Finite Group Databasebeta

Subgroups — #13130878

Order 1664.

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

normal subgroups only

This group has 312 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	#11719625	C₂	1	1	yes	maximal, characteristic
2	#11719429	C₂	1	1	yes	maximal, characteristic
3	#11719156	C₂	1	1	yes	maximal, characteristic
4	#11719153	C₂	1	2	yes	maximal, complemented
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	#88240	—	2	4	no	—
9	#88237	C₂²	1	2	yes	—
10	D₁₃⋊M₁₆	—	2	2	no	—
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	#88117	—	2	4	no	—
20	#88107	—	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	C₂×C₄	1	1	yes	characteristic
28	C₂×D₅₂	D₄	1	2	yes	—
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	—	2	1	no	—
33	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊#261	—	2	2	no	—
35	#6122	—	2	2	no	—
36	C₄⋊#261	—	2	2	no	—
37	C₄×#261	—	2	2	no	—
38	#6118	—	2	4	no	—
39	#6118	—	2	2	no	—
40	D₁₃⋊C₈	—	2	4	no	—
41	D₁₃⋊C₈	D₄	1	4	yes	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	#6103	—	2	2	no	—
45	#6103	—	2	2	no	—
46	#6103	—	2	1	no	—
47	#6101	—	2	2	no	—
48	#6101	—	2	1	no	—
49	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	#1909	—	13	1	no	maximal, Sylow
51	C₂²×D₁₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×#261	—	4	2	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	—	2	1	no	—
62	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
63	D₅₂	—	2	4	no	—
64	D₅₂	—	2	2	no	—
65	C₄×D₁₃	—	2	2	no	—
66	C₄×D₁₃	—	2	2	no	—
67	C₄×D₁₃	—	4	2	no	—
68	C₄×D₁₃	C₂×D₄	1	2	yes	—
69	C₄×D₁₃	#41	1	2	yes	—
70	Dic₂₆	—	2	2	no	—
71	Dic₂₆	—	4	1	no	—
72	Dic₂₆	—	4	1	no	—
73	#1057	—	2	4	no	—
74	#568	—	13	1	no	—
75	#536	—	13	1	no	—
76	#419	—	13	2	no	—
77	#410	—	13	1	no	—
78	H(3,ℤ/4)	—	13	2	no	—
79	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₁₃	—	2	2	no	—
81	C₂×D₁₃	—	4	1	no	—
82	C₂×D₁₃	—	4	2	no	—
83	C₂×D₁₃	#121	1	2	yes	—
84	#261	—	8	2	no	—
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	—	4	1	no	—
87	C₄×C₁₃	—	4	1	no	—
88	C₄×C₁₃	C₄×D₄	1	2	yes	—
89	Dic₁₃	—	4	1	no	—
90	Dic₁₃	—	4	1	no	—
91	Dic₁₃	#120	1	1	yes	characteristic
92	Dic₁₃	#123	1	1	yes	characteristic
93	C₂×#41	—	13	1	no	—
94	C₂²×Q₈	—	13	1	no	—
95	#131	—	26	4	no	—
96	C₂×M₁₆	—	13	2	no	—
97	C₂×M₁₆	—	26	2	no	—
98	C₄⋊Q₈	—	13	1	no	—
99	C₄⋊Q₈	#261	1	1	yes	characteristic, complemented
100	#124	—	13	1	no	—
101	#124	—	13	1	no	—
102	#123	—	26	2	no	—
103	C₂²⋊Q₈	—	26	2	no	—
104	#117	—	13	1	no	—
105	#117	—	13	2	no	—
106	#103	—	26	4	no	—
107	#99	—	26	2	no	—
108	C₂×C₁₃	—	2	1	no	—
109	C₂×C₁₃	#386	1	1	yes	characteristic
110	D₁₃	—	2	2	no	—
111	D₁₃	—	4	2	no	—
112	#41	—	26	2	no	—
113	#41	—	26	2	no	—
114	C₂×Q₈	—	13	1	no	—
115	C₂×Q₈	—	26	1	no	—
116	C₂×Q₈	—	26	2	no	—
117	C₂×Q₈	—	26	2	no	—
118	C₂×Q₈	C₂×#261	1	1	yes	characteristic
119	C₂×Q₈	C₂×#261	1	1	yes	characteristic
120	C₂×D₄	—	13	2	no	—
121	C₂×D₄	—	13	1	no	—
122	C₂²×C₄	—	13	1	no	—
123	C₂²×C₄	—	13	2	no	—
124	C₂²×C₄	—	26	1	no	—
125	M₁₆	—	26	2	no	—
126	M₁₆	—	26	2	no	—
127	M₁₆	—	26	4	no	—
128	C₂×C₈	—	13	4	no	—
129	C₂×C₈	—	26	4	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	26	1	no	—
132	C₄⋊C₄	—	26	2	no	—
133	C₄⋊C₄	—	26	2	no	—
134	C₂²⋊C₄	—	26	1	no	—
135	C₂²⋊C₄	—	26	2	no	—
136	C₂²⋊C₄	—	26	2	no	—
137	C₂²⋊C₄	—	26	2	no	—
138	C₄²	—	13	1	no	—
139	C₄²	—	26	2	no	—
140	C₄²	C₂×#261	1	1	yes	characteristic
141	C₁₃	#1909	1	1	yes	characteristic, complemented, Sylow
142	C₂³	—	13	1	no	—
143	C₂³	—	13	2	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	26	2	no	—
147	Q₈	—	52	1	no	—
148	Q₈	—	52	1	no	—
149	D₄	—	26	4	no	—
150	D₄	—	26	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	13	1	no	—
154	C₂×C₄	—	13	2	no	—
155	C₂×C₄	—	13	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	—	26	2	no	—
158	C₂×C₄	—	26	1	no	—
159	C₂×C₄	—	26	1	no	—
160	C₂×C₄	—	52	2	no	—
161	C₂×C₄	—	52	2	no	—
162	C₂×C₄	C₂²⋊#261	1	2	yes	—
163	C₂×C₄	C₂²×#261	1	1	yes	characteristic
164	C₈	—	26	4	no	—
165	C₂²	—	13	2	no	—
166	C₂²	—	26	2	no	—
167	C₂²	—	52	1	no	—
168	C₂²	—	52	2	no	—
169	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	13	1	no	—
174	C₄	—	13	1	no	—
175	C₄	—	52	1	no	—
176	C₄	—	52	1	no	—
177	C₄	—	104	2	no	—
178	C₄	D₄×#261	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	26	2	no	—
181	C₂	—	52	2	no	—
182	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	#1909
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
4. #11719153	C₂	semidirect
5. #11718235	C₂	semidirect
12. C₁₃×C₄⋊Q₈	C₄	semidirect
13. #88201	C₄	semidirect
14. #88198	C₄	semidirect
17. #88129	C₄	semidirect
99. C₄⋊Q₈	#261	semidirect
141. C₁₃	#1909	semidirect

Maximal subgroups

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

subgroup	index
#11719625	2
#11719429	2
#11719156	2
#11719153	2
#11718235	2
#1909	13