Subgroups — #13123785

Order 1664.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	2	yes	maximal, complemented
3	#11718685	C₂	1	4	yes	maximal, complemented
4	Q₈×#1055	C₂	1	2	yes	maximal, complemented
5	#11718660	C₂	1	2	yes	maximal
6	#11718658	C₂	1	2	yes	maximal
7	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
8	C₄×#6099	C₂	1	1	yes	maximal, characteristic
9	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₁₃×#125	C₄	1	2	yes	—
12	C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#117	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#6099	C₂²	1	1	yes	characteristic
19	C₂×#6099	C₂²	1	1	yes	characteristic
20	C₂²×#1055	C₂²	1	2	yes	—
21	C₄⋊#1055	C₂²	1	4	yes	—
22	C₄⋊#1055	C₂²	1	4	yes	—
23	C₄⋊#1055	C₂²	1	1	yes	characteristic
24	C₄⋊#1055	C₂²	1	2	yes	—
25	C₄⋊#1055	C₂²	1	1	yes	characteristic
26	#88043	C₂²	1	4	yes	—
27	#88043	C₂²	1	2	yes	—
28	C₄×#1055	C₂²	1	4	yes	—
29	C₄×#1055	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
42	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₄²×C₁₃	C₂³	1	4	yes	—
47	#6099	—	2	2	no	—
48	#6099	Q₈	1	4	yes	complemented
49	C₂×#1055	—	2	4	no	—
50	C₂×#1055	C₂³	1	2	yes	—
51	C₂×#1055	C₂³	1	2	yes	—
52	C₂×#1055	C₂³	1	2	yes	—
53	C₂×#1055	C₂³	1	2	yes	—
54	#2962	—	13	1	no	maximal, Sylow
55	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	Q₈×C₁₃	—	2	4	no	—
57	Q₈×C₁₃	—	4	2	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₃	#41	1	2	yes	—
69	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	#1055	—	2	2	no	—
71	#1055	—	2	2	no	—
72	#1055	—	2	2	no	—
73	#1055	C₂×Q₈	1	4	yes	—
74	#533	D₁₃	1	1	yes	characteristic
75	#446	—	13	2	no	—
76	#446	—	13	4	no	—
77	C₈×Q₈	—	13	2	no	—
78	#424	—	13	2	no	—
79	C₂×C₄⋊C₈	—	13	1	no	—
80	#405	—	13	2	no	—
81	C₄×M₁₆	—	13	1	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
85	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	—	2	4	no	—
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	4	no	—
91	C₄×C₁₃	C₄×Q₈	1	2	yes	—
92	C₄×C₁₃	#131	1	2	yes	—
93	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
95	C₂×M₁₆	—	13	1	no	—
96	C₂×M₁₆	—	13	1	no	—
97	C₂²×C₈	—	13	2	no	—
98	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
99	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
100	#125	Dic₁₃	1	2	yes	—
101	C₂²⋊Q₈	Dic₁₃	1	2	yes	—
102	C₂²⋊Q₈	Dic₁₃	1	2	yes	—
103	C₄×Q₈	C₂×D₁₃	1	2	yes	—
104	C₄×Q₈	C₂×D₁₃	1	2	yes	—
105	#117	C₂×D₁₃	1	2	yes	—
106	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
107	C₄⋊C₈	—	13	1	no	—
108	C₄⋊C₈	—	13	1	no	—
109	C₄⋊C₈	—	13	2	no	—
110	C₄⋊C₈	—	13	4	no	—
111	C₄⋊C₈	—	13	4	no	—
112	#97	—	13	2	no	—
113	#97	—	13	4	no	—
114	C₄×C₈	—	13	2	no	—
115	C₄×C₈	—	13	4	no	—
116	C₂×C₁₃	—	2	2	no	—
117	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
118	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₃	#568	1	1	yes	characteristic
120	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
121	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
122	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
125	M₁₆	—	13	4	no	—
126	M₁₆	—	26	2	no	—
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	—	13	2	no	—
129	C₂×C₈	—	13	2	no	—
130	C₂×C₈	—	13	2	no	—
131	C₂×C₈	—	26	4	no	—
132	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
133	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
134	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
135	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
136	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
137	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
138	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
139	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
141	C₄²	C₂²×D₁₃	1	1	yes	characteristic
142	C₄²	C₂²×D₁₃	1	4	yes	—
143	C₄²	C₂²×D₁₃	1	1	yes	characteristic
144	C₁₃	#2962	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	Q₈×D₁₃	1	2	yes	—
151	C₂×C₄	#6131	1	2	yes	—
152	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
154	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
155	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
156	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
157	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
159	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
160	C₈	—	13	4	no	—
161	C₈	—	26	2	no	—
162	C₈	—	26	2	no	—
163	C₈	—	26	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	Q₈×Dic₁₃	1	2	yes	—
167	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	Q₈×Dic₁₃	1	2	yes	—
174	C₄	#88202	1	2	yes	—
175	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
176	C₄	C₂×#6131	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
179	C₂	C₂×#88202	1	1	yes	characteristic
180	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#533	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#533	D₁₃
hypocenter	C₁₃	#2962
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. #11718685	C₂	semidirect
4. Q₈×#1055	C₂	semidirect
48. #6099	Q₈	semidirect
144. C₁₃	#2962	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11718685	2
Q₈×#1055	2
#11718660	2
#11718658	2
C₂×C₄⋊#1055	2
C₄×#6099	2
#2962	13