Subgroups — #15863121

Order 3328.

910 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117243	C₂	1	2	yes	maximal
2	#13114970	C₂	1	8	yes	maximal, complemented
3	#13114553	C₂	1	4	yes	maximal
4	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88180	C₂²	1	4	yes	—
8	C₂×C₁₃×#97	C₄	1	8	yes	complemented
9	C₂×C₁₃×#97	C₂²	1	4	yes	—
10	#11718873	—	2	4	no	—
11	#11718597	C₂²	1	4	yes	—
12	#11718596	C₂²	1	4	yes	—
13	C₄×C₄⋊Dic₁₃	—	2	4	no	—
14	#11718143	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₃	D₄	1	2	yes	—
16	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	8	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
24	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
25	#88180	—	2	16	no	—
26	C₂×C₄⋊Dic₁₃	—	2	8	no	—
27	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
28	C₂×C₄×Dic₁₃	—	2	4	no	—
29	C₁₃×#97	C₂×C₄	1	16	yes	—
30	#88071	—	2	8	no	—
31	#20762	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	4	4	no	—
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₁₃	#41	1	8	yes	—
40	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	8	no	—
42	C₂×C₈×C₁₃	—	2	16	no	—
43	C₂×C₈×C₁₃	—	2	8	no	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
45	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
46	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
47	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
48	C₂²⋊Dic₁₃	—	4	8	no	—
49	C₄⋊Dic₁₃	—	2	16	no	—
50	C₄⋊Dic₁₃	—	4	16	no	—
51	C₄×Dic₁₃	—	4	8	no	—
52	C₂²×#97	D₁₃	1	1	yes	characteristic
53	#2285	—	13	2	no	—
54	#1836	—	13	8	no	—
55	#1735	—	13	4	no	—
56	C₂³×C₁₃	—	2	4	no	—
57	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂³×C₁₃	#124	1	4	yes	—
59	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	8	no	—
61	C₂×C₄×C₁₃	—	4	8	no	—
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₃	#117	1	8	yes	—
65	C₂×C₄×C₁₃	#124	1	4	yes	—
66	C₂×C₄×C₁₃	#125	1	8	yes	—
67	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
70	C₂×Dic₁₃	—	4	8	no	—
71	C₂×Dic₁₃	—	4	16	no	—
72	C₈×C₁₃	—	2	8	no	—
73	C₂³×C₈	C₂×D₁₃	1	2	yes	—
74	C₂×#117	—	13	4	no	—
75	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
76	C₂×#97	Dic₁₃	1	8	yes	complemented
77	C₂×#97	C₂×D₁₃	1	4	yes	—
78	#384	—	13	4	no	—
79	#382	—	13	4	no	—
80	#380	—	26	4	no	—
81	C₄×C₄⋊C₄	—	26	4	no	—
82	#340	—	13	16	no	—
83	C₂²×C₁₃	—	2	8	no	—
84	C₂²×C₁₃	—	2	4	no	—
85	C₂²×C₁₃	#382	1	4	yes	—
86	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₃	C₂×#124	1	2	yes	—
88	C₂²×C₁₃	#575	1	4	yes	—
89	C₄×C₁₃	—	2	8	no	—
90	C₄×C₁₃	#382	1	4	yes	—
91	C₄×C₁₃	C₂×#117	1	2	yes	—
92	C₄×C₁₃	C₂×#125	1	2	yes	—
93	Dic₁₃	—	8	8	no	—
94	C₂³×C₄	D₅₂	1	2	yes	—
95	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
96	C₂²×C₈	—	2	8	no	—
97	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
98	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
99	#117	—	26	16	no	—
100	C₂×C₄⋊C₄	—	13	8	no	—
101	C₂×C₄⋊C₄	—	26	8	no	—
102	C₂×C₂²⋊C₄	—	26	4	no	—
103	C₂×C₄²	—	26	4	no	—
104	C₂×C₄²	Dic₂₆	1	4	yes	—
105	C₂×C₄²	D₅₂	1	2	yes	—
106	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
107	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
108	#97	C₂×Dic₁₃	1	16	yes	—
109	#95	—	26	8	no	—
110	C₂×C₁₃	—	2	4	no	—
111	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
112	C₂×C₁₃	#2911	1	2	yes	—
113	C₂×C₁₃	#3135	1	4	yes	—
114	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
115	C₂²×C₄	—	4	4	no	—
116	C₂²×C₄	—	26	8	no	—
117	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
118	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
119	C₂²×C₄	C₂×D₅₂	1	4	yes	—
120	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
121	C₂²×C₄	#6127	1	8	yes	—
122	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
123	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
124	C₂×C₈	—	2	16	no	—
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
127	C₄⋊C₄	—	26	16	no	—
128	C₄⋊C₄	—	52	16	no	—
129	C₂²⋊C₄	—	52	8	no	—
130	C₄²	—	52	8	no	—
131	C₄²	C₄⋊Dic₁₃	1	8	yes	—
132	C₄²	C₂×Dic₂₆	1	4	yes	—
133	C₄²	C₂²×Dic₁₃	1	4	yes	—
134	C₁₃	#20762	1	1	yes	characteristic, complemented, Sylow
135	C₂³	—	2	4	no	—
136	C₂³	#88129	1	4	yes	—
137	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
138	C₂³	C₂²×D₅₂	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	52	8	no	—
142	C₂×C₄	—	52	16	no	—
143	C₂×C₄	#88124	1	8	yes	—
144	C₂×C₄	#88129	1	4	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
146	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
147	C₂×C₄	#88180	1	8	yes	—
148	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
149	C₂×C₄	C₂×#6127	1	4	yes	—
150	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
151	C₈	—	2	8	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	#11718597	1	4	yes	—
155	C₂²	C₂×#88129	1	2	yes	—
156	C₂²	#11719453	1	4	yes	—
157	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
158	C₄	—	2	8	no	—
159	C₄	—	104	8	no	—
160	C₄	#11718597	1	4	yes	—
161	C₄	C₂×#88124	1	2	yes	—
162	C₄	C₂×#88180	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×#11718597	1	1	yes	characteristic
165	C₂	#13123426	1	2	yes	—
166	C₂	#13125182	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	#20762
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #13114970	C₂	semidirect
8. C₂×C₁₃×#97	C₄	semidirect
76. C₂×#97	Dic₁₃	semidirect
134. C₁₃	#20762	semidirect

Maximal subgroups

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

subgroup	index
#13117243	2
#13114970	2
#13114553	2
C₂²×C₁₃×#97	2
#20762	13