Finite Group Databasebeta

Subgroups — #15783544

Order 3328.

194 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 63 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	#13117148	C₂	1	1	yes	maximal, characteristic
2	#13116713	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113043	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113025	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113023	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2179	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#508	C₂²	1	1	yes	characteristic
9	C₁₃×#443	—	2	1	no	—
10	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
11	C₄×D₁₀₄	—	2	1	no	—
12	C₁₃×#368	C₂²	1	1	yes	characteristic
13	C₁₃×#359	—	2	1	no	—
14	C₁₃×#346	C₂²	1	1	yes	characteristic
15	#11718113	C₄	1	2	yes	complemented
16	#11718111	C₄	1	2	yes	complemented
17	#11718101	—	2	1	no	—
18	#11718097	C₂²	1	1	yes	characteristic
19	#11718091	C₂²	1	1	yes	characteristic
20	#11718091	C₂²	1	1	yes	characteristic
21	C₁₃×#135	—	2	1	no	—
22	C₁₃×#131	—	4	1	no	—
23	C₂×D₁₀₄	—	2	1	no	—
24	C₄×D₅₂	—	4	1	no	—
25	C₁₃×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
26	C₁₃×SD₃₂	—	2	1	no	—
27	C₁₃×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
30	C₁₃×#108	—	2	1	no	—
31	C₁₃×#104	—	4	1	no	—
32	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
33	#88061	—	4	1	no	—
34	#88058	—	2	1	no	—
35	D₂₀₈	—	2	1	no	—
36	D₂₀₈	C₂×C₄	1	2	yes	—
37	#88034	—	2	1	no	—
38	#88034	C₂×C₄	1	2	yes	complemented
39	#13658	—	13	1	no	maximal, Sylow
40	C₁₃×#41	—	4	1	no	—
41	C₂×D₅₂	—	4	1	no	—
42	C₂×C₄×D₁₃	—	8	1	no	—
43	C₁₃×Q₁₆	—	2	1	no	—
44	C₁₃×SD₁₆	—	4	1	no	—
45	C₁₃×D₈	—	2	1	no	—
46	C₁₃×M₁₆	—	4	1	no	—
47	C₁₃×M₁₆	—	8	1	no	—
48	C₂×C₈×C₁₃	—	8	1	no	—
49	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₁₃	D₈	1	1	yes	characteristic
52	#6103	—	8	1	no	—
53	C₄⋊Dic₁₃	—	4	1	no	—
54	D₁₀₄	—	2	2	no	—
55	D₁₀₄	—	4	1	no	—
56	C₁₃×C₁₆	—	2	1	no	—
57	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
58	C₁₃×C₁₆	#41	1	1	yes	characteristic
59	#2263	—	13	1	no	—
60	#2179	D₁₃	1	1	yes	characteristic, complemented
61	#2177	—	13	1	no	—
62	#1426	—	13	1	no	—
63	#1419	—	13	1	no	—
64	#1418	—	13	1	no	—
65	#1398	—	13	1	no	—
66	C₂²×D₁₃	—	8	1	no	—
67	Q₈×C₁₃	—	4	1	no	—
68	D₄×C₁₃	—	4	1	no	—
69	D₄×C₁₃	—	8	1	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	8	1	no	—
72	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
73	C₂×Dic₁₃	—	8	1	no	—
74	D₅₂	—	4	2	no	—
75	D₅₂	—	8	1	no	—
76	C₄×D₁₃	—	16	2	no	—
77	C₈×C₁₃	—	2	1	no	—
78	C₈×C₁₃	—	8	1	no	—
79	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
80	C₈×C₁₃	#135	1	1	yes	characteristic
81	#508	C₂×D₁₃	1	1	yes	characteristic
82	C₂×D₁₆	—	13	1	no	—
83	#443	—	2	1	no	—
84	C₄×D₈	—	26	1	no	—
85	SD₆₄	—	13	2	no	—
86	D₃₂	—	13	2	no	—
87	M₆₄	—	13	1	no	—
88	M₆₄	—	13	1	no	—
89	#368	C₂×D₁₃	1	1	yes	characteristic
90	#367	—	13	1	no	—
91	#359	—	2	1	no	—
92	#357	—	26	1	no	—
93	#346	C₂×D₁₃	1	1	yes	characteristic
94	C₂²×C₁₃	—	8	1	no	—
95	C₂²×C₁₃	#509	1	1	yes	characteristic
96	C₂×D₁₃	—	8	2	no	—
97	C₂×D₁₃	—	16	1	no	—
98	C₂×D₁₃	—	16	1	no	—
