Finite Group Databasebeta

Subgroups — #13131741

Order 1664.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 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	#11719433	C₂	1	1	yes	maximal, characteristic
2	#11719158	C₂	1	2	yes	maximal, complemented
3	Q₈×#1057	C₂	1	2	yes	maximal, complemented
4	#11719078	C₂	1	4	yes	maximal, complemented
5	#11719077	C₂	1	4	yes	maximal
6	#11719054	C₂	1	1	yes	maximal, characteristic
7	#11719049	C₂	1	1	yes	maximal, characteristic
8	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	#88200	C₄	1	2	yes	—
10	Q₈×Dic₁₃	C₂²	1	2	yes	—
11	#88151	C₄	1	2	yes	—
12	#88146	C₂²	1	2	yes	—
13	#88144	C₄	1	2	yes	—
14	#88141	C₂²	1	2	yes	—
15	C₄²×D₁₃	C₂²	1	1	yes	characteristic
16	#88123	C₄	1	1	yes	characteristic
17	#88112	C₂²	1	2	yes	—
18	#88112	C₂²	1	4	yes	—
19	#88111	C₂²	1	2	yes	—
20	#88111	C₂²	1	2	yes	—
21	#88110	C₂²	1	2	yes	—
22	#88110	C₂²	1	4	yes	—
23	C₄⋊#1057	C₂²	1	2	yes	—
24	C₄⋊#1057	C₂²	1	4	yes	—
25	C₄×#1057	C₂²	1	1	yes	characteristic
26	C₄×#1057	C₂²	1	1	yes	characteristic
27	C₄×#1057	C₂²	1	4	yes	—
28	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
33	C₂×#1057	C₂³	1	2	yes	—
34	C₂×#1057	C₂³	1	4	yes	—
35	C₂×#1057	C₂³	1	2	yes	—
36	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	#6103	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
43	#6101	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₄×Dic₁₃	C₂³	1	4	yes	—
47	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
48	#2985	—	13	1	no	maximal, Sylow
49	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
59	C₄×D₁₃	—	2	2	no	—
60	C₄×D₁₃	—	2	2	no	—
61	C₄×D₁₃	—	4	1	no	—
62	Dic₂₆	—	4	2	no	—
63	#1057	—	2	2	no	—
64	#1057	—	2	4	no	—
65	#1057	—	2	2	no	—
66	#533	—	13	1	no	—
67	#446	—	13	4	no	—
68	#446	—	13	2	no	—
69	C₈×Q₈	—	13	2	no	—
70	#433	—	13	4	no	—
71	#432	—	13	1	no	—
72	#431	—	13	1	no	—
73	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₃	—	2	1	no	—
75	C₂×D₁₃	—	2	1	no	—
76	C₂×D₁₃	—	2	1	no	—
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	—	2	4	no	—
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	#131	1	2	yes	—
81	Dic₁₃	—	2	1	no	—
82	Dic₁₃	—	2	4	no	—
83	Dic₁₃	—	2	1	no	—
84	Dic₁₃	2₋¹⁺⁴	1	2	yes	—
85	C₄⋊Q₈	—	13	1	no	—
86	C₄⋊Q₈	#261	1	1	yes	characteristic
87	#125	—	13	2	no	—
88	C₂²⋊Q₈	—	13	2	no	—
89	C₂²⋊Q₈	—	13	2	no	—
90	C₄×Q₈	—	13	2	no	—
91	C₄×Q₈	—	13	2	no	—
92	#117	—	13	2	no	—
93	C₂×C₄²	—	13	1	no	—
94	C₄⋊C₈	—	13	2	no	—
95	C₄⋊C₈	—	13	4	no	—
96	C₄⋊C₈	—	13	2	no	—
97	C₄⋊C₈	—	13	4	no	—
98	C₂²⋊C₈	—	13	2	no	—
99	C₂²⋊C₈	—	13	2	no	—
100	#97	—	13	2	no	—
101	#97	—	13	4	no	—
102	C₄×C₈	—	13	1	no	—
103	C₄×C₈	—	13	1	no	—
104	C₄×C₈	—	13	4	no	—
105	C₂×C₁₃	#519	1	1	yes	characteristic
106	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
107	C₂×C₁₃	#568	1	1	yes	characteristic
108	D₁₃	—	4	1	no	—
109	C₂×Q₈	—	13	2	no	—
110	C₂×Q₈	C₂×#261	1	2	yes	—
111	C₂²×C₄	—	13	1	no	—
112	C₂²×C₄	—	13	1	no	—
113	C₂²×C₄	—	13	1	no	—
114	C₂×C₈	—	13	2	no	—
115	C₂×C₈	—	13	2	no	—
116	C₂×C₈	—	13	4	no	—
117	C₄⋊C₄	—	13	2	no	—
118	C₄⋊C₄	—	13	4	no	—
119	C₄⋊C₄	—	13	4	no	—
120	C₄⋊C₄	—	13	2	no	—
121	C₄⋊C₄	C₂×#261	1	2	yes	—
122	C₄⋊C₄	C₂×#261	1	2	yes	—
123	C₂²⋊C₄	—	13	4	no	—
124	C₄²	—	13	1	no	—
125	C₄²	—	13	4	no	—
126	C₄²	—	13	1	no	—
127	C₄²	—	13	1	no	—
128	C₄²	C₂×#261	1	1	yes	characteristic
129	C₁₃	#2985	1	1	yes	characteristic, complemented, Sylow
130	C₂³	—	13	1	no	—
131	Q₈	—	2	4	no	—
132	Q₈	—	52	2	no	—
133	C₂×C₄	—	13	1	no	—
134	C₂×C₄	—	13	1	no	—
135	C₂×C₄	—	13	1	no	—
136	C₂×C₄	—	13	4	no	—
137	C₂×C₄	—	26	2	no	—
138	C₂×C₄	—	26	2	no	—
139	C₂×C₄	—	52	1	no	—
140	C₂×C₄	C₂²×#261	1	1	yes	characteristic
141	C₂×C₄	C₂²×#261	1	1	yes	characteristic
142	C₂×C₄	C₂²×#261	1	4	yes	—
143	C₂×C₄	C₂²×#261	1	1	yes	characteristic
144	C₈	—	26	2	no	—
145	C₈	—	26	2	no	—
146	C₈	—	26	4	no	—
147	C₂²	—	26	1	no	—
148	C₂²	—	26	1	no	—
149	C₂²	—	26	1	no	—
150	C₂²	C₂³×#261	1	1	yes	characteristic
151	C₄	—	2	4	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	13	2	no	—
155	C₄	—	26	1	no	—
156	C₄	—	26	1	no	—
157	C₄	—	26	4	no	—
158	C₄	#88238	1	2	yes	—
159	C₂	—	52	1	no	—
160	C₂	C₂×#88238	1	1	yes	characteristic
161	C₂	#11719625	1	1	yes	characteristic
162	C₂	#11719627	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×C₄⋊Q₈	C₄
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#2985
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #11719158	C₂	semidirect
3. Q₈×#1057	C₂	semidirect
4. #11719078	C₂	semidirect
129. C₁₃	#2985	semidirect

Maximal subgroups

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

subgroup	index
#11719433	2
#11719158	2
Q₈×#1057	2
#11719078	2
#11719077	2
#11719054	2
#11719049	2
#2985	13