Subgroups — #16791434

Order 3328.

202 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#88036	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#88036	C₂	1	2	yes	maximal
3	C₄×#88036	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×#6090	C₄	1	1	yes	characteristic
7	C₂²×#6090	C₂²	1	1	yes	characteristic
8	C₂×#88036	C₂²	1	4	yes	—
9	C₄×#6090	C₄	1	1	yes	characteristic
10	C₄×#6090	C₄	1	1	yes	characteristic
11	C₄×#6090	C₂²	1	1	yes	characteristic
12	C₄×#6090	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
16	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
17	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
18	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#6090	—	2	2	no	—
21	C₂×#6090	—	2	2	no	—
22	C₂×#6090	C₂×C₄	1	1	yes	characteristic
23	C₂×#6090	C₂×C₄	1	1	yes	characteristic
24	C₂×#6090	C₂×C₄	1	1	yes	characteristic
25	C₂×#6090	C₂³	1	1	yes	characteristic
26	#88036	—	2	4	no	—
27	#13639	—	13	1	no	maximal, Sylow
28	C₂²×C₄×C₁₃	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	—	2	1	no	—
33	C₂×C₈×C₁₃	—	2	1	no	—
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₁₃	C₁₆	1	1	yes	characteristic
39	C₄²×C₁₃	C₁₆	1	1	yes	characteristic
40	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
42	#6090	—	2	1	no	—
43	#6090	—	2	1	no	—
44	#6090	M₁₆	1	2	yes	—
45	#6090	#41	1	2	yes	—
46	C₂×C₄×C₁₆	—	13	1	no	—
47	C₄⋊C₃₂	—	13	2	no	—
48	C₂²⋊C₃₂	—	13	2	no	—
49	C₄×C₃₂	—	13	2	no	—
50	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	C₃₂	1	4	yes	complemented
55	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₁₃	—	2	1	no	—
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	M₃₂	1	2	yes	—
64	C₈×C₁₃	#117	1	1	yes	characteristic
65	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂²×C₁₆	—	13	1	no	—
67	C₂²×C₁₆	—	13	1	no	—
68	C₂×C₄×C₈	#261	1	1	yes	characteristic
69	C₂×C₃₂	—	13	4	no	—
70	C₄×C₁₆	—	13	1	no	—
71	C₄×C₁₆	—	13	1	no	—
72	C₄×C₁₆	—	13	1	no	—
73	C₄×C₁₆	—	13	1	no	—
74	C₂²×C₁₃	—	2	1	no	—
75	C₂²×C₁₃	—	2	1	no	—
76	C₂²×C₁₃	C₂×C₃₂	1	2	yes	—
77	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₁₃	—	2	1	no	—
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	C₂×C₃₂	1	2	yes	—
81	C₄×C₁₃	C₂×C₃₂	1	2	yes	—
82	C₄×C₁₃	M₆₄	1	2	yes	—
83	C₄×C₁₃	#431	1	1	yes	characteristic
84	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
85	C₂²×C₈	#1057	1	1	yes	characteristic
86	C₂²×C₈	C₂×#261	1	1	yes	characteristic
87	C₂×C₄²	#1057	1	1	yes	characteristic
88	C₂×C₁₆	—	13	1	no	—
89	C₂×C₁₆	—	13	1	no	—
90	C₂×C₁₆	—	13	1	no	—
91	C₂×C₁₆	—	13	1	no	—
92	C₂×C₁₆	—	26	2	no	—
93	C₂×C₁₆	—	26	2	no	—
94	C₄×C₈	#1057	1	1	yes	characteristic
95	C₄×C₈	#1057	1	1	yes	characteristic
96	C₄×C₈	C₂×#261	1	1	yes	characteristic
97	C₄×C₈	C₂×#261	1	1	yes	characteristic
98	C₃₂	—	26	4	no	—
99	C₂×C₁₃	—	2	2	no	—
100	C₂×C₁₃	#2162	1	1	yes	characteristic
101	C₂×C₁₃	C₂²×C₃₂	1	1	yes	characteristic
102	C₂×C₁₃	C₂×M₆₄	1	1	yes	characteristic
103	C₂²×C₄	#6092	1	1	yes	characteristic
104	C₂²×C₄	#6092	1	1	yes	characteristic
105	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂×#1057	1	1	yes	characteristic
110	C₂×C₈	C₂×#1057	1	1	yes	characteristic
111	C₂×C₈	C₂×#1057	1	1	yes	characteristic
112	C₂×C₈	C₂²×#261	1	1	yes	characteristic
113	C₄²	#6092	1	1	yes	characteristic
114	C₄²	#6092	1	1	yes	characteristic
115	C₄²	C₂×#1057	1	1	yes	characteristic
116	C₄²	C₂×#1057	1	1	yes	characteristic
117	C₁₆	—	13	2	no	—
118	C₁₆	—	13	2	no	—
119	C₁₆	—	26	1	no	—
120	C₁₆	—	26	1	no	—
121	C₁₃	#13639	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₂×#6092	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	#88036	1	4	yes	complemented
127	C₂×C₄	C₂×#6092	1	1	yes	characteristic
128	C₂×C₄	C₂×#6092	1	1	yes	characteristic
129	C₂×C₄	C₂×#6092	1	1	yes	characteristic
130	C₂×C₄	C₂×#6092	1	1	yes	characteristic
131	C₂×C₄	C₂×#6092	1	1	yes	characteristic
132	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	#88106	1	2	yes	—
136	C₈	D₁₃⋊M₁₆	1	1	yes	characteristic
137	C₈	#88237	1	1	yes	characteristic
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×#88036	1	2	yes	—
141	C₂²	C₂²×#6092	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	C₂×#88036	1	2	yes	—
145	C₄	C₂×#88036	1	2	yes	—
146	C₄	#11718249	1	2	yes	—
147	C₄	C₂×#88106	1	1	yes	characteristic
148	C₄	#11719119	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#13131193	1	1	yes	characteristic
151	C₂	C₂²×#88036	1	1	yes	characteristic
152	C₂	C₂×#11718249	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₂²×C₃₂
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₁₆
prime radical	C₂³×C₁₃	C₂×C₁₆
cosocle	C₂×#6090	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#13639
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₄⋊#88036	C₂	semidirect
3. C₄×#88036	C₂	semidirect
54. C₂×C₄×C₁₃	C₃₂	semidirect
121. C₁₃	#13639	semidirect
126. C₂×C₄	#88036	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#88036	2
C₂²⋊#88036	2
C₄×#88036	2
C₂×C₄×#6090	2
#13639	13