Subgroups — #16790202

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	2	yes	maximal
2	#13129874	C₂	1	4	yes	maximal
3	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6092	C₂²	1	4	yes	—
5	C₄³×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₄	1	2	yes	—
7	C₂×C₄×#1055	C₄	1	1	yes	characteristic
8	C₂×C₄×#1055	C₂²	1	2	yes	—
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₄⋊#6092	—	2	4	no	—
11	C₂×C₄²×C₁₃	C₈	1	2	yes	—
12	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
17	C₂²×#1055	C₂×C₄	1	2	yes	—
18	C₂²×#1055	C₂³	1	1	yes	characteristic
19	C₂×#6092	—	2	4	no	—
20	C₂×#6092	—	2	8	no	—
21	C₄×#1055	—	2	2	no	—
22	C₄×#1055	—	2	4	no	—
23	C₄×#1055	—	2	4	no	—
24	C₄×#1055	D₄	1	2	yes	—
25	C₄×#1055	Q₈	1	2	yes	—
26	#12879	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₁₃	—	2	4	no	—
33	C₄²×C₁₃	—	2	2	no	—
34	C₄²×C₁₃	—	2	2	no	—
35	C₄²×C₁₃	—	2	2	no	—
36	C₄²×C₁₃	C₁₆	1	4	yes	complemented
37	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
38	C₄²×C₁₃	M₁₆	1	2	yes	—
39	C₂×#1055	—	2	2	no	—
40	C₂×#1055	—	2	2	no	—
41	C₂×#1055	—	2	2	no	—
42	C₂×#1055	C₄⋊C₄	1	2	yes	—
43	C₂×#1055	M₁₆	1	4	yes	—
44	C₂×#1055	C₂×D₄	1	1	yes	characteristic
45	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
46	C₂×#1055	#41	1	4	yes	—
47	#6092	—	4	4	no	—
48	C₂×C₄⋊C₁₆	—	13	2	no	—
49	C₄²×C₈	—	13	1	no	—
50	#1374	—	13	4	no	—
51	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
60	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
62	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#117	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
67	#1055	—	2	2	no	—
68	#1055	—	2	4	no	—
69	#1055	#124	1	2	yes	—
70	#1055	#125	1	2	yes	—
71	C₂²×C₁₆	—	13	4	no	—
72	C₂×C₄×C₈	—	13	1	no	—
73	C₂×C₄×C₈	—	13	2	no	—
74	C₂×C₄×C₈	—	13	2	no	—
75	C₂×C₄×C₈	—	13	1	no	—
76	C₄³	#261	1	1	yes	characteristic
77	C₄⋊C₁₆	—	26	4	no	—
78	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₁₃	#431	1	2	yes	—
80	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
81	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
82	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	—	2	4	no	—
85	C₄×C₁₃	—	2	4	no	—
86	C₄×C₁₃	C₄⋊C₁₆	1	4	yes	—
87	C₄×C₁₃	#382	1	1	yes	characteristic
88	C₄×C₁₃	#423	1	1	yes	characteristic
89	C₄×C₁₃	#432	1	2	yes	—
90	C₂²×C₈	—	13	1	no	—
91	C₂²×C₈	—	13	2	no	—
92	C₂²×C₈	—	13	1	no	—
93	C₂×C₄²	#1057	1	2	yes	—
94	C₂×C₄²	#1057	1	1	yes	characteristic
95	C₂×C₄²	#1057	1	1	yes	characteristic
96	C₂×C₄²	C₂×#261	1	2	yes	—
97	C₂×C₄²	C₂×#261	1	1	yes	characteristic
98	C₂×C₁₆	—	26	4	no	—
99	C₂×C₁₆	—	26	8	no	—
100	C₄×C₈	—	13	2	no	—
101	C₄×C₈	—	13	2	no	—
102	C₄×C₈	—	26	4	no	—
103	C₄×C₈	—	26	4	no	—
104	C₄×C₈	—	26	2	no	—
105	C₂×C₁₃	#1831	1	1	yes	characteristic
106	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
107	C₂×C₁₃	#2150	1	1	yes	characteristic
108	C₂×C₁₃	#2162	1	2	yes	—
109	C₂×C₁₃	#2163	1	2	yes	—
110	C₂²×C₄	C₂×#1057	1	2	yes	—
111	C₂²×C₄	C₂×#1057	1	2	yes	—
112	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
113	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
114	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
115	C₂×C₈	—	13	1	no	—
116	C₂×C₈	—	13	1	no	—
117	C₂×C₈	—	13	4	no	—
118	C₂×C₈	—	13	4	no	—
119	C₂×C₈	—	13	2	no	—
120	C₂×C₈	—	26	2	no	—
121	C₂×C₈	—	26	2	no	—
122	C₂×C₈	—	26	2	no	—
123	C₄²	—	2	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#6092	1	4	yes	complemented
128	C₄²	#6118	1	2	yes	—
129	C₄²	C₄⋊#261	1	2	yes	—
130	C₁₆	—	52	4	no	—
131	C₁₃	#12879	1	1	yes	characteristic, complemented, Sylow
132	C₂³	C₂²×#1057	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂×#6092	1	4	yes	—
139	C₂×C₄	C₂×#6092	1	2	yes	—
140	C₂×C₄	#88106	1	4	yes	—
141	C₂×C₄	#88106	1	2	yes	—
142	C₂×C₄	C₄⋊#1057	1	2	yes	—
143	C₂×C₄	C₄⋊#1057	1	2	yes	—
144	C₂×C₄	C₂×#6118	1	1	yes	characteristic
145	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
147	C₂×C₄	#88237	1	2	yes	—
148	C₈	—	13	2	no	—
149	C₈	—	13	2	no	—
150	C₈	—	26	2	no	—
151	C₈	—	26	4	no	—
152	C₂²	C₂²×#6092	1	1	yes	characteristic
153	C₂²	C₂×#88106	1	1	yes	characteristic
154	C₂²	C₂×#88106	1	2	yes	—
155	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
156	C₂²	#11719119	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₄⋊#6092	1	4	yes	—
161	C₄	#11719053	1	1	yes	characteristic
162	C₄	#11719054	1	2	yes	—
163	C₄	#11719060	1	1	yes	characteristic
164	C₂	#13130751	1	1	yes	characteristic
165	C₂	C₂×C₄⋊#6092	1	1	yes	characteristic
166	C₂	#13131168	1	1	yes	characteristic
167	C₂	#13131193	1	2	yes	—
168	C₂	#13131197	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₄³×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₄³	#261
hypocenter	C₁₃	#12879
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
36. C₄²×C₁₃	C₁₆	semidirect
127. C₄²	#6092	semidirect
131. C₁₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
#13129874	2
C₄²×#1055	2
#12879	13