Finite Group Databasebeta

Subgroups — #16786673

Order 3328.

250 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 108 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	#13131197	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718231	C₂	1	1	yes	maximal, characteristic
3	#13129754	C₂	1	2	yes	maximal, complemented
4	#13129754	C₂	1	2	yes	maximal, complemented
5	#13113126	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6092	C₄	1	2	yes	—
7	C₁₃×#431	C₄	1	1	yes	characteristic
8	C₄×#6099	C₄	1	1	yes	characteristic
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₂²⋊#6092	C₄	1	2	yes	—
11	#11718231	C₄	1	2	yes	—
12	#11718231	C₂²	1	1	yes	characteristic
13	#11718231	C₂²	1	1	yes	characteristic
14	#11718231	C₂²	1	2	yes	—
15	C₄⋊#6092	C₄	1	2	yes	—
16	#11718047	C₄	1	2	yes	—
17	#11718046	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#6099	—	2	1	no	—
20	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
22	C₂×#6092	—	2	4	no	—
23	C₂×#6092	C₂×C₄	1	4	yes	—
24	C₂×#6092	C₂×C₄	1	2	yes	—
25	C₂×#6092	C₂×C₄	1	2	yes	—
26	C₁₃×C₄⋊C₈	—	2	1	no	—
27	C₁₃×C₂²⋊C₈	—	2	1	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	#88043	—	2	1	no	—
30	C₄×#1055	C₂×C₄	1	2	yes	—
31	C₄×#1055	C₂×C₄	1	1	yes	characteristic
32	C₄×#1055	C₂×C₄	1	1	yes	characteristic
33	C₄×#1055	C₂×C₄	1	1	yes	characteristic
34	C₄×#1055	C₂³	1	1	yes	characteristic
35	#11413	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₄²×C₁₃	C₄²	1	1	yes	characteristic
41	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
42	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
43	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	#6099	—	4	2	no	—
45	C₂×#1055	—	2	2	no	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	C₄²	1	1	yes	characteristic
49	C₂×#1055	C₄²	1	1	yes	characteristic
50	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
51	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
52	#6092	—	2	2	no	—
53	#6092	—	2	4	no	—
54	#6092	M₁₆	1	4	yes	complemented
55	#2163	—	13	1	no	—
56	C₂×#346	—	13	1	no	—
57	#1451	—	13	1	no	—
58	#1313	—	13	2	no	—
59	#1313	—	13	2	no	—
60	C₂³×C₁₃	#97	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	4	1	no	—
65	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₁₃	—	4	2	no	—
72	#1055	—	2	1	no	—
73	#1055	—	2	1	no	—
74	#1055	—	4	2	no	—
75	#1055	C₂×M₁₆	1	2	yes	—
76	#1055	#131	1	2	yes	—
77	C₂²×C₁₆	—	13	2	no	—
78	#431	#261	1	1	yes	characteristic
79	C₄×M₁₆	—	13	1	no	—
80	C₂×C₄×C₈	—	13	1	no	—
81	C₄⋊C₁₆	—	13	2	no	—
82	C₂²⋊C₁₆	—	13	2	no	—
83	#346	—	13	1	no	—
84	#346	—	13	1	no	—
85	#346	—	13	2	no	—
86	#346	—	13	2	no	—
87	#322	—	13	2	no	—
88	#322	—	13	2	no	—
89	C₂²×C₁₃	—	2	1	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	#347	1	2	yes	—
92	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
93	C₄×C₁₃	—	2	1	no	—
94	C₄×C₁₃	—	2	1	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	—	2	1	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
99	C₄×C₁₃	#405	1	1	yes	characteristic
100	C₂×M₁₆	—	26	1	no	—
101	C₂²×C₈	—	13	1	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₂×C₄²	C₂×#261	1	1	yes	characteristic
104	C₂×C₁₆	—	13	2	no	—
105	C₂×C₁₆	—	13	2	no	—
106	C₂×C₁₆	—	13	4	no	—
107	C₂×C₁₆	—	26	4	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₂²⋊C₈	—	2	1	no	—
110	#97	—	26	1	no	—
111	C₄×C₈	—	13	1	no	—
112	C₄×C₈	—	13	1	no	—
113	C₄×C₈	—	13	1	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	13	2	no	—
116	C₄×C₈	C₂×#261	1	2	yes	—
117	C₂×C₁₃	—	2	2	no	—
118	C₂×C₁₃	#1451	1	1	yes	characteristic
119	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
120	C₂×C₁₃	#2110	1	1	yes	characteristic
121	C₂²×C₄	#6118	1	1	yes	characteristic
122	C₂²×C₄	C₄×#261	1	1	yes	characteristic
123	C₂²×C₄	#6122	1	1	yes	characteristic
124	M₁₆	—	52	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	13	1	no	—
127	C₂×C₈	—	13	1	no	—
128	C₂×C₈	—	13	1	no	—
129	C₂×C₈	—	13	1	no	—
130	C₂×C₈	—	26	2	no	—
131	C₂×C₈	—	26	2	no	—
132	C₂×C₈	—	26	2	no	—
133	C₄²	#6118	1	1	yes	characteristic
134	C₄²	C₄×#261	1	1	yes	characteristic
135	C₄²	#6122	1	1	yes	characteristic
136	C₄²	C₂²×#261	1	1	yes	characteristic
137	C₁₆	—	13	4	no	—
138	C₁₆	—	26	2	no	—
139	C₁₆	—	26	4	no	—
140	C₁₃	#11413	1	1	yes	characteristic, complemented, Sylow
141	C₂³	#88110	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#88110	1	1	yes	characteristic
147	C₂×C₄	#88110	1	1	yes	characteristic
148	C₂×C₄	#88110	1	1	yes	characteristic
149	C₂×C₄	C₂×#6118	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
151	C₂×C₄	C₂×#6122	1	1	yes	characteristic
152	C₈	—	4	2	no	—
153	C₈	—	13	2	no	—
154	C₈	—	13	2	no	—
155	C₈	—	26	1	no	—
156	C₈	—	26	1	no	—
157	C₈	—	52	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#11718262	1	2	yes	—
161	C₂²	C₂×#88110	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	M₁₆×#261	1	1	yes	characteristic
168	C₄	#11719099	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#13130043	1	1	yes	characteristic
171	C₂	C₂×#11718262	1	1	yes	characteristic
172	C₂	#13131057	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#431	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	#97
cosocle	C₄×#1055	C₂³
hypercenter	#431	#261
hypocenter	C₁₃	#11413
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #13129754	C₂	semidirect
4. #13129754	C₂	semidirect
54. #6092	M₁₆	semidirect
140. C₁₃	#11413	semidirect

Maximal subgroups

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

subgroup	index
#13131197	2
C₂×#11718231	2
#13129754	2
#13113126	2
#11413	13