Subgroups — #13123783

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	#11719175	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#11718783	C₂	1	2	yes	maximal
4	#11718780	C₂	1	2	yes	maximal
5	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
6	#11718362	C₂	1	4	yes	maximal, complemented
7	#11718340	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	#88180	C₂²	1	2	yes	—
13	#88178	C₄	1	2	yes	—
14	#88178	C₄	1	2	yes	—
15	C₂²×#1055	C₂²	1	2	yes	—
16	#88124	C₄	1	2	yes	—
17	#88123	C₄	1	1	yes	characteristic
18	#88123	C₄	1	1	yes	characteristic
19	C₄×Dic₂₆	C₂²	1	2	yes	—
20	C₄×Dic₂₆	C₂²	1	2	yes	—
21	C₁₃×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₁₃	C₂²	1	2	yes	—
23	#88055	C₂²	1	4	yes	—
24	#88054	C₂²	1	4	yes	—
25	#88054	C₂²	1	4	yes	—
26	C₈×Dic₁₃	C₂²	1	4	yes	—
27	C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₄⋊#1055	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₁₃×M₁₆	—	2	2	no	—
36	C₁₃×M₁₆	Q₈	1	4	yes	complemented
37	C₂×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂×C₈×C₁₃	C₂³	1	2	yes	—
39	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
48	#6101	C₂×C₄	1	4	yes	—
49	#6101	C₂×C₄	1	4	yes	—
50	C₄×Dic₁₃	C₂³	1	4	yes	—
51	C₂×#1055	—	2	4	no	—
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	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	4	1	no	—
58	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₃	#41	1	2	yes	—
65	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
67	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
68	Dic₂₆	—	2	4	no	—
69	Dic₂₆	—	4	2	no	—
70	C₈×C₁₃	—	2	2	no	—
71	C₈×C₁₃	C₂×Q₈	1	4	yes	—
72	#1055	—	2	2	no	—
73	#1055	—	2	2	no	—
74	#533	—	13	1	no	—
75	#446	—	13	4	no	—
76	#446	—	13	2	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	D₁₃	1	1	yes	characteristic
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	1	no	—
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	C₄×Q₈	1	2	yes	—
90	C₄×C₁₃	#131	1	2	yes	—
91	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
92	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
93	Dic₁₃	—	2	4	no	—
94	Dic₁₃	—	2	4	no	—
95	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
96	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
97	C₂²×C₈	—	13	2	no	—
98	C₄⋊Q₈	—	13	1	no	—
99	C₄⋊Q₈	—	13	1	no	—
100	#125	—	13	2	no	—
101	C₂²⋊Q₈	—	13	2	no	—
102	C₂²⋊Q₈	—	13	2	no	—
103	C₄×Q₈	—	13	2	no	—
104	C₄×Q₈	—	13	2	no	—
105	#117	—	13	2	no	—
106	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
107	C₄⋊C₈	—	13	4	no	—
108	C₄⋊C₈	—	13	4	no	—
109	C₄⋊C₈	—	13	1	no	—
110	C₄⋊C₈	—	13	1	no	—
111	C₄⋊C₈	—	13	2	no	—
112	#97	—	13	4	no	—
113	#97	C₂×D₁₃	1	2	yes	—
114	C₄×C₈	—	13	4	no	—
115	C₄×C₈	C₂×D₁₃	1	2	yes	—
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	—	13	2	no	—
121	C₂×Q₈	—	13	2	no	—
122	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
125	M₁₆	—	2	2	no	—
126	M₁₆	Dic₂₆	1	4	yes	complemented
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	—	13	2	no	—
129	C₂×C₈	—	26	4	no	—
130	C₂×C₈	C₂²×D₁₃	1	2	yes	—
131	C₂×C₈	C₂²×D₁₃	1	2	yes	—
132	C₄⋊C₄	—	13	2	no	—
133	C₄⋊C₄	—	13	2	no	—
134	C₄⋊C₄	—	13	4	no	—
135	C₄⋊C₄	—	13	4	no	—
136	C₄⋊C₄	—	13	2	no	—
137	C₄⋊C₄	—	13	2	no	—
138	C₂²⋊C₄	—	13	4	no	—
139	C₄²	—	13	4	no	—
140	C₄²	C₄×D₁₃	1	1	yes	characteristic
141	C₄²	C₄×D₁₃	1	1	yes	characteristic
142	C₄²	C₂²×D₁₃	1	1	yes	characteristic
143	C₄²	C₂²×D₁₃	1	1	yes	characteristic
144	C₁₃	#2962	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
146	Q₈	—	26	4	no	—
147	Q₈	—	52	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	13	4	no	—
151	C₂×C₄	—	13	4	no	—
152	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
153	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
158	C₂×C₄	#6127	1	2	yes	—
159	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	26	2	no	—
162	C₈	—	26	2	no	—
163	C₈	C₂×Dic₂₆	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₂₆	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	26	4	no	—
172	C₄	—	26	4	no	—
173	C₄	C₄×Dic₂₆	1	2	yes	—
174	C₄	#88161	1	2	yes	—
175	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
176	C₄	C₂×#6127	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₂₆	1	1	yes	characteristic
179	C₂	C₂×#88161	1	1	yes	characteristic
180	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×M₁₆	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#2962
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
6. #11718362	C₂	semidirect
7. #11718340	C₂	semidirect
8. C₈×Dic₂₆	C₂	semidirect
36. C₁₃×M₁₆	Q₈	semidirect
126. M₁₆	Dic₂₆	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
#11719175	2
C₄×C₁₃×M₁₆	2
#11718783	2
#11718780	2
C₂×C₄⋊#1055	2
#11718362	2
#11718340	2
C₈×Dic₂₆	2
#2962	13