Subgroups — #15794111

Order 3328.

1710 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718596	C₂	1	2	yes	maximal
2	C₂×#11718597	C₂	1	4	yes	maximal
3	#13113066	C₂	1	8	yes	maximal
4	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
9	C₂²×C₄⋊Dic₁₃	C₂²	1	8	yes	—
10	C₄³×C₁₃	C₄	1	8	yes	complemented
11	C₂×#88071	C₂²	1	16	yes	—
12	#11718597	—	2	16	no	—
13	#11718596	—	2	8	no	—
14	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
17	C₂³×Dic₁₃	C₂³	1	8	yes	—
18	C₂×C₄²×C₁₃	—	2	16	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	16	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
22	C₂×C₄²×C₁₃	D₄	1	4	yes	—
23	C₂×C₄²×C₁₃	D₄	1	8	yes	—
24	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
25	C₂×C₄²×C₁₃	Q₈	1	8	yes	—
26	C₂×C₄⋊Dic₁₃	—	2	16	no	—
27	C₂×C₄⋊Dic₁₃	—	2	8	no	—
28	C₂×C₄⋊Dic₁₃	—	2	16	no	—
29	C₂×C₄⋊Dic₁₃	—	2	32	no	—
30	#88071	—	2	32	no	—
31	#14687	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	2	16	no	—
34	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
35	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
42	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₁₃	#41	1	16	yes	—
46	C₂²×Dic₁₃	—	2	16	no	—
47	C₂²×Dic₁₃	—	2	8	no	—
48	C₂²×Dic₁₃	—	2	32	no	—
49	C₄²×C₁₃	—	2	32	no	—
50	C₄²×C₁₃	C₄⋊C₄	1	16	yes	—
51	C₄²×C₁₃	C₄⋊C₄	1	32	yes	complemented
52	C₄⋊Dic₁₃	—	4	32	no	—
53	C₄⋊Dic₁₃	—	4	16	no	—
54	C₂×#384	—	13	2	no	—
55	C₂×#382	—	13	4	no	—
56	C₂×C₄³	D₁₃	1	1	yes	characteristic
57	C₄⋊#95	—	13	8	no	—
58	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
59	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
60	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
61	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
62	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
63	C₂³×C₁₃	C₂×#41	1	8	yes	—
64	C₂×C₄×C₁₃	—	2	16	no	—
65	C₂×C₄×C₁₃	—	2	16	no	—
66	C₂×C₄×C₁₃	—	2	16	no	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₁₃	#117	1	16	yes	—
72	C₂×C₄×C₁₃	#124	1	16	yes	—
73	C₂×C₄×C₁₃	#125	1	16	yes	—
74	C₂×C₄×C₁₃	#127	1	4	yes	—
75	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
77	C₂×Dic₁₃	—	4	16	no	—
78	C₂×Dic₁₃	—	4	8	no	—
79	C₂×Dic₁₃	—	4	32	no	—
80	C₂²×C₄⋊C₄	—	13	8	no	—
81	C₂²×C₄⋊C₄	—	13	4	no	—
82	C₂²×C₄²	C₂×D₁₃	1	4	yes	—
83	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
84	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
85	#384	—	26	8	no	—
86	#382	—	26	16	no	—
87	C₂×#95	—	13	16	no	—
88	C₄³	Dic₁₃	1	8	yes	complemented
89	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
91	C₂²×C₁₃	C₂×#117	1	4	yes	—
92	C₂²×C₁₃	C₂×#124	1	4	yes	—
93	C₂²×C₁₃	C₂×#125	1	4	yes	—
94	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
95	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
96	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₁₃	#532	1	8	yes	—
98	C₂²×C₁₃	#533	1	8	yes	—
99	C₄×C₁₃	—	2	16	no	—
100	C₄×C₁₃	#382	1	16	yes	complemented
101	C₄×C₁₃	#384	1	8	yes	—
102	Dic₁₃	—	8	8	no	—
103	C₂³×C₄	—	13	8	no	—
104	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
105	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
106	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
107	C₂×C₄⋊C₄	—	26	16	no	—