99	C₄×C₁₃	—	4	1	no	—
100	C₄×C₁₃	—	8	1	no	—
101	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
102	C₄×C₁₃	#509	1	1	yes	characteristic
103	Dic₁₃	—	16	1	no	—
104	#135	—	2	1	no	—
105	C₂×D₈	—	26	1	no	—
106	#131	—	4	1	no	—
107	C₄×D₄	—	52	1	no	—
108	Q₃₂	C₄×D₁₃	1	1	yes	characteristic, complemented
109	SD₃₂	—	2	1	no	—
110	D₁₆	—	13	2	no	—
111	D₁₆	—	26	1	no	—
112	D₁₆	C₄×D₁₃	1	1	yes	characteristic, complemented
113	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
114	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
115	#108	—	2	1	no	—
116	#107	—	26	1	no	—
117	#104	—	4	1	no	—
118	#102	—	52	1	no	—
119	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
120	C₃₂	—	13	2	no	—
121	C₃₂	—	26	1	no	—
122	C₂×C₁₃	—	2	1	no	—
123	C₂×C₁₃	—	16	1	no	—
124	C₂×C₁₃	#2177	1	1	yes	characteristic
125	D₁₃	—	16	2	no	—
126	#41	—	4	1	no	—
127	C₂×D₄	—	52	1	no	—
128	C₂²×C₄	—	104	1	no	—
129	Q₁₆	—	2	1	no	—
130	SD₁₆	—	4	1	no	—
131	D₈	—	2	1	no	—
132	D₈	—	26	2	no	—
133	D₈	—	52	1	no	—
134	M₁₆	—	4	1	no	—
135	M₁₆	—	8	1	no	—
136	C₂×C₈	—	8	1	no	—
137	C₂×C₈	#6104	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
139	C₄⋊C₄	—	52	1	no	—
140	C₂²⋊C₄	—	104	1	no	—
141	C₄²	#6104	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	C₂×C₄×D₁₃	1	1	yes	characteristic
144	C₁₆	#6127	1	1	yes	characteristic
145	C₁₃	#13658	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	104	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	52	2	no	—
151	D₄	—	104	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	104	1	no	—
155	C₂×C₄	—	208	2	no	—
156	C₂×C₄	C₂×#6104	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	8	1	no	—
159	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
160	C₈	#88204	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	104	2	no	—
163	C₂²	—	208	1	no	—
164	C₂²	—	208	1	no	—
165	C₂²	#11718854	1	1	yes	characteristic
166	C₄	—	4	1	no	—
167	C₄	—	8	1	no	—
168	C₄	—	208	1	no	—
169	C₄	C₄×#6104	1	1	yes	characteristic
170	C₄	#11718808	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	208	2	no	—
174	C₂	#13116710	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13116710
commutator	C₁₃×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₁₃×#2179	C₂
socle	C₂×C₁₃	#2177
prime radical	#13117148	C₂
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	#2179	D₁₃
hypocenter	C₁₃	#13658
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #2179

Complements

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

N	H ≅ G/N	type
2. #13116713	C₂	semidirect
3. #13113043	C₂	semidirect
4. #13113025	C₂	semidirect
5. #13113023	C₂	semidirect
6. #13112972	C₂	semidirect
7. C₁₃×#2179	C₂	semidirect
15. #11718113	C₄	semidirect
16. #11718111	C₄	semidirect
25. C₁₃×Q₃₂	C₂×C₄	semidirect
27. C₁₃×D₁₆	C₂×C₄	semidirect
38. #88034	C₂×C₄	semidirect
60. #2179	D₁₃	semidirect
108. Q₃₂	C₄×D₁₃	semidirect
112. D₁₆	C₄×D₁₃	semidirect
145. C₁₃	#13658	semidirect

Maximal subgroups

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

subgroup	index
#13117148	2
#13116713	2
#13113043	2
#13113025	2
#13113023	2
#13112972	2
C₁₃×#2179	2
#13658	13