Subgroups — #16790250

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718260	C₂	1	4	yes	maximal
2	C₂×C₄⋊#6092	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6092	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
7	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
8	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	#11718260	—	2	8	no	—
11	C₄⋊#6092	—	2	4	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
16	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
18	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	C₂³	1	1	yes	characteristic
20	C₂×#6092	—	2	4	no	—
21	C₂×#6092	—	2	8	no	—
22	C₄×C₈×C₁₃	—	2	2	no	—
23	C₈×Dic₁₃	—	2	4	no	—
24	C₈×Dic₁₃	D₄	1	4	yes	—
25	C₈×Dic₁₃	Q₈	1	4	yes	—
26	C₄×#1055	D₄	1	2	yes	—
27	C₄×#1055	Q₈	1	2	yes	—
28	#12898	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
33	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	2	2	no	—
36	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
37	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
38	C₄²×C₁₃	M₁₆	1	2	yes	—
39	C₄×Dic₁₃	—	2	2	no	—
40	C₄×Dic₁₃	—	2	4	no	—
41	C₄×Dic₁₃	C₁₆	1	4	yes	complemented
42	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
43	C₄×Dic₁₃	M₁₆	1	4	yes	—
44	C₂×#1055	—	2	2	no	—
45	C₂×#1055	C₄⋊C₄	1	2	yes	—
46	C₂×#1055	C₂×D₄	1	1	yes	characteristic
47	C₂×#1055	C₂×D₄	1	2	yes	—
48	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
49	C₂×#1055	C₂×Q₈	1	2	yes	—
50	#6092	—	4	4	no	—
51	C₂×C₄⋊C₁₆	—	13	2	no	—
52	C₂×C₄⋊C₁₆	—	13	4	no	—
53	C₄²×C₈	—	13	1	no	—
54	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
59	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×Dic₁₃	—	2	2	no	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
67	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
68	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
69	C₂×Dic₁₃	M₃₂	1	4	yes	—
70	C₈×C₁₃	—	2	4	no	—
71	#1055	—	2	2	no	—
72	#1055	#127	1	1	yes	characteristic
73	#1055	C₄⋊Q₈	1	1	yes	characteristic
74	#1055	C₄⋊Q₈	1	2	yes	—
75	C₂²×C₁₆	—	13	4	no	—
76	C₂×C₄×C₈	—	13	2	no	—
77	C₂×C₄×C₈	—	13	1	no	—
78	C₂×C₄×C₈	—	13	2	no	—
79	C₂×C₄×C₈	#261	1	1	yes	characteristic
80	C₄³	—	13	1	no	—
81	C₄⋊C₁₆	—	26	4	no	—
82	C₄⋊C₁₆	—	26	8	no	—
83	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
84	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
86	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
87	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	C₄⋊C₁₆	1	4	yes	complemented
90	C₄×C₁₃	#384	1	1	yes	characteristic
91	C₄×C₁₃	#423	1	1	yes	characteristic
92	C₄×C₁₃	#423	1	2	yes	—
93	Dic₁₃	—	2	4	no	—
94	Dic₁₃	C₄⋊C₁₆	1	8	yes	complemented
95	C₂²×C₈	—	13	1	no	—
96	C₂²×C₈	—	13	1	no	—
97	C₂²×C₈	C₂×#261	1	2	yes	—
98	C₂×C₄²	—	13	1	no	—
99	C₂×C₄²	—	13	2	no	—
100	C₂×C₄²	—	13	2	no	—
101	C₂×C₄²	—	13	1	no	—
102	C₂×C₄²	C₂×#261	1	1	yes	characteristic
103	C₂×C₁₆	—	26	4	no	—
104	C₂×C₁₆	—	26	8	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	13	4	no	—
107	C₄×C₈	—	13	4	no	—
108	C₄×C₈	—	13	2	no	—
109	C₄×C₈	—	13	2	no	—
110	C₄×C₈	—	26	4	no	—
111	C₂×C₁₃	#1842	1	1	yes	characteristic
112	C₂×C₁₃	C₂×C₄⋊C₁₆	1	2	yes	—
113	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
114	C₂×C₁₃	#2150	1	2	yes	—
115	C₂×C₁₃	#2150	1	1	yes	characteristic
116	C₂²×C₄	—	13	2	no	—
117	C₂²×C₄	—	13	2	no	—
118	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	13	2	no	—
125	C₂×C₈	—	13	1	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	C₄⋊#261	1	4	yes	—
130	C₄²	—	13	4	no	—
131	C₄²	—	13	4	no	—
132	C₄²	—	13	4	no	—
133	C₄²	—	26	4	no	—
134	C₄²	—	26	2	no	—
135	C₄²	C₄⋊#261	1	2	yes	—
136	C₄²	#6122	1	2	yes	—
137	C₁₆	—	52	4	no	—
138	C₁₃	#12898	1	1	yes	characteristic, complemented, Sylow
139	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	13	4	no	—
142	C₂×C₄	—	13	4	no	—
143	C₂×C₄	—	13	4	no	—
144	C₂×C₄	—	13	4	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	—	26	2	no	—
147	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
148	C₂×C₄	#88098	1	2	yes	—
149	C₂×C₄	#88112	1	2	yes	—
150	C₂×C₄	#88112	1	2	yes	—
151	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
154	C₂×C₄	C₂×#6122	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	13	1	no	—
157	C₈	—	13	1	no	—
158	C₈	—	13	2	no	—
159	C₈	—	26	2	no	—
160	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
161	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
162	C₂²	C₂×#88098	1	1	yes	characteristic
163	C₂²	D₁₃⋊M₃₂	1	2	yes	—
164	C₂²	C₂×#88112	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	13	8	no	—
167	C₄	—	26	4	no	—
168	C₄	#11718260	1	4	yes	complemented
169	C₄	#11719053	1	1	yes	characteristic
170	C₄	#11719059	1	1	yes	characteristic
171	C₄	#11719129	1	2	yes	—
172	C₂	#13130769	1	1	yes	characteristic
173	C₂	D₁₃⋊(C₄⋊C₁₆)	1	2	yes	—
174	C₂	C₂×#11718260	1	1	yes	characteristic
175	C₂	#13131169	1	2	yes	—
176	C₂	#13131170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#12898
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
41. C₄×Dic₁₃	C₁₆	semidirect
89. C₄×C₁₃	C₄⋊C₁₆	semidirect
94. Dic₁₃	C₄⋊C₁₆	semidirect
138. C₁₃	#12898	semidirect
168. C₄	#11718260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718260	2
C₂×C₄⋊#6092	2
C₄×C₈×Dic₁₃	2
#12898	13