Finite Group Databasebeta

Subgroups — #16786626

Order 3328.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₄×#88106	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718231	C₂	1	1	yes	maximal, characteristic
3	#13129754	C₂	1	4	yes	maximal, complemented
4	#13113120	C₂	1	1	yes	maximal, characteristic
5	C₂×#88106	C₄	1	2	yes	—
6	C₂²×#6092	C₄	1	2	yes	—
7	C₁₃×#431	C₄	1	1	yes	characteristic
8	#11718586	C₄	1	1	yes	characteristic
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	#11718231	C₄	1	2	yes	—
11	#11718231	C₂²	1	2	yes	—
12	#11718231	C₂²	1	2	yes	—
13	C₄×#6092	C₄	1	2	yes	—
14	#11718046	C₂²	1	2	yes	—
15	C₈×#1055	C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
18	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
19	#88106	C₈	1	8	yes	complemented
20	C₂×#6092	—	2	4	no	—
21	C₂×#6092	C₂×C₄	1	4	yes	—
22	C₂×#6092	C₂×C₄	1	4	yes	—
23	C₁₃×C₄⋊C₈	—	2	1	no	—
24	C₁₃×C₂²⋊C₈	—	2	1	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²⋊#1055	—	2	1	no	—
27	C₄⋊#1055	—	2	1	no	—
28	C₄×#1055	C₂×C₄	1	2	yes	—
29	C₄×#1055	C₂×C₄	1	1	yes	characteristic
30	C₄×#1055	C₂×C₄	1	1	yes	characteristic
31	C₄×#1055	C₂×C₄	1	1	yes	characteristic
32	C₄×#1055	C₂³	1	1	yes	characteristic
33	#11406	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	C₄²	1	1	yes	characteristic
39	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
40	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
41	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×#1055	—	2	2	no	—
43	C₂×#1055	C₄²	1	1	yes	characteristic
44	C₂×#1055	C₄²	1	1	yes	characteristic
45	C₂×#1055	C₂×C₈	1	4	yes	—
46	C₂×#1055	M₁₆	1	4	yes	—
47	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
48	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
49	#6092	—	2	4	no	—
50	#6092	C₂×C₈	1	8	yes	—
51	C₄×M₃₂	—	13	1	no	—
52	C₂×#346	—	13	1	no	—
53	#1450	—	13	1	no	—
54	#1313	—	13	4	no	—
55	C₂³×C₁₃	#97	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	4	1	no	—
57	C₂×C₄×C₁₃	—	4	1	no	—
58	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#97	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₁₃	—	4	2	no	—
67	#1055	—	4	2	no	—
68	#1055	C₄×C₈	1	2	yes	—
69	#1055	#97	1	2	yes	—
70	#1055	C₂²×C₈	1	2	yes	—
71	#1055	C₂×M₁₆	1	2	yes	—
72	C₂×M₃₂	—	13	2	no	—
73	C₂²×C₁₆	—	13	2	no	—
74	#431	—	13	1	no	—
75	#431	#261	1	1	yes	characteristic
76	C₂×C₄×C₈	—	13	1	no	—
77	#346	—	13	2	no	—
78	#346	—	13	2	no	—
79	#346	—	13	2	no	—
80	C₄×C₁₆	—	13	2	no	—
81	#322	—	13	2	no	—
82	C₈²	—	13	2	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	—	2	1	no	—
85	C₂²×C₁₃	#322	1	2	yes	—
86	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	#322	1	2	yes	—
92	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
94	C₂²×C₈	—	13	1	no	—
95	C₂²×C₈	—	13	1	no	—
96	C₂×C₄²	C₂×#261	1	1	yes	characteristic
97	M₃₂	—	13	8	no	—
98	C₂×C₁₆	—	13	4	no	—
99	C₂×C₁₆	—	13	4	no	—
100	C₂×C₁₆	—	26	4	no	—
101	C₄⋊C₈	—	2	1	no	—
102	C₄⋊C₈	—	26	1	no	—
103	C₂²⋊C₈	—	2	1	no	—
104	C₂²⋊C₈	—	26	1	no	—
105	C₄×C₈	—	13	1	no	—
106	C₄×C₈	—	13	1	no	—
107	C₄×C₈	—	13	1	no	—
108	C₄×C₈	—	13	1	no	—
109	C₄×C₈	—	13	2	no	—
110	C₄×C₈	C₂×#261	1	2	yes	—
111	C₂×C₁₃	—	2	2	no	—
112	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
113	C₂×C₁₃	#2110	1	1	yes	characteristic
114	C₂×C₁₃	#2110	1	1	yes	characteristic
115	C₂²×C₄	#6118	1	1	yes	characteristic
116	C₂²×C₄	C₄×#261	1	1	yes	characteristic
117	C₂²×C₄	#6122	1	1	yes	characteristic
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	13	4	no	—
120	C₂×C₈	—	13	4	no	—
121	C₂×C₈	—	13	1	no	—
122	C₂×C₈	—	13	1	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	26	2	no	—
126	C₄²	#6118	1	1	yes	characteristic
127	C₄²	C₄×#261	1	1	yes	characteristic
128	C₄²	#6122	1	1	yes	characteristic
129	C₄²	C₂²×#261	1	1	yes	characteristic
130	C₁₆	—	13	8	no	—
131	C₁₆	—	26	4	no	—
132	C₁₃	#11406	1	1	yes	characteristic, complemented, Sylow
133	C₂³	#88110	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₈×#261	1	2	yes	—
137	C₂×C₄	#88100	1	2	yes	—
138	C₂×C₄	#88110	1	1	yes	characteristic
139	C₂×C₄	#88110	1	1	yes	characteristic
140	C₂×C₄	#88110	1	1	yes	characteristic
141	C₂×C₄	C₂×#6118	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
143	C₂×C₄	C₂×#6122	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	13	2	no	—
146	C₈	—	13	2	no	—
147	C₈	—	13	2	no	—
148	C₈	—	13	2	no	—
149	C₈	—	52	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#1057⋊C₈	1	2	yes	—
153	C₂²	C₂×#88110	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#1057⋊C₈	1	2	yes	—
159	C₄	C₂×C₈×#261	1	1	yes	characteristic
160	C₄	C₂×#88100	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#1057⋊C₈	1	1	yes	characteristic
163	C₂	#13131057	1	1	yes	characteristic
164	C₂	#13131059	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
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	#11406
LCS	G ▷ 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
19. #88106	C₈	semidirect
132. C₁₃	#11406	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88106	2
C₂×#11718231	2
#13129754	2
#13113120	2
#11406	13