Finite Group Databasebeta

Subgroups — #13131518

Order 1664.

392 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₂×#88146	C₂	1	1	yes	maximal, characteristic
2	#11719129	C₂	1	2	yes	maximal, complemented
3	#11719126	C₂	1	2	yes	maximal
4	C₄×#6122	C₂	1	2	yes	maximal, complemented
5	#11719077	C₂	1	8	yes	maximal, complemented
6	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×#6122	C₂²	1	4	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×#6103	C₄	1	1	yes	characteristic
10	C₂×#6103	C₄	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
14	#88146	C₄	1	4	yes	—
15	#88146	C₂²	1	4	yes	—
16	#88112	C₂²	1	4	yes	complemented
17	#88111	C₂²	1	8	yes	—
18	#88110	C₂²	1	4	yes	complemented
19	C₄⋊#1057	C₂²	1	4	yes	complemented
20	C₄×#1057	C₂²	1	4	yes	complemented
21	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₁₃	—	2	4	no	—
29	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₁₃	C₂³	1	2	yes	—
31	#6122	—	2	8	no	—
32	C₂×#1057	C₂³	1	8	yes	—
33	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	#6103	C₂×C₄	1	4	yes	—
35	#6103	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
39	C₄×Dic₁₃	C₂³	1	2	yes	—
40	C₄×Dic₁₃	C₂³	1	2	yes	—
41	#2921	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×D₁₃	—	2	2	no	—
44	C₂²×D₁₃	—	4	1	no	—
45	C₂²×D₁₃	—	4	1	no	—
46	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	4	1	no	—
49	C₂×C₄×C₁₃	—	4	1	no	—
50	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂×Dic₁₃	—	4	1	no	—
54	C₂×Dic₁₃	—	4	1	no	—
55	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
56	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
57	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₁₃	#41	1	4	yes	—
59	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
60	C₄×D₁₃	—	2	4	no	—
61	C₄×D₁₃	—	2	4	no	—
62	#1057	—	2	8	no	—
63	C₂×#117	—	13	1	no	—
64	#433	—	13	8	no	—
65	#423	—	13	2	no	—
66	#407	—	13	2	no	—
67	C₄×M₁₆	—	13	2	no	—
68	C₂²×C₁₃	—	2	1	no	—
69	C₂²×C₁₃	—	2	1	no	—
70	C₂²×C₁₃	#117	1	2	yes	—
71	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×D₁₃	—	2	2	no	—
73	C₂×D₁₃	—	2	2	no	—
74	C₂×D₁₃	—	2	2	no	—
75	C₂×D₁₃	—	4	4	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	—	2	2	no	—
79	Dic₁₃	—	2	2	no	—
80	Dic₁₃	—	2	2	no	—
81	Dic₁₃	#117	1	2	yes	—
82	Dic₁₃	C₂×#41	1	2	yes	—
83	C₂³×C₄	—	13	1	no	—
84	C₂×M₁₆	—	13	4	no	—
85	#117	—	13	4	no	—
86	#117	—	13	4	no	—
87	C₂×C₄⋊C₄	—	13	1	no	—
88	C₂×C₄⋊C₄	#261	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	13	1	no	—
90	C₂×C₂²⋊C₄	—	13	1	no	—
91	C₂×C₄²	—	13	1	no	—
92	C₂×C₄²	—	13	1	no	—
93	C₄⋊C₈	—	13	4	no	—
94	C₄⋊C₈	—	13	4	no	—
95	C₂²⋊C₈	—	13	8	no	—
96	#97	—	13	4	no	—
97	C₄×C₈	—	13	4	no	—
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
100	C₂×C₁₃	#568	1	1	yes	characteristic
101	C₂×C₁₃	#568	1	1	yes	characteristic
102	D₁₃	—	4	2	no	—
103	C₂⁴	—	13	1	no	—
104	C₂²×C₄	—	13	1	no	—
105	C₂²×C₄	—	13	1	no	—
106	C₂²×C₄	—	13	1	no	—
107	C₂²×C₄	—	13	2	no	—
108	C₂²×C₄	—	13	2	no	—
109	C₂²×C₄	—	26	4	no	—
110	C₂²×C₄	C₂×#261	1	1	yes	characteristic
111	C₂²×C₄	C₂×#261	1	1	yes	characteristic
112	C₂²×C₄	C₂×#261	1	1	yes	characteristic
113	M₁₆	—	26	8	no	—
114	C₂×C₈	—	13	8	no	—
115	C₄⋊C₄	—	13	4	no	—
116	C₄⋊C₄	C₂×#261	1	4	yes	—
117	C₂²⋊C₄	—	13	4	no	—
118	C₂²⋊C₄	—	13	4	no	—
119	C₄²	—	13	2	no	—
120	C₄²	—	13	2	no	—
121	C₄²	—	13	2	no	—
122	C₄²	—	13	2	no	—
123	C₁₃	#2921	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	13	2	no	—
125	C₂³	—	26	2	no	—
126	C₂³	—	52	1	no	—
127	C₂³	—	52	1	no	—
128	C₂³	C₂²×#261	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	13	4	no	—
133	C₂×C₄	—	13	1	no	—
134	C₂×C₄	—	13	1	no	—
135	C₂×C₄	—	13	2	no	—
136	C₂×C₄	—	13	2	no	—
137	C₂×C₄	—	26	4	no	—
138	C₂×C₄	—	26	4	no	—
139	C₂×C₄	—	52	1	no	—
140	C₂×C₄	—	52	1	no	—
141	C₂×C₄	C₂²×#261	1	2	yes	—
142	C₂×C₄	C₂²×#261	1	2	yes	—
143	C₂×C₄	C₂²×#261	1	2	yes	—
144	C₈	—	26	8	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	26	2	no	—
148	C₂²	—	26	2	no	—
149	C₂²	—	26	2	no	—
150	C₂²	—	52	4	no	—
151	C₂²	#88237	1	2	yes	—
152	C₂²	C₂³×#261	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	13	2	no	—
157	C₄	—	13	2	no	—
158	C₄	—	26	2	no	—
159	C₄	—	26	2	no	—
160	C₂	—	2	2	no	—
161	C₂	—	52	2	no	—
162	C₂	C₂×#88237	1	1	yes	characteristic
163	C₂	#11719621	1	1	yes	characteristic
164	C₂	#11719625	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₄⋊C₄	C₄
prime radical	C₂³×D₁₃	C₂×C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₄	#261
hypocenter	C₁₃	#2921
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #11719129	C₂	semidirect
4. C₄×#6122	C₂	semidirect
5. #11719077	C₂	semidirect
16. #88112	C₂²	semidirect
18. #88110	C₂²	semidirect
19. C₄⋊#1057	C₂²	semidirect
20. C₄×#1057	C₂²	semidirect
123. C₁₃	#2921	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88146	2
#11719129	2
#11719126	2
C₄×#6122	2
#11719077	2
#2921	13