀₉	1	1	yes	characteristic
93	#446	—	109	2	no	—
94	#446	—	109	4	no	—
95	C₈×Q₈	—	109	2	no	—
96	#424	—	109	2	no	—
97	C₂×C₄⋊C₈	—	109	1	no	—
98	#405	—	109	2	no	—
99	C₄×M₁₆	—	109	1	no	—
100	C₂×M₁₆	—	109	1	no	—
101	C₂×M₁₆	—	109	1	no	—
102	C₂²×C₈	—	109	2	no	—
103	C₄⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
104	C₄⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
105	#125	Dic₁₀₉	1	2	yes	—
106	C₂²⋊Q₈	Dic₁₀₉	1	2	yes	—
107	C₂²⋊Q₈	Dic₁₀₉	1	2	yes	—
108	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
109	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
110	#117	C₂×D₁₀₉	1	2	yes	—
111	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
112	C₄⋊C₈	—	109	1	no	—
113	C₄⋊C₈	—	109	1	no	—
114	C₄⋊C₈	—	109	2	no	—
115	C₄⋊C₈	—	109	4	no	—
116	C₄⋊C₈	—	109	4	no	—
117	#97	—	109	2	no	—
118	#97	—	109	4	no	—
119	C₄×C₈	—	109	2	no	—
120	C₄×C₈	—	109	4	no	—
121	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
122	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
123	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
126	M₁₆	—	109	4	no	—
127	M₁₆	—	218	2	no	—
128	C₂×C₈	—	109	2	no	—
129	C₂×C₈	—	109	2	no	—
130	C₂×C₈	—	109	2	no	—
131	C₂×C₈	—	109	2	no	—
132	C₂×C₈	—	218	4	no	—
133	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
134	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
135	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
136	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
137	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
139	C₂²⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
140	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
141	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
142	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
143	C₄²	C₂²×D₁₀₉	1	4	yes	—
144	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
145	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
152	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
154	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
155	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
156	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
157	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
158	C₂×C₄	#13182030	1	2	yes	—
159	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
160	C₈	—	109	4	no	—
161	C₈	—	218	2	no	—
162	C₈	—	218	2	no	—
163	C₈	—	218	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	Q₈×Dic₁₀₉	1	2	yes	—
167	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	Q₈×Dic₁₀₉	1	2	yes	—
174	C₄	#16812706	1	2	yes	—
175	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
176	C₄	C₂×#13182030	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×Q₈×Dic₁₀₉	1	1	yes	characteristic
179	C₂	C₂×#16812706	1	1	yes	characteristic
180	C₂	#20295793	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#533×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#2962
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #20295157	C₂	semidirect
2. #20295157	C₂	semidirect
3. Q₈×#11724784	C₂	semidirect
31. #13182013	Q₈	semidirect
91. C₁₀₉	#2962	semidirect

Maximal subgroups

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

subgroup	index
#20295157	2
Q₈×#11724784	2
#20295053	2
C₂×C₄⋊#11724784	2
#20294982	2
C₄×#13182013	2
#533×C₁₀₉	2
#2962	109