Finite Group Databasebeta

Subgroups — #13130995

Order 1664.

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

normal subgroups only

This group has 284 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	C₂×#88173	C₂	1	1	yes	maximal, characteristic
2	#11719156	C₂	1	2	yes	maximal, complemented
3	#11719155	C₂	1	2	yes	maximal, complemented
4	#11719092	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719089	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6131	C₂²	1	2	yes	—
7	#88237	C₂²	1	2	yes	—
8	#88237	C₂²	1	1	yes	characteristic
9	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×#6106	C₄	1	2	yes	complemented
12	#88173	—	2	1	no	—
13	#88173	—	2	1	no	—
14	#88173	—	2	1	no	—
15	#88173	—	2	1	no	—
16	C₂×D₁₀₄	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
18	#88118	—	2	4	no	—
19	#88107	—	2	4	no	—
20	#88102	—	2	1	no	—
21	#88101	—	2	1	no	—
22	#88100	—	2	1	no	—
23	C₈×#261	—	2	1	no	—
24	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
25	#6131	—	2	2	no	—
26	#6131	—	2	2	no	—
27	#6131	—	2	2	no	—
28	#6131	—	2	2	no	—
29	C₂×D₅₂	—	2	2	no	—
30	C₂×D₅₂	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₃	—	2	2	no	—
32	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
33	C₂²⋊#261	—	2	2	no	—
34	C₂²⋊#261	D₄	1	2	yes	—
35	#6122	—	2	1	no	—
36	C₄⋊#261	—	2	2	no	—
37	C₄⋊#261	—	2	1	no	—
38	C₄×#261	—	2	1	no	—
39	C₄×#261	—	2	2	no	—
40	#6118	D₄	1	2	yes	—
41	D₁₃⋊C₈	—	2	1	no	—
42	C₁₃×Q₁₆	—	2	1	no	—
43	C₁₃×Q₁₆	—	2	1	no	—
44	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	#6106	—	2	2	no	—
46	#6106	—	2	2	no	—
47	C₂×#1055	C₂×C₄	1	1	yes	characteristic
48	D₁₀₄	—	2	1	no	—
49	D₁₀₄	—	2	1	no	—
50	C₈×D₁₃	D₄	1	1	yes	characteristic
51	C₈×D₁₃	D₄	1	1	yes	characteristic
52	C₈×D₁₃	D₄	1	1	yes	characteristic
53	C₈×D₁₃	D₄	1	1	yes	characteristic
54	#1976	—	13	1	no	maximal, Sylow
55	C₂²×D₁₃	—	2	2	no	—
56	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×#261	—	2	2	no	—
58	C₂×#261	—	4	2	no	—
59	Q₈×C₁₃	—	2	2	no	—
60	Q₈×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×Dic₁₃	#41	1	1	yes	characteristic
64	D₅₂	—	2	2	no	—
65	D₅₂	—	2	2	no	—
66	D₅₂	—	4	2	no	—
67	D₅₂	—	4	2	no	—
68	C₄×D₁₃	—	4	2	no	—
69	C₄×D₁₃	—	4	2	no	—
70	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄×D₁₃	#41	1	1	yes	characteristic
74	#1057	—	2	2	no	—
75	C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	#1055	C₂²⋊C₄	1	1	yes	characteristic
78	#1055	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×#135	—	13	1	no	—
80	#427	—	13	1	no	—
81	#421	—	13	2	no	—
82	#410	—	13	2	no	—
83	#405	—	13	1	no	—
84	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₂×D₁₃	—	4	1	no	—
86	C₂×D₁₃	—	4	2	no	—
87	C₂×D₁₃	—	8	2	no	—
88	C₂×D₁₃	#121	1	1	yes	characteristic
89	C₂×D₁₃	#127	1	1	yes	characteristic
90	#261	—	4	2	no	—
91	#261	—	8	2	no	—
92	C₄×C₁₃	—	4	2	no	—
93	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
95	Dic₁₃	#121	1	1	yes	characteristic
96	Dic₁₃	#124	1	1	yes	characteristic
