Finite Group Databasebeta

Subgroups — #13131164

Order 1664.

220 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#88103	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718282	C₂	1	2	yes	maximal, complemented
5	#11718278	C₂	1	2	yes	maximal, complemented
6	#41×D₁₃	—	2	1	no	—
7	C₂×#6118	—	2	1	no	—
8	C₁₃×#135	C₄	1	1	yes	characteristic
9	#88204	—	2	1	no	—
10	#88173	C₄	1	1	yes	characteristic
11	Q₁₆×D₁₃	C₂²	1	1	yes	characteristic
12	#88170	—	2	1	no	—
13	SD₁₆×D₁₃	—	2	1	no	—
14	#88166	C₄	1	1	yes	characteristic
15	D₈×D₁₃	C₂²	1	1	yes	characteristic
16	#88158	C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
18	#88106	—	2	1	no	—
19	#88103	—	2	1	no	—
20	#88103	C₂²	1	2	yes	—
21	#88098	C₂²	1	2	yes	—
22	D₁₃⋊C₁₆	—	2	1	no	—
23	C₁₃×#41	—	2	1	no	—
24	#6131	—	2	1	no	—
25	Q₈×D₁₃	—	2	1	no	—
26	#6129	—	2	1	no	—
27	#6129	—	4	1	no	—
28	D₄×D₁₃	—	2	1	no	—
29	D₄×D₁₃	—	4	1	no	—
30	#6127	—	2	1	no	—
31	#6127	—	4	1	no	—
32	C₂×C₄×D₁₃	—	4	1	no	—
33	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
34	C₂×#1057	—	4	1	no	—
35	#6118	—	2	2	no	—
36	#6118	—	4	1	no	—
37	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₁₃×SD₁₆	—	2	1	no	—
39	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	#6107	—	2	1	no	—
42	#6106	—	2	1	no	—
43	#6105	—	2	1	no	—
44	#6104	—	2	1	no	—
45	C₂×#1055	D₄	1	1	yes	characteristic
46	Dic₅₂	C₂×C₄	1	1	yes	characteristic
47	D₁₀₄	C₂×C₄	1	1	yes	characteristic
48	#6095	—	2	1	no	—
49	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
50	C₈×D₁₃	D₄	1	1	yes	characteristic
51	C₈×D₁₃	D₄	1	1	yes	characteristic
52	C₈×D₁₃	C₂³	1	1	yes	characteristic
53	#6092	—	2	2	no	—
54	#2148	—	13	1	no	maximal, Sylow
55	C₂²×D₁₃	—	4	1	no	—
56	C₂²×D₁₃	SD₁₆	1	1	yes	characteristic
57	Q₈×C₁₃	—	2	1	no	—
58	D₄×C₁₃	—	2	1	no	—
59	D₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	—	4	1	no	—
61	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²⋊D₁₃	—	4	1	no	—
63	C₂²⋊D₁₃	—	4	1	no	—
64	C₂²⋊D₁₃	—	4	1	no	—
65	C₂×Dic₁₃	—	4	1	no	—
66	C₂×Dic₁₃	D₈	1	1	yes	characteristic
67	D₅₂	—	2	1	no	—
68	D₅₂	—	4	1	no	—
69	C₄×D₁₃	—	4	1	no	—
70	C₄×D₁₃	—	4	1	no	—
71	C₄×D₁₃	—	4	1	no	—
72	C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄×D₁₃	D₈	1	1	yes	characteristic
74	C₄×D₁₃	SD₁₆	1	1	yes	characteristic
75	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
76	Dic₂₆	—	2	1	no	—
77	Dic₂₆	—	4	1	no	—
78	#1057	—	4	2	no	—
79	C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	#1055	C₂²⋊C₄	1	1	yes	characteristic
82	#1055	C₂×D₄	1	1	yes	characteristic
83	C₂×#135	—	13	1	no	—
84	C₂×M₃₂	—	13	1	no	—
85	C₂×#108	—	13	1	no	—
86	#362	—	13	2	no	—
87	#361	—	13	2	no	—
88	C₂²×C₁₃	—	4	1	no	—
89	C₂²×C₁₃	#102	1	1	yes	characteristic
90	C₂×D₁₃	—	4	1	no	—
91	C₂×D₁₃	—	4	1	no	—
92	C₂×D₁₃	—	8	2	no	—
93	C₂×D₁₃	#102	1	1	yes	characteristic
94	C₂×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₄×C₁₃	—	4	1	no	—
96	C₄×C₁₃	#102	1	1	yes	characteristic
97	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	Dic₁₃	—	4	1	no	—
99	Dic₁₃	#102	1	1	yes	characteristic
