Subgroups — #15881871

Order 3328.

822 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
2	#13115595	C₂	1	6	yes	maximal
3	#13114964	C₂	1	3	yes	maximal
4	C₄×#88058	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	—	2	2	no	—
8	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
11	C₂×#88058	C₂²	1	6	yes	—
12	#11718646	—	2	6	no	—
13	#11718596	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	6	yes	—
15	#11718143	C₂²	1	3	yes	—
16	#11718143	C₂²	1	6	yes	—
17	#11718143	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
21	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
22	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
23	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
25	Q₈×Dic₁₃	—	4	8	no	—
26	C₂×C₄⋊Dic₁₃	—	2	6	no	—
27	C₂×C₄⋊Dic₁₃	—	2	6	no	—
28	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
29	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
30	C₂×C₄×Dic₁₃	—	2	6	no	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
33	#88071	—	2	6	no	—
34	#88058	—	2	12	no	—
35	#22740	—	13	1	no	maximal, Sylow
36	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
37	C₂×Q₈×C₁₃	D₈	1	4	yes	—
38	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₂²×Dic₁₃	—	2	2	no	—
43	C₂²×Dic₁₃	—	2	6	no	—
44	C₂×C₈×C₁₃	—	2	6	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
48	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
49	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
50	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
51	C₄⋊Dic₁₃	—	2	2	no	—
52	C₄⋊Dic₁₃	—	2	6	no	—
53	C₄⋊Dic₁₃	—	2	6	no	—
54	C₄⋊Dic₁₃	—	4	12	no	—
55	C₄⋊Dic₁₃	—	8	6	no	—
56	C₄⋊Dic₁₃	#41	1	4	yes	—
57	C₄×Dic₁₃	—	4	12	no	—
58	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
59	Q₈×C₄⋊C₄	—	13	2	no	—
60	#1947	—	13	6	no	—
61	#1834	—	13	3	no	—
62	C₄×#107	—	13	3	no	—
63	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
64	Q₈×C₁₃	#107	1	16	yes	complemented
65	C₂×C₄×C₁₃	—	2	6	no	—
66	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
69	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
70	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
73	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
74	C₂×Dic₁₃	—	2	4	no	—
75	C₂×Dic₁₃	—	4	2	no	—
76	C₂×Dic₁₃	—	4	2	no	—
77	C₂×Dic₁₃	—	4	6	no	—
78	C₂×Dic₁₃	—	4	6	no	—
79	C₂×Dic₁₃	—	4	6	no	—
80	C₈×C₁₃	—	2	6	no	—
81	C₈×C₁₃	—	2	2	no	—
82	C₂×C₄×Q₈	—	26	2	no	—
83	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
84	C₈×Q₈	Dic₁₃	1	8	yes	complemented
85	C₂×#107	—	13	6	no	—
86	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
87	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
88	#389	—	26	6	no	—
89	#384	—	13	6	no	—
90	C₄×C₄⋊C₄	—	13	6	no	—
91	#340	—	13	3	no	—
92	#340	—	13	6	no	—
93	#340	—	13	1	no	—
94	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₁₃	#548	1	2	yes	—
96	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
97	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
98	C₂²×C₁₃	#577	1	2	yes	—
99	C₄×C₁₃	—	2	6	no	—
100	C₄×C₁₃	C₂×#107	1	12	yes	—
101	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
102	C₄×C₁₃	#519	1	1	yes	characteristic
103	C₄×C₁₃	#557	1	2	yes	—
104	Dic₁₃	—	4	4	no	—
105	Dic₁₃	—	8	6	no	—
106	C₂²×Q₈	D₅₂	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
109	C₄×Q₈	—	52	8	no	—
110	C₄×Q₈	Dic₂₆	1	4	yes	—
111	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
112	C₂×C₄⋊C₄	—	13	2	no	—
113	C₂×C₄⋊C₄	—	13	6	no	—
114	C₂×C₄⋊C₄	—	26	6	no	—
115	C₂×C₄⋊C₄	—	26	6	no	—
116	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
117	C₂×C₄²	—	26	6	no	—
118	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
119	#107	—	26	12	no	—
120	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
121	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
122	#95	—	26	6	no	—
123	C₂×C₁₃	#2351	1	1	yes	characteristic
124	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
125	C₂×C₁₃	#2912	1	1	yes	characteristic
126	C₂×C₁₃	#3358	1	2	yes	—
127	C₂×C₁₃	#3363	1	2	yes	—
128	C₂×Q₈	D₁₀₄	1	4	yes	—
129	C₂×Q₈	Dic₅₂	1	4	yes	—
130	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
131	C₂²×C₄	—	26	2	no	—
132	C₂²×C₄	—	26	6	no	—
133	C₂²×C₄	C₂×D₅₂	1	3	yes	—
134	C₂²×C₄	C₂×D₅₂	1	3	yes	—
135	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
139	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
140	C₄⋊C₄	—	13	4	no	—
141	C₄⋊C₄	—	26	2	no	—
142	C₄⋊C₄	—	26	6	no	—
143	C₄⋊C₄	—	26	6	no	—
144	C₄⋊C₄	—	52	12	no	—
145	C₄⋊C₄	—	104	6	no	—
146	C₄⋊C₄	C₄⋊Dic₁₃	1	12	yes	—
147	C₄²	—	52	12	no	—
148	C₄²	C₂×Dic₂₆	1	6	yes	—
149	C₄²	C₂²×Dic₁₃	1	6	yes	—
150	C₁₃	#22740	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂²×D₅₂	1	1	yes	characteristic
152	Q₈	#88058	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	—	52	2	no	—
156	C₂×C₄	—	52	2	no	—
157	C₂×C₄	—	52	6	no	—
158	C₂×C₄	—	52	6	no	—
159	C₂×C₄	—	52	6	no	—
160	C₂×C₄	C₂×D₁₀₄	1	6	yes	—
161	C₂×C₄	C₂×Dic₅₂	1	6	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
164	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
165	C₂×C₄	#88254	1	2	yes	—
166	C₂×C₄	#41×D₁₃	1	2	yes	—
167	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	6	no	—
170	C₂²	#11719281	1	2	yes	—
171	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
172	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
173	C₂²	#11719456	1	2	yes	—
174	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
175	C₄	—	2	6	no	—
176	C₄	—	52	4	no	—
177	C₄	—	104	6	no	—
178	C₄	C₂×#88058	1	12	yes	—
179	C₄	#11719274	1	2	yes	—
180	C₄	#11719517	1	1	yes	characteristic
181	C₄	#41×Dic₁₃	1	1	yes	characteristic
182	C₂	#13117757	1	1	yes	characteristic
183	C₂	C₂²×#88058	1	1	yes	characteristic
184	C₂	#13123431	1	1	yes	characteristic
185	C₂	#13127472	1	2	yes	—
186	C₂	#13127515	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#22740
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₃	C₄	semidirect
64. Q₈×C₁₃	#107	semidirect
84. C₈×Q₈	Dic₁₃	semidirect
150. C₁₃	#22740	semidirect
152. Q₈	#88058	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₁₃	2
#13115595	2
#13114964	2
C₄×#88058	2
C₂×C₈×Q₈×C₁₃	2
#22740	13