108	C₂×C₄⋊C₄	—	26	8	no	—
109	C₂×C₄⋊C₄	—	26	16	no	—
110	C₂×C₄⋊C₄	—	26	32	no	—
111	C₂×C₄²	—	2	16	no	—
112	C₂×C₄²	Dic₂₆	1	4	yes	—
113	C₂×C₄²	Dic₂₆	1	8	yes	—
114	C₂×C₄²	D₅₂	1	4	yes	—
115	C₂×C₄²	D₅₂	1	8	yes	—
116	C₂×C₄²	C₂×Dic₁₃	1	16	yes	—
117	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
118	C₂×C₄²	C₂×Dic₁₃	1	8	yes	—
119	#95	—	26	32	no	—
120	C₂×C₁₃	C₂×#382	1	2	yes	—
121	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
122	C₂×C₁₃	#2285	1	4	yes	—
123	C₂×C₁₃	#2858	1	2	yes	—
124	C₂×C₁₃	#2859	1	4	yes	—
125	C₂×C₁₃	#2862	1	2	yes	—
126	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
127	C₂²×C₄	—	2	16	no	—
128	C₂²×C₄	—	26	16	no	—
129	C₂²×C₄	—	26	8	no	—
130	C₂²×C₄	—	26	32	no	—
131	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
132	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
133	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
134	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
135	C₂²×C₄	C₂×D₅₂	1	8	yes	—
136	C₂²×C₄	C₂×D₅₂	1	2	yes	—
137	C₂²×C₄	C₂×D₅₂	1	4	yes	—
138	C₂²×C₄	C₂×D₅₂	1	4	yes	—
139	C₂²×C₄	#6127	1	16	yes	—
140	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
143	C₄⋊C₄	—	52	16	no	—
144	C₄⋊C₄	—	52	32	no	—
145	C₄²	—	2	32	no	—
146	C₄²	C₄⋊Dic₁₃	1	16	yes	—
147	C₄²	C₄⋊Dic₁₃	1	32	yes	complemented
148	C₁₃	#14687	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
150	C₂³	C₂²×Dic₂₆	1	2	yes	—
151	C₂³	C₂²×D₅₂	1	1	yes	characteristic
152	C₂³	C₂²×D₅₂	1	2	yes	—
153	C₂³	C₂×#6127	1	8	yes	—
154	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	52	8	no	—
159	C₂×C₄	—	52	16	no	—
160	C₂×C₄	—	52	32	no	—
161	C₂×C₄	#88123	1	8	yes	—
162	C₂×C₄	#88123	1	4	yes	—
163	C₂×C₄	#88124	1	16	yes	—
164	C₂×C₄	#88128	1	4	yes	—
165	C₂×C₄	#88129	1	16	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₁₃	1	16	yes	—
167	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
168	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
169	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
170	C₂×C₄	#88180	1	16	yes	—
171	C₂²	C₂×#88123	1	2	yes	—
172	C₂²	C₂×#88123	1	1	yes	characteristic
173	C₂²	C₂×#88124	1	4	yes	—
174	C₂²	#11719175	1	8	yes	—
175	C₂²	C₂×#88128	1	1	yes	characteristic
176	C₂²	C₂×#88129	1	4	yes	—
177	C₂²	#11719182	1	8	yes	—
178	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
179	C₂²	C₂²×C₄⋊Dic₁₃	1	2	yes	—
180	C₂²	C₂×#88180	1	4	yes	—
181	C₄	—	2	16	no	—
182	C₄	—	104	8	no	—
183	C₄	#11718596	1	8	yes	—
184	C₄	#11718597	1	16	yes	complemented
185	C₂	C₂×#11718597	1	2	yes	—
186	C₂	C₂×#11718596	1	1	yes	characteristic
187	C₂	#13117243	1	4	yes	—
188	C₂	#13123193	1	2	yes	—
189	C₂	#13123199	1	4	yes	—
190	C₂	#13123214	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₁₃
commutator	C₂³×C₁₃	C₂³×C₄
Fitting	C₂×C₄³×C₁₃	C₂
socle, prime radical, cosocle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	#14687
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
10. C₄³×C₁₃	C₄	semidirect
51. C₄²×C₁₃	C₄⋊C₄	semidirect
88. C₄³	Dic₁₃	semidirect
100. C₄×C₁₃	#382	semidirect
147. C₄²	C₄⋊Dic₁₃	semidirect
148. C₁₃	#14687	semidirect
184. C₄	#11718597	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718596	2
C₂×#11718597	2
#13113066	2
C₂×C₄³×C₁₃	2
#14687	13