100	Dic₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×#41	—	26	1	no	—
102	#135	—	13	1	no	—
103	#135	—	13	1	no	—
104	#135	—	13	1	no	—
105	#135	—	26	1	no	—
106	#135	—	26	1	no	—
107	#135	#261	1	1	yes	characteristic
108	C₂×Q₁₆	—	13	1	no	—
109	C₂×SD₁₆	—	26	1	no	—
110	C₂×D₈	—	13	1	no	—
111	C₂×M₁₆	—	26	1	no	—
112	C₂²×C₈	—	13	1	no	—
113	M₃₂	—	13	2	no	—
114	M₃₂	—	26	1	no	—
115	C₂×C₁₆	—	26	1	no	—
116	#108	—	13	2	no	—
117	#108	—	26	1	no	—
118	C₂×C₁₃	—	2	1	no	—
119	C₂×C₁₃	—	8	1	no	—
120	C₂×C₁₃	C₂×#102	1	1	yes	characteristic
121	D₁₃	—	2	1	no	—
122	D₁₃	—	2	1	no	—
123	D₁₃	—	8	1	no	—
124	#41	—	2	1	no	—
125	#41	—	26	1	no	—
126	#41	—	26	1	no	—
127	#41	—	26	1	no	—
128	#41	—	52	1	no	—
129	#41	—	52	1	no	—
130	C₂×Q₈	—	26	1	no	—
131	C₂×D₄	—	26	1	no	—
132	C₂×D₄	—	52	1	no	—
133	C₂²×C₄	—	13	1	no	—
134	C₂²×C₄	—	52	1	no	—
135	Q₁₆	—	13	1	no	—
136	Q₁₆	—	26	1	no	—
137	Q₁₆	C₂×#261	1	1	yes	characteristic
138	SD₁₆	—	2	1	no	—
139	SD₁₆	—	26	1	no	—
140	SD₁₆	—	26	1	no	—
141	SD₁₆	—	26	1	no	—
142	D₈	—	13	1	no	—
143	D₈	—	26	1	no	—
144	D₈	C₂×#261	1	1	yes	characteristic
145	M₁₆	—	26	2	no	—
146	M₁₆	—	52	1	no	—
147	C₂×C₈	—	13	1	no	—
148	C₂×C₈	—	13	1	no	—
149	C₂×C₈	—	13	1	no	—
150	C₂×C₈	—	13	1	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	52	1	no	—
153	C₂×C₈	C₂×#261	1	1	yes	characteristic
154	C₁₆	—	26	2	no	—
155	C₁₃	#2148	1	1	yes	characteristic, complemented, Sylow
156	C₂³	—	13	1	no	—
157	C₂³	—	52	1	no	—
158	Q₈	—	2	1	no	—
159	Q₈	—	26	1	no	—
160	Q₈	—	52	1	no	—
161	D₄	—	2	1	no	—
162	D₄	—	4	1	no	—
163	D₄	—	26	1	no	—
164	D₄	—	52	1	no	—
165	D₄	—	52	1	no	—
166	D₄	—	52	1	no	—
167	D₄	—	52	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	13	1	no	—
170	C₂×C₄	—	13	1	no	—
171	C₂×C₄	—	13	1	no	—
172	C₂×C₄	—	13	1	no	—
173	C₂×C₄	—	13	1	no	—
174	C₂×C₄	—	52	1	no	—
175	C₂×C₄	—	52	1	no	—
176	C₂×C₄	—	52	1	no	—
177	C₂×C₄	—	52	1	no	—
178	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
179	C₈	—	13	1	no	—
180	C₈	—	13	1	no	—
181	C₈	—	52	2	no	—
182	C₈	C₂²⋊#261	1	1	yes	characteristic
183	C₈	C₂²×#261	1	1	yes	characteristic
184	C₂²	—	4	1	no	—
185	C₂²	—	13	1	no	—
186	C₂²	—	13	1	no	—
187	C₂²	—	52	1	no	—
188	C₂²	—	52	1	no	—
189	C₂²	—	104	2	no	—
190	C₂²	#88115	1	1	yes	characteristic
191	C₄	—	4	1	no	—
192	C₄	—	13	1	no	—
193	C₄	—	13	1	no	—
194	C₄	—	52	1	no	—
195	C₄	#88115	1	1	yes	characteristic
196	C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	8	1	no	—
199	C₂	—	26	1	no	—
200	C₂	—	26	1	no	—
201	C₂	—	104	1	no	—
202	C₂	C₂×#88115	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#88115
commutator	C₈×C₁₃	C₂²×C₄
Frattini	C₈	C₂²×#261
Fitting	C₁₃×#135	C₄
socle	C₂×C₁₃	C₂×#102
prime radical	D₁₃×#135	C₂
cosocle	C₈×D₁₃	C₂³
hypercenter	#135	#261
hypocenter	C₁₃	#2148
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. C₂×#88103	C₂	semidirect
3. D₁₃⋊M₃₂	C₂	semidirect
4. #11718282	C₂	semidirect
5. #11718278	C₂	semidirect
155. C₁₃	#2148	semidirect

Maximal subgroups

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

subgroup	index
D₁₃×#135	2
C₂×#88103	2
D₁₃⋊M₃₂	2
#11718282	2
#11718278	2
#2148	13