Finite Group Databasebeta

Subgroups — #13130971

Order 1664.

360 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 74 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	#11719373	C₂	1	1	yes	maximal, characteristic
2	C₂×#88118	C₂	1	2	yes	maximal, complemented
3	C₂×#88116	C₂	1	2	yes	maximal, complemented
4	C₄²×#261	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719053	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6131	C₂²	1	1	yes	characteristic
7	C₂×#6129	C₂²	1	1	yes	characteristic
8	C₂×C₄×#261	—	2	2	no	—
9	C₂×C₄×#261	C₂²	1	2	yes	—
10	C₂×#6118	C₂²	1	2	yes	—
11	C₁₃×#124	C₄	1	1	yes	characteristic, complemented
12	#88198	C₄	1	1	yes	characteristic, complemented
13	#88194	C₄	1	1	yes	characteristic, complemented
14	#88138	—	2	2	no	—
15	#88135	—	2	2	no	—
16	#88129	C₄	1	1	yes	characteristic, complemented
17	C₄²×D₁₃	C₂²	1	1	yes	characteristic
18	#88118	—	2	4	no	—
19	#88116	—	2	4	no	—
20	#88112	—	2	2	no	—
21	C₂²×#261	—	2	2	no	—
22	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₂²⋊D₁₃	—	2	1	no	—
25	C₂×C₂²⋊D₁₃	—	2	2	no	—
26	C₂²×Dic₁₃	—	2	1	no	—
27	#6131	—	2	2	no	—
28	#6131	—	2	2	no	—
29	#6129	—	2	2	no	—
30	#6129	—	2	2	no	—
31	C₂×D₅₂	—	2	1	no	—
32	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₃	—	2	1	no	—
34	C₂×C₄×D₁₃	D₄	1	2	yes	—
35	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
36	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#1057	—	2	2	no	—
38	C₄×#261	—	2	8	no	—
39	C₄×#261	—	2	2	no	—
40	C₄×#261	D₄	1	4	yes	complemented
41	#6118	—	2	2	no	—
42	#6118	—	2	2	no	—
43	C₁₃×C₂²⋊C₄	—	2	2	no	—
44	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
45	#6103	—	2	2	no	—
46	#6101	—	2	2	no	—
47	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₃	D₄	1	2	yes	—
49	#1962	—	13	1	no	maximal, Sylow
50	C₂³×C₁₃	—	2	1	no	—
51	C₂²×D₁₃	—	2	1	no	—
52	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×#261	—	2	8	no	—
54	C₂×#261	—	2	4	no	—
55	Q₈×C₁₃	—	2	2	no	—
56	D₄×C₁₃	—	2	2	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	C₂²⋊D₁₃	—	4	2	no	—
61	C₂²⋊D₁₃	—	4	4	no	—
62	C₂×Dic₁₃	—	2	1	no	—
63	C₂×Dic₁₃	—	4	1	no	—
64	C₂×Dic₁₃	—	4	1	no	—
65	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
67	D₅₂	—	2	2	no	—
68	D₅₂	—	4	2	no	—
69	C₄×D₁₃	—	2	4	no	—
70	C₄×D₁₃	—	4	1	no	—
71	C₄×D₁₃	—	4	1	no	—
72	C₄×D₁₃	C₂×D₄	1	2	yes	—
73	C₄×D₁₃	#41	1	2	yes	—
74	Dic₂₆	—	2	2	no	—
75	#1057	—	4	2	no	—
76	#532	—	13	1	no	—
77	#423	—	13	1	no	—
78	C₂×#104	—	13	2	no	—
79	C₂×#104	—	13	2	no	—
80	C₄³	—	13	1	no	—
81	C₂²×C₁₃	—	4	1	no	—
82	C₂²×C₁₃	—	4	2	no	—
83	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×D₁₃	—	2	2	no	—
85	C₂×D₁₃	—	4	1	no	—
86	C₂×D₁₃	—	4	2	no	—
87	C₂×D₁₃	#121	1	2	yes	—
88	#261	—	2	8	no	—
89	C₄×C₁₃	—	2	2	no	—
90	C₄×C₁₃	—	4	1	no	—
91	C₄×C₁₃	C₄×D₄	1	2	yes	—
92	Dic₁₃	—	4	1	no	—
93	Dic₁₃	#104	1	4	yes	complemented
94	Dic₁₃	#124	1	1	yes	characteristic
95	Dic₁₃	#127	1	1	yes	characteristic
96	C₂×#41	—	13	1	no	—
97	C₂×#41	—	13	1	no	—
