Subgroups — #16727312

Order 3328.

1810 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	#13123777	C₂	1	12	yes	maximal, complemented
2	#13123771	C₂	1	12	yes	maximal, complemented
3	#13123400	C₂	1	3	yes	maximal
4	C₂×#11718658	C₂	1	3	yes	maximal
5	C₁₃×#3458	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#544	C₄	1	12	yes	—
7	C₁₃×#543	C₄	1	4	yes	—
8	C₁₃×#533	C₂²	1	12	yes	—
9	C₂×C₁₃×#117	C₂²	1	3	yes	—
10	C₂²×#6099	C₂²	1	3	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	#11718685	C₂²	1	48	yes	complemented
13	Q₈×#1055	C₂²	1	16	yes	complemented
14	#11718660	C₂²	1	24	yes	—
15	#11718658	C₂²	1	12	yes	—
16	#11718658	C₂²	1	12	yes	—
17	#11718585	C₂²	1	12	yes	—
18	C₂×#88043	C₂²	1	6	yes	—
19	C₂×C₄×#1055	C₂²	1	6	yes	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×C₄⋊Q₈	C₂×C₄	1	24	yes	—
22	C₁₃×#125	C₂×C₄	1	24	yes	—
23	C₁₃×C₂²⋊Q₈	C₂×C₄	1	48	yes	—
24	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
25	C₁₃×#117	C₂×C₄	1	12	yes	—
26	C₁₃×#117	C₂³	1	12	yes	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
28	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
30	C₂×#6099	—	2	12	no	—
31	C₂×#6099	C₂³	1	12	yes	—
32	C₂²×#1055	Q₈	1	8	yes	complemented
33	C₂²×#1055	C₂³	1	6	yes	—
34	C₂²×#1055	C₂³	1	3	yes	—
35	C₂²×#1055	C₂³	1	3	yes	—
36	C₄⋊#1055	C₂³	1	48	yes	—
37	#88043	C₂³	1	24	yes	—
38	C₄×#1055	C₂³	1	12	yes	—
39	C₄×#1055	C₂³	1	12	yes	—
40	#60191	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₁₃	#41	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
48	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
49	C₁₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
50	C₁₃×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
52	C₄²×C₁₃	C₂⁴	1	12	yes	—
53	#6099	—	2	24	no	—
54	C₂×#1055	—	2	12	no	—
55	C₂×#1055	C₂×Q₈	1	12	yes	—
56	C₂×#1055	C₂×Q₈	1	12	yes	—
57	C₂×#1055	C₂⁴	1	12	yes	—
58	C₂×#1055	C₂⁴	1	1	yes	characteristic
59	C₂×#1055	C₂⁴	1	3	yes	—
60	#3458	D₁₃	1	1	yes	characteristic
61	#2961	—	13	12	no	—
62	#2960	—	13	12	no	—
63	#2905	—	13	3	no	—
64	C₂×#405	—	13	3	no	—
65	C₂³×C₁₃	—	4	2	no	—
66	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
67	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
68	Q₈×C₁₃	—	4	16	no	—
69	C₂×C₄×C₁₃	—	4	12	no	—
70	C₂×C₄×C₁₃	C₄×Q₈	1	12	yes	—
71	C₂×C₄×C₁₃	C₂³×C₄	1	24	yes	—
72	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
73	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
74	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
75	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
76	#1055	—	2	12	no	—
77	#1055	C₂²×Q₈	1	2	yes	—
78	#1055	C₂²×Q₈	1	6	yes	—
79	C₂²×M₁₆	—	13	3	no	—
80	C₂³×C₈	—	13	1	no	—
81	#544	Dic₁₃	1	12	yes	—
82	#543	Dic₁₃	1	4	yes	—
83	#533	C₂×D₁₃	1	12	yes	—
84	C₂×#117	C₂×D₁₃	1	3	yes	—
85	#446	—	13	48	no	—
86	C₈×Q₈	—	13	16	no	—
87	#424	—	13	24	no	—
88	#423	—	13	12	no	—
89	#405	—	13	12	no	—
90	#405	—	13	12	no	—
91	C₂×#97	—	13	6	no	—
92	C₂×C₄×C₈	—	13	6	no	—
93	C₂²×C₁₃	—	2	6	no	—
94	C₂²×C₁₃	—	4	4	no	—
95	C₂²×C₁₃	C₂×C₄×Q₈	1	6	yes	—
96	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
97	C₄×C₁₃	—	2	24	no	—
98	C₄×C₁₃	C₂×C₄×Q₈	1	6	yes	—
