Subgroups — #15861959

Order 3328.

990 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	2	yes	maximal
2	C₄⋊C₄×#1055	C₂	1	6	yes	maximal
3	C₄×C₄⋊#1055	C₂	1	3	yes	maximal
4	#13114582	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	C₄	1	6	yes	—
9	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
10	C₄³×C₁₃	C₂²	1	3	yes	—
11	Q₈×#1055	C₄	1	16	yes	complemented
12	C₂×C₄⋊#1055	C₂²	1	6	yes	—
13	C₂×C₄×#1055	C₂²	1	6	yes	—
14	C₂×C₄×#1055	C₂²	1	6	yes	—
15	#11718116	C₂²	1	3	yes	—
16	#11718116	C₂²	1	6	yes	—
17	#11718116	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Q₈×C₁₃	C₈	1	16	yes	complemented
20	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
28	C₂²×#1055	C₂³	1	6	yes	—
29	C₂²×#1055	C₂³	1	2	yes	—
30	C₁₃×#95	C₂×C₄	1	6	yes	—
31	C₄⋊#1055	C₂×C₄	1	24	yes	—
32	C₄×#1055	—	4	6	no	—
33	C₄×#1055	C₂×C₄	1	24	yes	—
34	#20569	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
36	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₁₃×C₄⋊C₄	C₄²	1	12	yes	—
43	C₁₃×C₄⋊C₄	C₂×C₈	1	24	yes	—
44	C₄²×C₁₃	—	2	3	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	2	3	no	—
47	C₄²×C₁₃	—	4	6	no	—
48	C₄²×C₁₃	C₂×C₈	1	24	yes	—
49	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
50	C₂×#1055	—	2	6	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	—	2	2	no	—
53	C₂×#1055	—	2	6	no	—
54	C₂×#1055	C₂²×C₄	1	4	yes	—
55	C₂×#1055	C₂²×C₄	1	12	yes	—
56	C₂×C₈×Q₈	—	13	2	no	—
57	C₄²×Q₈	D₁₃	1	1	yes	characteristic
58	C₈×C₄⋊C₄	—	13	6	no	—
59	C₄×C₄⋊C₈	—	13	3	no	—
60	#1744	—	13	3	no	—
61	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
62	Q₈×C₁₃	C₄×C₈	1	16	yes	complemented
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	6	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	6	no	—
67	C₂×C₄×C₁₃	—	2	6	no	—
68	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
69	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
70	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
71	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
72	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
73	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
74	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
75	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
76	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
77	#1055	—	2	4	no	—
78	#1055	—	2	12	no	—
79	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
80	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
81	C₈×Q₈	—	13	16	no	—
82	C₂×C₄⋊C₈	—	13	6	no	—
83	C₂×C₄×C₈	—	13	6	no	—
84	C₂×C₄×C₈	—	13	6	no	—
85	C₄×C₄⋊C₄	Dic₁₃	1	6	yes	—
86	C₄×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
87	C₄³	C₂×D₁₃	1	3	yes	—
88	#336	—	13	1	no	—
89	#336	—	13	3	no	—
90	#336	—	13	6	no	—
91	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
92	C₂²×C₁₃	#519	1	1	yes	characteristic
93	C₂²×C₁₃	#520	1	1	yes	characteristic
94	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
95	C₂²×C₁₃	#568	1	2	yes	—
96	C₄×C₁₃	—	2	6	no	—
97	C₄×C₁₃	—	2	4	no	—
98	C₄×C₁₃	—	2	12	no	—
99	C₄×C₁₃	C₂×C₄×C₈	1	12	yes	—
100	C₄×C₁₃	#519	1	2	yes	—
101	C₄×C₁₃	#520	1	2	yes	—
102	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
103	C₂²×C₈	—	13	2	no	—
104	C₂²×C₈	—	13	6	no	—
105	C₄×Q₈	#1055	1	16	yes	complemented
106	C₄×Q₈	C₄×D₁₃	1	8	yes	—
107	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
108	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
109	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
110	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
111	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
112	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
113	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
114	C₄⋊C₈	—	13	24	no	—
115	C₄×C₈	—	13	24	no	—
116	C₄×C₈	—	52	6	no	—
117	#95	C₂×Dic₁₃	1	6	yes	—
118	C₂×C₁₃	#2276	1	1	yes	characteristic
119	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
120	C₂×C₁₃	#2875	1	1	yes	characteristic
121	C₂×C₁₃	#2982	1	2	yes	—
122	C₂×C₁₃	#2988	1	2	yes	—
123	C₂×Q₈	C₂×#1055	1	8	yes	—
124	C₂×Q₈	C₄×Dic₁₃	1	4	yes	—
125	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
126	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
127	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
128	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
129	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
130	C₂×C₈	—	13	4	no	—
131	C₂×C₈	—	13	12	no	—
132	C₂×C₈	—	26	2	no	—
133	C₂×C₈	—	26	6	no	—
134	C₂×C₈	—	26	6	no	—
135	C₂×C₈	—	26	2	no	—
136	C₄⋊C₄	C₂×#1055	1	24	yes	—
137	C₄⋊C₄	C₄×Dic₁₃	1	12	yes	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	3	no	—
140	C₄²	—	2	3	no	—
141	C₄²	—	4	6	no	—
142	C₄²	C₂×#1055	1	24	yes	—
143	C₄²	C₂×C₄×D₁₃	1	12	yes	—
144	C₁₃	#20569	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
146	Q₈	C₄×#1055	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	C₂²×#1055	1	4	yes	—
153	C₂×C₄	C₂²×#1055	1	12	yes	—
154	C₂×C₄	C₂²×#1055	1	12	yes	—
155	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
156	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
157	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
158	C₂×C₄	#88254	1	2	yes	—
159	C₂×C₄	#88256	1	1	yes	characteristic
160	C₂×C₄	#88257	1	1	yes	characteristic
161	C₈	—	26	4	no	—
162	C₈	—	26	12	no	—
163	C₂²	C₂³×#1055	1	2	yes	—
164	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
165	C₂²	#11719517	1	1	yes	characteristic
166	C₂²	#11719524	1	2	yes	—
167	C₂²	#11719532	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	12	no	—
170	C₄	—	2	6	no	—
171	C₄	C₂×C₄×#1055	1	12	yes	—
172	C₄	#11719277	1	2	yes	—
173	C₄	#11719279	1	2	yes	—
174	C₂	#13117201	1	1	yes	characteristic
175	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
176	C₂	#13123264	1	1	yes	characteristic
177	C₂	#13123917	1	2	yes	—
178	C₂	#13123947	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₄²×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	#20569
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
11. Q₈×#1055	C₄	semidirect
19. C₄×Q₈×C₁₃	C₈	semidirect
62. Q₈×C₁₃	C₄×C₈	semidirect
105. C₄×Q₈	#1055	semidirect
144. C₁₃	#20569	semidirect
146. Q₈	C₄×#1055	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
C₄⋊C₄×#1055	2
C₄×C₄⋊#1055	2
#13114582	2
C₄²×Q₈×C₁₃	2
#20569	13