Subgroups — #13131637

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×#88141	C₂	1	1	yes	maximal, characteristic
2	#11719129	C₂	1	1	yes	maximal, characteristic
3	#11719129	C₂	1	2	yes	maximal
4	C₄×#6122	C₂	1	2	yes	maximal
5	C₄×#6122	C₂	1	1	yes	maximal, characteristic
6	#11719080	C₂	1	2	yes	maximal, complemented
7	#11719078	C₂	1	4	yes	maximal, complemented
8	#11719074	C₂	1	2	yes	maximal, complemented
9	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
10	C₂×#6122	C₂²	1	2	yes	—
11	C₂×#6122	C₂²	1	1	yes	characteristic
12	C₂×#6122	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₂×#6101	C₄	1	2	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
17	#88141	C₄	1	2	yes	—
18	#88141	C₄	1	1	yes	characteristic
19	#88141	C₄	1	1	yes	characteristic
20	#88141	C₂²	1	1	yes	characteristic
21	#88141	C₂²	1	2	yes	—
22	#88141	C₂²	1	1	yes	characteristic
23	#88112	C₂²	1	4	yes	—
24	#88112	C₂²	1	2	yes	—
25	#88110	C₂²	1	2	yes	—
26	#88110	C₂²	1	4	yes	—
27	C₄⋊#1057	C₂²	1	2	yes	—
28	C₄⋊#1057	C₂²	1	4	yes	—
29	C₄×#1057	C₂²	1	4	yes	—
30	C₄×#1057	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
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	#6122	—	2	4	no	—
41	#6122	—	2	2	no	—
42	#6122	Q₈	1	4	yes	complemented
43	C₂×#1057	C₂³	1	4	yes	—
44	C₂×#1057	C₂³	1	2	yes	—
45	C₂×#1057	C₂³	1	2	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₁₃×C₄⋊C₄	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	1	yes	characteristic
54	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₄×Dic₁₃	C₂³	1	2	yes	—
59	C₄×Dic₁₃	C₂³	1	2	yes	—
60	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
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	2	no	—
65	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₁₃	—	2	4	no	—
70	C₂×Dic₁₃	—	4	1	no	—
71	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
77	C₂×Dic₁₃	#41	1	2	yes	—
78	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
79	Dic₂₆	—	2	2	no	—
80	Dic₂₆	—	2	4	no	—
81	Dic₂₆	M₁₆	1	4	yes	complemented
82	#1057	—	2	4	no	—
83	#1057	—	2	2	no	—
84	#1057	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	1	no	—
90	#423	—	13	2	no	—
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	2	no	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
101	Dic₁₃	—	2	2	no	—
102	Dic₁₃	—	2	1	no	—
103	Dic₁₃	—	2	1	no	—
104	Dic₁₃	C₄×Q₈	1	2	yes	—
105	Dic₁₃	C₂×M₁₆	1	4	yes	—
106	Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
107	Dic₁₃	C₂×#41	1	1	yes	characteristic
108	C₂²×Q₈	—	13	1	no	—
109	C₂×M₁₆	—	13	1	no	—
110	C₂×M₁₆	—	13	2	no	—
111	C₂×M₁₆	—	13	1	no	—
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	#261	1	1	yes	characteristic
120	C₂×C₄²	—	13	2	no	—
121	C₂×C₄²	—	13	1	no	—
122	C₄⋊C₈	—	13	2	no	—
123	C₄⋊C₈	—	13	4	no	—
124	C₄⋊C₈	—	13	4	no	—
125	C₄⋊C₈	—	13	2	no	—
126	#97	—	13	2	no	—
127	#97	—	13	4	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	1	no	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	—	13	1	no	—
141	C₂²×C₄	C₂×#261	1	1	yes	characteristic
142	C₂²×C₄	C₂×#261	1	2	yes	—
143	M₁₆	—	13	4	no	—
144	M₁₆	—	26	4	no	—
145	M₁₆	—	26	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	—	13	2	no	—
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	C₂×#261	1	2	yes	—
154	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
155	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
156	C₄²	—	13	2	no	—
157	C₄²	—	13	1	no	—
158	C₄²	—	13	1	no	—
159	C₄²	—	13	2	no	—
160	C₄²	—	13	2	no	—
161	C₄²	—	13	1	no	—
162	C₄²	—	13	1	no	—
163	C₄²	—	13	2	no	—
164	C₁₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂²×#261	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	2	no	—
171	C₂×C₄	—	13	2	no	—
172	C₂×C₄	—	13	2	no	—
173	C₂×C₄	—	13	1	no	—
174	C₂×C₄	—	13	1	no	—
175	C₂×C₄	—	13	2	no	—
176	C₂×C₄	—	13	1	no	—
177	C₂×C₄	—	13	1	no	—
178	C₂×C₄	—	13	2	no	—
179	C₂×C₄	—	26	4	no	—
180	C₂×C₄	—	52	1	no	—
181	C₂×C₄	C₂²×#261	1	2	yes	—
182	C₂×C₄	C₂²×#261	1	2	yes	—
183	C₂×C₄	C₂²×#261	1	1	yes	characteristic
184	C₂×C₄	C₂²×#261	1	1	yes	characteristic
185	C₈	—	13	4	no	—
186	C₈	—	26	4	no	—
187	C₈	—	26	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	Q₈×#261	1	2	yes	—
191	C₂²	C₂³×#261	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	13	4	no	—
196	C₄	—	13	2	no	—
197	C₄	—	13	1	no	—
198	C₄	—	13	1	no	—
199	C₄	—	26	2	no	—
200	C₄	—	26	1	no	—
201	C₄	—	26	1	no	—
202	C₄	D₁₃⋊M₁₆	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
205	C₂	#11719621	1	1	yes	characteristic
206	C₂	C₂×Q₈×#261	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₄⋊C₄	C₄
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₄	#261
hypocenter	C₁₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #11719080	C₂	semidirect
7. #11719078	C₂	semidirect
8. #11719074	C₂	semidirect
42. #6122	Q₈	semidirect
81. Dic₂₆	M₁₆	semidirect
84. #1057	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×#88141	2
#11719129	2
C₄×#6122	2
#11719080	2
#11719078	2
#11719074	2
Q₈×M₁₆	13