99	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
100	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
101	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
102	C₂×M₁₆	—	13	12	no	—
103	C₂×M₁₆	—	26	12	no	—
104	C₂²×C₈	—	13	8	no	—
105	C₂²×C₈	—	13	3	no	—
106	C₂²×C₈	—	13	3	no	—
107	C₂²×C₈	—	13	6	no	—
108	C₄⋊Q₈	C₂×Dic₁₃	1	24	yes	—
109	#125	C₂×Dic₁₃	1	24	yes	—
110	C₂²⋊Q₈	C₂×Dic₁₃	1	48	yes	—
111	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
112	#117	C₂×Dic₁₃	1	12	yes	—
113	#117	C₂²×D₁₃	1	12	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
115	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	6	yes	—
116	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
117	C₄⋊C₈	—	13	48	no	—
118	#97	—	13	24	no	—
119	C₄×C₈	—	13	12	no	—
120	C₄×C₈	—	13	12	no	—
121	C₂×C₁₃	—	2	6	no	—
122	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
123	C₂×C₁₃	#3573	1	2	yes	—
124	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
125	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
126	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
127	C₂²×C₄	#6131	1	4	yes	—
128	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
129	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
130	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
131	M₁₆	—	26	24	no	—
132	C₂×C₈	—	13	12	no	—
133	C₂×C₈	—	13	12	no	—
134	C₂×C₈	—	13	1	no	—
135	C₂×C₈	—	13	3	no	—
136	C₂×C₈	—	13	12	no	—
137	C₂×C₈	—	26	12	no	—
138	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
139	C₄⋊C₄	C₂²×Dic₁₃	1	48	yes	—
140	C₂²⋊C₄	C₂²×Dic₁₃	1	24	yes	—
141	C₄²	C₂²×Dic₁₃	1	12	yes	—
142	C₄²	C₂³×D₁₃	1	12	yes	—
143	C₁₃	#60191	1	1	yes	characteristic, complemented, Sylow
144	C₂³	—	4	2	no	—
145	C₂³	Q₈×Dic₁₃	1	4	yes	—
146	C₂³	C₂³×Dic₁₃	1	3	yes	—
147	Q₈	—	4	16	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	Q₈×Dic₁₃	1	12	yes	—
150	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
151	C₂×C₄	C₂×#6131	1	6	yes	—
152	C₂×C₄	C₂³×Dic₁₃	1	24	yes	—
153	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
154	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
155	C₈	—	13	2	no	—
156	C₈	—	13	6	no	—
157	C₈	—	26	12	no	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×Q₈×Dic₁₃	1	6	yes	—
161	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×Q₈×Dic₁₃	1	6	yes	—
164	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
165	C₄	C₂²×#6131	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×Q₈×Dic₁₃	1	1	yes	characteristic
168	C₂	#13129548	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₁₃
commutator, socle	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₁₃×#3458	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	#3458	D₁₃
hypocenter	C₁₃	#60191
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #13123777	C₂	semidirect
2. #13123771	C₂	semidirect
12. #11718685	C₂²	semidirect
13. Q₈×#1055	C₂²	semidirect
32. C₂²×#1055	Q₈	semidirect
143. C₁₃	#60191	semidirect

Maximal subgroups

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

subgroup	index
#13123777	2
#13123771	2
#13123400	2
C₂×#11718658	2
C₁₃×#3458	2
#60191	13