Subgroups — #15778228

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6090	C₂	1	2	yes	maximal
2	C₂×#11718053	C₂	1	2	yes	maximal
3	#13112914	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₄³×C₁₃	C₄	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	4	yes	—
13	C₄⋊#6090	C₄	1	8	yes	complemented
14	#11718053	—	2	4	no	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₈	1	2	yes	—
20	C₂×C₄²×C₁₃	C₈	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	—	2	2	no	—
25	C₄×C₈×C₁₃	—	2	2	no	—
26	C₄×C₈×C₁₃	—	2	2	no	—
27	C₄×C₈×C₁₃	C₈	1	4	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	C₂×#6090	—	2	4	no	—
32	C₂×#6090	—	2	4	no	—
33	C₂×#6090	C₂×C₄	1	8	yes	—
34	#12653	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	C₄²	1	2	yes	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	#41	1	2	yes	—
53	C₄²×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	—	2	4	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	C₄²	1	2	yes	—
58	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₄²×C₁₃	C₂×C₈	1	2	yes	—
60	C₄²×C₁₃	M₁₆	1	2	yes	—
61	#6090	—	2	8	no	—
62	C₂×C₄⋊C₁₆	—	13	2	no	—
63	C₂×#346	—	13	2	no	—
64	C₄²×C₈	D₁₃	1	1	yes	characteristic
65	#1374	—	13	2	no	—
66	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
75	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
76	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
86	C₈×C₁₃	C₄×D₄	1	2	yes	—
87	C₈×C₁₃	C₄×Q₈	1	2	yes	—
88	C₂²×C₁₆	—	13	4	no	—
89	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
90	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
91	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
95	C₄³	Dic₁₃	1	1	yes	characteristic
96	C₄⋊C₁₆	—	13	8	no	—
97	#346	—	26	4	no	—
98	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
99	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₃	#431	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
102	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	2	4	no	—
105	C₄×C₁₃	—	2	4	no	—
106	C₄×C₁₃	#346	1	4	yes	complemented
107	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₁₃	#431	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×C₄²	#1055	1	2	yes	—
116	C₂×C₄²	#1055	1	2	yes	—
117	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
120	C₂×C₁₆	—	13	8	no	—
121	C₂×C₁₆	—	26	4	no	—
122	C₂×C₁₆	—	26	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	#1055	1	4	yes	—
127	C₄×C₈	Dic₂₆	1	2	yes	—
128	C₄×C₈	C₄×D₁₃	1	4	yes	—
129	C₄×C₈	D₅₂	1	2	yes	—
130	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
132	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
133	C₂×C₁₃	#2150	1	2	yes	—
134	C₂×C₁₃	#2163	1	2	yes	—
135	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
136	C₂²×C₄	C₂×#1055	1	2	yes	—
137	C₂²×C₄	C₂×#1055	1	2	yes	—
138	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	C₂×#1055	1	2	yes	—
145	C₂×C₈	C₂×#1055	1	2	yes	—
146	C₂×C₈	#6099	1	2	yes	—
147	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
149	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
151	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
152	C₂×C₈	#6127	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	C₂×#1055	1	2	yes	—
158	C₄²	#6099	1	2	yes	—
159	C₄²	C₄×Dic₁₃	1	2	yes	—
160	C₄²	C₄⋊Dic₁₃	1	2	yes	—
161	C₁₆	—	26	8	no	—
162	C₁₃	#12653	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂²×#1055	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₄×#1055	1	2	yes	—
170	C₂×C₄	C₄×#1055	1	2	yes	—
171	C₂×C₄	C₄⋊#1055	1	4	yes	—
172	C₂×C₄	#88052	1	4	yes	—
173	C₂×C₄	#88052	1	4	yes	—
174	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
175	C₂×C₄	C₂×#6099	1	1	yes	characteristic
176	C₂×C₄	C₂×#6099	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
179	C₂×C₄	#88180	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	C₄⋊#1055	1	4	yes	—
183	C₈	C₄×Dic₂₆	1	2	yes	—
184	C₈	C₄×D₅₂	1	2	yes	—
185	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
187	C₂²	#11718586	1	1	yes	characteristic
188	C₂²	C₂×#88052	1	2	yes	—
189	C₂²	C₂×#88052	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	#11718053	1	4	yes	complemented
194	C₄	C₄×#6099	1	1	yes	characteristic
195	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
196	C₄	#11718586	1	1	yes	characteristic
197	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
198	C₂	C₄×C₄⋊#1055	1	1	yes	characteristic
199	C₂	C₂×#11718053	1	1	yes	characteristic
200	C₂	C₄×#88052	1	1	yes	characteristic
201	C₂	#13116592	1	2	yes	—
202	C₂	#13116639	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#12653
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
13. C₄⋊#6090	C₄	semidirect
106. C₄×C₁₃	#346	semidirect
162. C₁₃	#12653	semidirect
193. C₄	#11718053	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6090	2
C₂×#11718053	2
#13112914	2
C₄²×C₈×C₁₃	2
#12653	13