Finite Group Databasebeta

Subgroups — #13116391

Order 1664.

210 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₁₃×#421	C₂	1	1	yes	maximal, characteristic
2	#11718897	C₂	1	1	yes	maximal, characteristic
3	#11718895	C₂	1	1	yes	maximal, characteristic
4	#11718785	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718780	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718123	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#117	C₂²	1	1	yes	characteristic
11	#88180	C₂²	1	1	yes	characteristic
12	C₂×#6099	C₂²	1	1	yes	characteristic
13	C₂×#6099	C₂²	1	1	yes	characteristic
14	C₂²×#1055	C₂²	1	1	yes	characteristic
15	C₁₃×#104	—	2	2	no	—
16	#88077	—	2	2	no	—
17	C₂²⋊#1055	—	2	2	no	—
18	#88062	—	2	2	no	—
19	#88054	—	2	2	no	—
20	C₄⋊#1055	—	2	2	no	—
21	C₁₃×#41	—	2	1	no	—
22	C₁₃×#41	—	2	1	no	—
23	C₁₃×#41	—	2	1	no	—
24	C₁₃×#41	—	2	1	no	—
25	C₂×Q₈×C₁₃	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₁₃	—	2	1	no	—
27	C₂×D₄×C₁₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	—	2	1	no	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×M₁₆	—	2	1	no	—
31	C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₈×C₁₃	—	2	1	no	—
33	C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₁₃×C₂²⋊C₄	—	2	1	no	—
35	C₄²×C₁₃	D₄	1	2	yes	—
36	C₂²⋊Dic₁₃	—	2	1	no	—
37	C₄⋊Dic₁₃	—	2	1	no	—
38	C₄×Dic₁₃	D₄	1	2	yes	—
39	#6099	—	2	1	no	—
40	#6099	—	2	1	no	—
41	#6099	—	2	1	no	—
42	#6099	—	2	1	no	—
43	C₂×#1055	—	2	1	no	—
44	C₂×#1055	—	2	1	no	—
45	C₂×#1055	—	2	1	no	—
46	C₂×#1055	—	2	1	no	—
47	C₂×#1055	D₄	1	2	yes	—
48	#2098	—	13	1	no	maximal, Sylow
49	C₂³×C₁₃	—	2	1	no	—
50	C₂³×C₁₃	#41	1	1	yes	characteristic
51	Q₈×C₁₃	—	2	1	no	—
52	Q₈×C₁₃	—	2	1	no	—
53	D₄×C₁₃	—	2	1	no	—
54	D₄×C₁₃	—	2	1	no	—
55	D₄×C₁₃	—	4	1	no	—
56	D₄×C₁₃	—	4	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	—	4	1	no	—
61	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂×Dic₁₃	—	2	2	no	—
68	C₈×C₁₃	—	4	1	no	—
69	#1055	—	2	2	no	—
70	#1055	—	4	1	no	—
71	#1055	—	4	1	no	—
72	#430	—	13	1	no	—
73	#424	—	13	1	no	—
74	#424	—	13	1	no	—
75	#421	—	13	1	no	—
76	#421	D₁₃	1	1	yes	characteristic
77	#408	—	13	1	no	—
78	H(3,ℤ/4)	—	13	1	no	—
79	C₂²×C₁₃	—	4	1	no	—
80	C₂²×C₁₃	—	4	2	no	—
81	C₂²×C₁₃	—	4	1	no	—
82	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂²×C₁₃	#123	1	1	yes	characteristic
84	C₂²×C₁₃	#123	1	1	yes	characteristic
85	C₄×C₁₃	—	4	2	no	—
86	C₄×C₁₃	—	4	1	no	—
87	C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
89	C₄×C₁₃	#123	1	1	yes	characteristic
90	C₄×C₁₃	#127	1	1	yes	characteristic
91	Dic₁₃	—	4	2	no	—
92	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
93	C₂×M₁₆	—	13	1	no	—
94	C₂×M₁₆	—	13	1	no	—
95	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
96	C₂²×C₈	—	13	1	no	—
97	#117	—	13	1	no	—
98	#117	C₂×D₁₃	1	1	yes	characteristic
99	#108	—	26	2	no	—
100	C₄⋊C₈	—	26	2	no	—
101	C₄⋊C₈	—	26	2	no	—
102	#104	—	2	2	no	—
103	#104	—	26	2	no	—
104	C₂²⋊C₈	—	26	2	no	—
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	—	2	1	no	—
107	C₂×C₁₃	—	2	1	no	—
108	C₂×C₁₃	—	8	1	no	—
109	C₂×C₁₃	#399	1	1	yes	characteristic
110	#41	—	2	1	no	—
111	#41	—	2	1	no	—
112	#41	—	2	1	no	—
113	#41	—	2	1	no	—
114	C₂×Q₈	Dic₂₆	1	1	yes	characteristic
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	Dic₂₆	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
119	M₁₆	—	2	1	no	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	26	1	no	—
122	M₁₆	—	26	1	no	—
123	M₁₆	—	26	1	no	—
124	M₁₆	—	26	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	26	1	no	—
128	C₂×C₈	—	26	1	no	—
129	C₂×C₈	—	26	1	no	—
130	C₂×C₈	—	26	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	26	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	26	1	no	—
135	C₄²	—	13	2	no	—
136	C₄²	C₂²⋊D₁₃	1	2	yes	—
137	C₁₃	#2098	1	1	yes	characteristic, complemented, Sylow
138	C₂³	—	2	1	no	—
139	C₂³	#6129	1	1	yes	characteristic
140	Q₈	—	2	1	no	—
141	Q₈	—	2	1	no	—
142	D₄	—	2	1	no	—
143	D₄	—	2	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	26	2	no	—
151	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
152	C₂×C₄	#6127	1	1	yes	characteristic
153	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
155	C₂×C₄	#6129	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	26	2	no	—
159	C₈	—	52	1	no	—
160	C₈	—	52	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	#88132	1	1	yes	characteristic
165	C₂²	#88137	1	1	yes	characteristic
166	C₂²	#88189	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	52	2	no	—
170	C₄	#88132	1	1	yes	characteristic
171	C₄	#88137	1	1	yes	characteristic
172	C₄	#88178	1	1	yes	characteristic
173	C₄	#88194	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#11718610	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88194
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#421	C₂
socle	C₂×C₁₃	#399
prime radical	C₂×C₁₃×#41	C₂²
hypercenter	#421	D₁₃
hypocenter	C₁₃	#2098
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
4. #11718785	C₂	semidirect
5. #11718780	C₂	semidirect
6. #11718660	C₂	semidirect
7. #11718123	C₂	semidirect
137. C₁₃	#2098	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#421	2
#11718897	2
#11718895	2
#11718785	2
#11718780	2
#11718660	2
#11718123	2
#2098	13