Finite Group Databasebeta

Subgroups — #13130883

Order 1664.

314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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	#11719627	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719429	C₂	1	1	yes	maximal, characteristic
3	D₁₃⋊#101	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719155	C₂	1	2	yes	maximal
5	#11718235	C₂	1	2	yes	maximal, complemented
6	C₂×#6131	C₂²	1	1	yes	characteristic
7	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
8	Q₈×#261	—	2	4	no	—
9	#88237	—	2	2	no	—
10	#88237	C₂²	1	2	yes	—
11	D₁₃⋊M₁₆	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
13	#88201	C₄	1	1	yes	characteristic, complemented
14	#88198	C₄	1	1	yes	characteristic, complemented
15	#88150	—	2	2	no	—
16	#88148	—	2	2	no	—
17	#88129	C₄	1	1	yes	characteristic, complemented
18	#88126	C₂²	1	1	yes	characteristic
19	#88118	—	2	4	no	—
20	#88113	—	2	2	no	—
21	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	#6131	—	2	2	no	—
24	#6131	—	2	2	no	—
25	Q₈×D₁₃	—	2	2	no	—
26	Q₈×D₁₃	—	2	2	no	—
27	C₂×D₅₂	—	2	1	no	—
28	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₁₃	—	2	1	no	—
30	C₂×C₄×D₁₃	D₄	1	2	yes	—
31	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₂₆	D₄	1	2	yes	—
34	C₂²⋊#261	—	2	2	no	—
35	#6122	—	2	2	no	—
36	C₄⋊#261	—	2	4	no	—
37	C₄⋊#261	—	2	2	no	—
38	C₄×#261	—	2	4	no	—
39	C₄×#261	D₄	1	4	yes	—
40	#6118	—	2	2	no	—
41	D₁₃⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	#6103	—	2	1	no	—
45	#6103	—	2	2	no	—
46	#6103	—	2	2	no	—
47	#6101	—	2	2	no	—
48	#6101	—	2	1	no	—
49	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	#1912	—	13	1	no	maximal, Sylow
51	C₂²×D₁₃	—	2	1	no	—
52	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×#261	—	2	4	no	—
54	Q₈×C₁₃	—	2	2	no	—
55	Q₈×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×Dic₁₃	—	2	1	no	—
61	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
63	D₅₂	—	2	2	no	—
64	D₅₂	—	4	1	no	—
65	D₅₂	—	4	1	no	—
66	C₄×D₁₃	—	2	2	no	—
67	C₄×D₁₃	—	2	2	no	—
68	C₄×D₁₃	—	4	2	no	—
69	C₄×D₁₃	C₂×D₄	1	2	yes	—
70	C₄×D₁₃	#41	1	2	yes	—
71	Dic₂₆	—	2	2	no	—
72	Dic₂₆	—	2	4	no	—
73	#1057	—	4	2	no	—
74	#536	—	13	1	no	—
75	#519	—	13	1	no	—
76	#421	—	13	2	no	—
77	C₂×#101	—	13	1	no	—
78	H(3,ℤ/4)	—	13	2	no	—
79	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₁₃	—	4	1	no	—
81	C₂×D₁₃	—	4	2	no	—
82	C₂×D₁₃	—	4	1	no	—
83	C₂×D₁₃	#120	1	1	yes	characteristic
84	C₂×D₁₃	#123	1	1	yes	characteristic
85	#261	—	4	4	no	—
86	C₄×C₁₃	—	2	2	no	—
87	C₄×C₁₃	—	4	1	no	—
88	C₄×C₁₃	—	4	1	no	—
89	C₄×C₁₃	C₄×D₄	1	2	yes	—
90	Dic₁₃	—	2	2	no	—
91	Dic₁₃	—	4	1	no	—
92	Dic₁₃	#121	1	2	yes	—
93	C₂×#41	—	13	1	no	—
94	C₂²×Q₈	—	13	1	no	—
95	C₂×M₁₆	—	13	2	no	—
96	C₄⋊Q₈	—	13	1	no	—
97	C₄⋊Q₈	#261	1	1	yes	characteristic, complemented
