Subgroups — #13131638

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	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#11719158	C₂	1	2	yes	maximal, complemented
3	#11719078	C₂	1	4	yes	maximal, complemented
4	#11719074	C₂	1	2	yes	maximal, complemented
5	#11719053	C₂	1	2	yes	maximal
6	#11719052	C₂	1	1	yes	maximal, characteristic
7	C₄×#6118	C₂	1	2	yes	maximal
8	C₄×#6118	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×D₁₃	C₄	1	1	yes	characteristic
10	C₂×#6118	C₂²	1	2	yes	—
11	C₂×#6118	C₂²	1	1	yes	characteristic
12	C₂×#6118	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
14	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
16	C₄⋊C₄×D₁₃	C₄	1	2	yes	—
17	#88141	C₂²	1	1	yes	characteristic
18	#88141	C₂²	1	2	yes	—
19	C₄²×D₁₃	C₂²	1	1	yes	characteristic
20	C₄²×D₁₃	C₂²	1	2	yes	—
21	C₄×Dic₂₆	C₄	1	2	yes	—
22	C₄×Dic₂₆	C₄	1	1	yes	characteristic
23	#88112	C₂²	1	4	yes	—
24	#88112	C₂²	1	4	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	2	yes	—
29	C₄×#1057	C₂²	1	4	yes	—
30	C₄×#1057	C₂²	1	2	yes	—
31	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	Q₈×D₁₃	—	2	4	no	—
33	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
38	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
40	C₂×#1057	C₂³	1	4	yes	—
41	C₂×#1057	C₂³	1	2	yes	—
42	C₂×#1057	C₂³	1	2	yes	—
43	#6118	—	2	4	no	—
44	#6118	—	2	2	no	—
45	#6118	Q₈	1	4	yes	complemented
46	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×C₄	1	2	yes	—
49	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
52	#6101	C₂×C₄	1	2	yes	—
53	#6101	C₂×C₄	1	2	yes	—
54	#6101	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
60	C₄×Dic₁₃	C₂³	1	2	yes	—
61	Q₈×M₁₆	—	13	1	no	maximal, Sylow
62	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
63	Q₈×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
71	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
74	C₄×D₁₃	—	2	2	no	—
75	C₄×D₁₃	—	2	4	no	—
76	C₄×D₁₃	—	4	1	no	—
77	C₄×D₁₃	—	4	2	no	—
78	C₄×D₁₃	C₂×Q₈	1	2	yes	—
79	C₄×D₁₃	#41	1	2	yes	—
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	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₃	—	2	1	no	—
95	C₂×D₁₃	—	2	1	no	—
96	C₂×D₁₃	C₄×Q₈	1	2	yes	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	C₄×Q₈	1	2	yes	—
101	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
102	Dic₁₃	—	2	1	no	—
103	Dic₁₃	—	2	1	no	—
104	Dic₁₃	—	2	2	no	—
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	1	no	—
114	C₄×Q₈	—	13	2	no	—
115	C₄×Q₈	—	13	1	no	—
116	C₄×Q₈	—	13	2	no	—
117	C₄×Q₈	#261	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	13	2	no	—
119	C₂×C₄⋊C₄	—	13	1	no	—
120	C₂×C₄²	—	13	1	no	—
121	C₂×C₄²	—	13	2	no	—
122	C₄⋊C₈	—	13	2	no	—
123	C₄⋊C₈	—	13	2	no	—
124	C₄⋊C₈	—	13	4	no	—
125	C₄⋊C₈	—	13	4	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	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
132	C₂×C₁₃	#568	1	1	yes	characteristic
133	D₁₃	—	2	2	no	—
134	C₂×Q₈	—	13	1	no	—
135	C₂×Q₈	—	13	2	no	—
136	C₂×Q₈	—	26	4	no	—
137	C₂×Q₈	C₂×#261	1	1	yes	characteristic
138	C₂²×C₄	—	13	1	no	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	—	13	1	no	—
141	C₂²×C₄	—	13	1	no	—
142	C₂²×C₄	—	13	2	no	—
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	—	13	1	no	—
154	C₄⋊C₄	C₂×#261	1	2	yes	—
155	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
156	C₄²	—	13	2	no	—
157	C₄²	—	13	1	no	—
158	C₄²	—	13	2	no	—
159	C₄²	—	13	1	no	—
160	C₄²	—	13	2	no	—
161	C₄²	—	13	1	no	—
162	C₄²	C₂×#261	1	2	yes	—
163	C₄²	C₂×#261	1	1	yes	characteristic
164	C₁₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	13	1	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	13	4	no	—
168	Q₈	—	26	4	no	—
169	C₂×C₄	—	13	1	no	—
170	C₂×C₄	—	13	2	no	—
171	C₂×C₄	—	13	2	no	—
172	C₂×C₄	—	13	2	no	—
173	C₂×C₄	—	13	1	no	—
174	C₂×C₄	—	13	2	no	—
175	C₂×C₄	—	13	1	no	—
176	C₂×C₄	—	26	4	no	—
177	C₂×C₄	—	26	2	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	—	52	1	no	—
180	C₂×C₄	C₂²×#261	1	2	yes	—
181	C₂×C₄	C₂²×#261	1	1	yes	characteristic
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	—	13	2	no	—
189	C₂²	—	26	1	no	—
190	C₂²	—	26	1	no	—
191	C₂²	C₂³×#261	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	13	4	no	—
196	C₄	—	13	1	no	—
197	C₄	—	13	1	no	—
198	C₄	—	26	1	no	—
199	C₄	—	26	1	no	—
200	C₄	—	26	2	no	—
201	C₄	D₁₃⋊M₁₆	1	2	yes	—
202	C₄	Q₈×#261	1	2	yes	—
203	C₂	—	26	2	no	—
204	C₂	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
205	C₂	C₂×Q₈×#261	1	1	yes	characteristic
206	C₂	#11719629	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×Q₈×C₁₃	C₄
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₄×Q₈	#261
hypocenter	C₁₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #11719158	C₂	semidirect
3. #11719078	C₂	semidirect
4. #11719074	C₂	semidirect
45. #6118	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
C₄×Q₈×D₁₃	2
#11719158	2
#11719078	2
#11719074	2
#11719053	2
#11719052	2
C₄×#6118	2
Q₈×M₁₆	13