98	C₂×M₁₆	—	13	2	no	—
99	C₄⋊Q₈	—	13	1	no	—
100	#127	—	13	1	no	—
101	#124	—	13	1	no	—
102	#124	#261	1	1	yes	characteristic, complemented
103	#121	—	26	2	no	—
104	C₄×D₄	—	26	2	no	—
105	C₂×C₄²	—	13	1	no	—
106	C₂×C₄²	—	13	2	no	—
107	C₂×C₄²	—	26	2	no	—
108	C₄⋊C₈	—	26	2	no	—
109	#104	—	26	4	no	—
110	#104	—	26	4	no	—
111	C₂×C₁₃	—	8	1	no	—
112	C₂×C₁₃	#390	1	1	yes	characteristic
113	C₂×C₁₃	C₂×#104	1	2	yes	—
114	D₁₃	—	2	2	no	—
115	D₁₃	—	8	1	no	—
116	#41	—	26	2	no	—
117	#41	—	26	2	no	—
118	#41	—	26	2	no	—
119	#41	—	26	2	no	—
120	C₂×Q₈	—	13	1	no	—
121	C₂×Q₈	C₂×#261	1	1	yes	characteristic
122	C₂×D₄	—	13	1	no	—
123	C₂×D₄	—	26	1	no	—
124	C₂×D₄	—	26	2	no	—
125	C₂×D₄	—	26	1	no	—
126	C₂×D₄	C₂×#261	1	1	yes	characteristic
127	C₂²×C₄	—	13	1	no	—
128	C₂²×C₄	—	13	2	no	—
129	C₂²×C₄	—	26	1	no	—
130	C₂²×C₄	—	26	1	no	—
131	C₂²×C₄	—	26	2	no	—
132	M₁₆	—	26	2	no	—
133	M₁₆	—	26	2	no	—
134	C₂×C₈	—	26	2	no	—
135	C₄⋊C₄	—	26	2	no	—
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	26	2	no	—
138	C₄²	—	13	2	no	—
139	C₄²	—	13	1	no	—
140	C₄²	—	13	4	no	—
141	C₄²	—	26	2	no	—
142	C₄²	—	26	8	no	—
143	C₄²	C₂×#261	1	1	yes	characteristic
144	C₁₃	#1962	1	1	yes	characteristic, complemented, Sylow
145	C₂³	—	2	1	no	—
146	C₂³	—	13	1	no	—
147	C₂³	—	26	1	no	—
148	Q₈	—	2	2	no	—
149	Q₈	—	26	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	26	2	no	—
152	D₄	—	52	2	no	—
153	D₄	—	52	4	no	—
154	D₄	—	52	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	13	2	no	—
157	C₂×C₄	—	13	1	no	—
158	C₂×C₄	—	13	2	no	—
159	C₂×C₄	—	13	2	no	—
160	C₂×C₄	—	26	4	no	—
161	C₂×C₄	—	26	4	no	—
162	C₂×C₄	—	26	8	no	—
163	C₂×C₄	—	26	1	no	—
164	C₂×C₄	—	52	1	no	—
165	C₂×C₄	—	52	1	no	—
166	C₂×C₄	—	52	1	no	—
167	C₂×C₄	—	52	1	no	—
168	C₂×C₄	C₂²⋊#261	1	2	yes	—
169	C₂×C₄	C₂²×#261	1	1	yes	characteristic
170	C₈	—	52	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	13	2	no	—
174	C₂²	—	26	2	no	—
175	C₂²	—	52	1	no	—
176	C₂²	—	52	2	no	—
177	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	13	1	no	—
181	C₄	—	13	1	no	—
182	C₄	—	13	4	no	—
183	C₄	—	26	8	no	—
184	C₄	—	52	1	no	—
185	C₄	D₄×#261	1	2	yes	—
186	C₂	—	8	1	no	—
187	C₂	—	26	2	no	—
188	C₂	—	104	1	no	—
189	C₂	#11719151	1	1	yes	characteristic
190	C₂	D₁₃⋊#104	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×#124	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	#11719373	C₂
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	#124	#261
hypocenter	C₁₃	#1962
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. C₂×#88118	C₂	semidirect
3. C₂×#88116	C₂	semidirect
4. C₄²×#261	C₂	semidirect
5. #11719053	C₂	semidirect
11. C₁₃×#124	C₄	semidirect
12. #88198	C₄	semidirect
13. #88194	C₄	semidirect
16. #88129	C₄	semidirect
40. C₄×#261	D₄	semidirect
93. Dic₁₃	#104	semidirect
102. #124	#261	semidirect
144. C₁₃	#1962	semidirect

Maximal subgroups

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

subgroup	index
#11719373	2
C₂×#88118	2
C₂×#88116	2
C₄²×#261	2
#11719053	2
#1962	13