98	#124	—	13	1	no	—
99	#124	—	13	1	no	—
100	#123	—	26	2	no	—
101	C₂²⋊Q₈	—	26	2	no	—
102	C₄×Q₈	—	26	4	no	—
103	#117	—	13	2	no	—
104	#117	—	13	1	no	—
105	#117	—	26	2	no	—
106	#104	—	26	4	no	—
107	#101	—	26	2	no	—
108	C₂×C₁₃	—	2	1	no	—
109	C₂×C₁₃	#386	1	1	yes	characteristic
110	D₁₃	—	2	1	no	—
111	D₁₃	—	2	1	no	—
112	D₁₃	—	8	1	no	—
113	#41	—	26	2	no	—
114	#41	—	26	2	no	—
115	C₂×Q₈	—	13	2	no	—
116	C₂×Q₈	—	13	1	no	—
117	C₂×Q₈	—	26	2	no	—
118	C₂×Q₈	—	26	2	no	—
119	C₂×Q₈	C₂×#261	1	1	yes	characteristic
120	C₂×Q₈	C₂×#261	1	1	yes	characteristic
121	C₂×D₄	—	13	1	no	—
122	C₂×D₄	—	26	1	no	—
123	C₂²×C₄	—	13	1	no	—
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	—	26	1	no	—
126	M₁₆	—	26	2	no	—
127	M₁₆	—	26	2	no	—
128	C₂×C₈	—	26	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	26	1	no	—
131	C₄⋊C₄	—	26	2	no	—
132	C₄⋊C₄	—	26	2	no	—
133	C₄⋊C₄	—	26	4	no	—
134	C₂²⋊C₄	—	26	2	no	—
135	C₂²⋊C₄	—	26	1	no	—
136	C₂²⋊C₄	—	26	2	no	—
137	C₂²⋊C₄	—	26	2	no	—
138	C₄²	—	13	4	no	—
139	C₄²	—	13	1	no	—
140	C₄²	—	26	4	no	—
141	C₄²	C₂×#261	1	1	yes	characteristic
142	C₁₃	#1912	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	13	1	no	—
144	C₂³	—	26	1	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	2	2	no	—
147	Q₈	—	26	4	no	—
148	Q₈	—	26	2	no	—
149	D₄	—	26	2	no	—
150	D₄	—	52	1	no	—
151	D₄	—	52	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	13	1	no	—
155	C₂×C₄	—	13	2	no	—
156	C₂×C₄	—	13	2	no	—
157	C₂×C₄	—	13	2	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	2	no	—
160	C₂×C₄	—	26	4	no	—
161	C₂×C₄	—	26	1	no	—
162	C₂×C₄	—	52	2	no	—
163	C₂×C₄	C₂²⋊#261	1	2	yes	—
164	C₂×C₄	C₂²×#261	1	1	yes	characteristic
165	C₈	—	52	2	no	—
166	C₂²	—	13	1	no	—
167	C₂²	—	13	1	no	—
168	C₂²	—	52	1	no	—
169	C₂²	—	52	2	no	—
170	C₂²	—	52	1	no	—
171	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	13	2	no	—
176	C₄	—	26	2	no	—
177	C₄	—	52	1	no	—
178	C₄	—	52	4	no	—
179	C₄	D₄×#261	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	26	1	no	—
182	C₂	—	26	1	no	—
183	C₂	—	104	1	no	—
184	C₂	#11719150	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11719150
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×C₄⋊Q₈	C₄
socle	C₂×C₁₃	#386
prime radical	C₂×#6131	C₂²
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#1912
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #11719627	C₂	semidirect
3. D₁₃⋊#101	C₂	semidirect
5. #11718235	C₂	semidirect
12. C₁₃×C₄⋊Q₈	C₄	semidirect
13. #88201	C₄	semidirect
14. #88198	C₄	semidirect
17. #88129	C₄	semidirect
97. C₄⋊Q₈	#261	semidirect
142. C₁₃	#1912	semidirect

Maximal subgroups

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

subgroup	index
#11719627	2
#11719429	2
D₁₃⋊#101	2
#11719155	2
#11718235	2
#1912	13