97	C₂×#41	—	13	2	no	—
98	#135	—	26	1	no	—
99	#135	—	26	1	no	—
100	#135	—	26	1	no	—
101	#135	—	26	1	no	—
102	C₂×Q₁₆	#261	1	1	yes	characteristic, complemented
103	C₂×SD₁₆	—	13	2	no	—
104	C₂×D₈	—	13	1	no	—
105	C₂×M₁₆	—	13	1	no	—
106	C₂²×C₈	—	13	1	no	—
107	#117	—	13	1	no	—
108	#117	—	13	2	no	—
109	#107	—	26	1	no	—
110	#106	—	26	1	no	—
111	#104	—	26	4	no	—
112	#99	—	26	4	no	—
113	#97	—	26	1	no	—
114	C₄×C₈	—	26	1	no	—
115	C₂×C₁₃	—	2	1	no	—
116	C₂×C₁₃	#390	1	1	yes	characteristic
117	D₁₃	—	2	1	no	—
118	D₁₃	—	2	1	no	—
119	D₁₃	—	8	2	no	—
120	#41	—	26	2	no	—
121	#41	—	26	2	no	—
122	#41	—	26	2	no	—
123	#41	—	26	2	no	—
124	C₂×Q₈	C₂×#261	1	2	yes	—
125	C₂×D₄	—	13	2	no	—
126	C₂×D₄	—	26	2	no	—
127	C₂²×C₄	—	13	1	no	—
128	C₂²×C₄	—	26	2	no	—
129	Q₁₆	—	2	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	26	2	no	—
132	SD₁₆	—	26	2	no	—
133	D₈	—	26	1	no	—
134	D₈	—	26	1	no	—
135	M₁₆	—	13	2	no	—
136	M₁₆	—	26	1	no	—
137	C₂×C₈	—	13	1	no	—
138	C₂×C₈	—	13	1	no	—
139	C₂×C₈	—	13	1	no	—
140	C₂×C₈	—	13	1	no	—
141	C₂×C₈	—	13	1	no	—
142	C₂×C₈	—	26	1	no	—
143	C₂×C₈	C₂×#261	1	1	yes	characteristic
144	C₄⋊C₄	—	26	1	no	—
145	C₄⋊C₄	—	26	2	no	—
146	C₂²⋊C₄	—	13	2	no	—
147	C₂²⋊C₄	—	26	2	no	—
148	C₄²	—	26	1	no	—
149	C₄²	—	26	2	no	—
150	C₁₃	#1976	1	1	yes	characteristic, complemented, Sylow
151	C₂³	—	13	1	no	—
152	C₂³	—	26	2	no	—
153	Q₈	—	2	2	no	—
154	Q₈	—	2	2	no	—
155	D₄	—	26	2	no	—
156	D₄	—	26	2	no	—
157	D₄	—	52	2	no	—
158	D₄	—	52	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	13	1	no	—
161	C₂×C₄	—	13	1	no	—
162	C₂×C₄	—	13	1	no	—
163	C₂×C₄	—	13	1	no	—
164	C₂×C₄	—	13	1	no	—
165	C₂×C₄	—	26	2	no	—
166	C₂×C₄	—	52	2	no	—
167	C₂×C₄	—	52	2	no	—
168	C₂×C₄	—	52	2	no	—
169	C₂×C₄	C₂²×#261	1	1	yes	characteristic
170	C₈	—	13	1	no	—
171	C₈	—	13	1	no	—
172	C₈	—	26	2	no	—
173	C₈	C₂²⋊#261	1	1	yes	characteristic
174	C₈	C₂²⋊#261	1	1	yes	characteristic
175	C₂²	—	13	1	no	—
176	C₂²	—	13	1	no	—
177	C₂²	—	52	1	no	—
178	C₂²	—	52	2	no	—
179	C₂²	—	104	2	no	—
180	C₂²	D₄×#261	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	13	1	no	—
183	C₄	—	13	1	no	—
184	C₄	—	52	2	no	—
185	C₄	—	104	2	no	—
186	C₄	D₄×#261	1	1	yes	characteristic
187	C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	26	1	no	—
190	C₂	—	26	1	no	—
191	C₂	—	104	2	no	—
192	C₂	#11719151	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11719151
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₂×C₁₃×Q₁₆	C₄
socle	C₂×C₁₃	#390
prime radical	C₂×#88173	C₂
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	C₂×Q₁₆	#261
hypocenter	C₁₃	#1976
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #11719156	C₂	semidirect
3. #11719155	C₂	semidirect
4. #11719092	C₂	semidirect
5. #11719089	C₂	semidirect
10. C₂×C₁₃×Q₁₆	C₄	semidirect
11. C₂×#6106	C₄	semidirect
16. C₂×D₁₀₄	C₄	semidirect
102. C₂×Q₁₆	#261	semidirect
150. C₁₃	#1976	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88173	2
#11719156	2
#11719155	2
#11719092	2
#11719089	2
#1976	13