Subgroups — #13123787

Order 1664.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#423	C₂	1	1	yes	maximal, characteristic
3	#11718779	C₂	1	2	yes	maximal
4	M₁₆×Dic₁₃	C₂	1	2	yes	maximal
5	C₄×#6099	C₂	1	1	yes	maximal, characteristic
6	#11718492	C₂	1	2	yes	maximal, complemented
7	#11718491	C₂	1	2	yes	maximal, complemented
8	#11718489	C₂	1	4	yes	maximal, complemented
9	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×#6101	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
15	C₂×#6099	C₂²	1	1	yes	characteristic
16	C₂×#6099	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₆	C₄	1	1	yes	characteristic
18	C₄×Dic₂₆	C₄	1	2	yes	—
19	C₄×Dic₂₆	C₄	1	1	yes	characteristic
20	C₄×Dic₂₆	C₂²	1	2	yes	—
21	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
25	#88055	C₂²	1	4	yes	—
26	#88054	C₂²	1	4	yes	—
27	#88054	C₂²	1	4	yes	—
28	C₈×Dic₁₃	C₂²	1	4	yes	—
29	#88043	C₂²	1	2	yes	—
30	C₄×#1055	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
36	C₂×Dic₂₆	—	2	4	no	—
37	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
40	C₁₃×M₁₆	—	2	4	no	—
41	C₂×C₈×C₁₃	C₂³	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	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	#6101	C₂×C₄	1	2	yes	—
50	#6101	C₂×C₄	1	2	yes	—
51	#6101	C₂×C₄	1	2	yes	—
52	#6101	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃	C₂³	1	2	yes	—
56	C₄×Dic₁₃	C₂³	1	2	yes	—
57	#6099	—	2	2	no	—
58	#6099	Q₈	1	4	yes	complemented
59	C₂×#1055	C₂³	1	2	yes	—
60	C₂×#1055	C₂³	1	2	yes	—
61	Q₈×M₁₆	—	13	1	no	maximal, Sylow
62	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	4	1	no	—
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₁₃	#41	1	2	yes	—
72	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₁₃	—	2	4	no	—
74	C₂×Dic₁₃	—	4	2	no	—
75	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
79	Dic₂₆	—	2	2	no	—
80	Dic₂₆	—	2	4	no	—
81	Dic₂₆	M₁₆	1	4	yes	complemented
82	C₈×C₁₃	—	2	4	no	—
83	#1055	—	2	2	no	—
84	#1055	C₂×Q₈	1	4	yes	complemented
85	C₂×C₄×Q₈	—	13	1	no	—
86	#446	—	13	4	no	—
87	#446	—	13	2	no	—
88	C₈×Q₈	—	13	2	no	—
89	#423	—	13	2	no	—
90	#423	D₁₃	1	1	yes	characteristic
91	C₄×M₁₆	—	13	2	no	—
92	C₄×M₁₆	—	13	1	no	—
93	C₂²×C₁₃	—	2	1	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
96	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	C₄×Q₈	1	2	yes	—
101	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
102	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
103	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
104	Dic₁₃	—	2	2	no	—
105	Dic₁₃	—	2	2	no	—
106	Dic₁₃	—	2	2	no	—
107	Dic₁₃	C₂×M₁₆	1	4	yes	—
108	C₂²×Q₈	—	13	1	no	—
109	C₂×M₁₆	—	13	1	no	—
110	C₂×M₁₆	—	13	1	no	—
111	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
112	C₄×Q₈	—	13	1	no	—
113	C₄×Q₈	—	13	2	no	—
114	C₄×Q₈	—	13	1	no	—
115	C₄×Q₈	—	13	1	no	—
116	C₄×Q₈	—	13	2	no	—
117	C₄×Q₈	—	13	1	no	—
118	C₂×C₄⋊C₄	—	13	2	no	—
119	C₂×C₄⋊C₄	—	13	1	no	—
120	C₂×C₄²	—	13	2	no	—
121	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
122	C₄⋊C₈	—	13	4	no	—
123	C₄⋊C₈	—	13	4	no	—
124	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
125	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
126	#97	—	13	4	no	—
127	#97	—	13	2	no	—
128	C₄×C₈	—	13	4	no	—
129	C₄×C₈	—	13	2	no	—
130	C₂×C₁₃	—	2	2	no	—
131	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
133	C₂×C₁₃	#568	1	1	yes	characteristic
134	C₂×Q₈	—	13	1	no	—
135	C₂×Q₈	—	13	2	no	—
136	C₂×Q₈	—	13	1	no	—
137	C₂×Q₈	—	26	4	no	—
138	C₂²×C₄	—	13	2	no	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
141	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
143	M₁₆	—	2	4	no	—
144	M₁₆	—	13	4	no	—
145	M₁₆	—	26	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	13	2	no	—
148	C₂×C₈	C₂²×D₁₃	1	4	yes	—
149	C₄⋊C₄	—	13	2	no	—
150	C₄⋊C₄	—	13	2	no	—
151	C₄⋊C₄	—	13	2	no	—
152	C₄⋊C₄	—	13	2	no	—
153	C₄⋊C₄	—	13	2	no	—
154	C₄⋊C₄	—	13	1	no	—
155	C₄⋊C₄	—	13	1	no	—
156	C₄²	—	13	2	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	13	2	no	—
159	C₄²	—	13	2	no	—
160	C₄²	C₄×D₁₃	1	1	yes	characteristic
161	C₄²	C₄×D₁₃	1	1	yes	characteristic
162	C₄²	C₂²×D₁₃	1	1	yes	characteristic
163	C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	C₁₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
166	Q₈	—	13	4	no	—
167	Q₈	—	26	4	no	—
168	Q₈	—	26	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	13	2	no	—
172	C₂×C₄	—	13	2	no	—
173	C₂×C₄	—	13	2	no	—
174	C₂×C₄	—	13	2	no	—
175	C₂×C₄	—	26	4	no	—
176	C₂×C₄	—	52	2	no	—
177	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂×C₄	#6129	1	2	yes	—
183	C₂×C₄	Q₈×D₁₃	1	2	yes	—
184	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	13	4	no	—
187	C₈	—	26	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#88141	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	13	4	no	—
196	C₄	—	26	2	no	—
197	C₄	—	26	2	no	—
198	C₄	—	26	2	no	—
199	C₄	#88141	1	2	yes	—
200	C₄	M₁₆×D₁₃	1	2	yes	—
201	C₄	C₂×#6129	1	1	yes	characteristic
202	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#88141	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
206	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#423	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#423	D₁₃
hypocenter	C₁₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
6. #11718492	C₂	semidirect
7. #11718491	C₂	semidirect
8. #11718489	C₂	semidirect
58. #6099	Q₈	semidirect
81. Dic₂₆	M₁₆	semidirect
84. #1055	C₂×Q₈	semidirect
164. C₁₃	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆	2
C₁₃×#423	2
#11718779	2
M₁₆×Dic₁₃	2
C₄×#6099	2
#11718492	2
#11718491	2
#11718489	2
Q₈×M₁₆	13