Subgroups — #15881924

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6099	C₂	1	2	yes	maximal
2	#13115612	C₂	1	6	yes	maximal
3	#13114974	C₂	1	3	yes	maximal
4	C₄×#88059	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	C₈×Q₈×C₁₃	C₄	1	6	yes	—
8	C₈×Q₈×C₁₃	C₄	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
11	C₂×#88059	C₂²	1	6	yes	—
12	#11718685	—	2	6	no	—
13	Q₈×#1055	—	2	2	no	—
14	#11718585	C₂²	1	6	yes	—
15	C₄×#6099	C₂²	1	6	yes	—
16	#11718144	C₂²	1	3	yes	—
17	#11718144	C₂²	1	6	yes	—
18	#11718144	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
22	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	D₄	1	3	yes	—
26	C₂×C₁₃×C₄⋊C₄	Q₈	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
28	C₂×#6099	C₂³	1	6	yes	—
29	C₂×#6099	C₂³	1	2	yes	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
34	#88059	—	2	12	no	—
35	C₄⋊#1055	—	2	6	no	—
36	C₄⋊#1055	—	2	6	no	—
37	#88043	—	2	6	no	—
38	C₄×#1055	—	2	6	no	—
39	#22746	—	13	1	no	maximal, Sylow
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	C₄⋊C₄	1	3	yes	—
42	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
46	C₂×C₈×C₁₃	—	2	6	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
50	C₁₃×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
51	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
52	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
53	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
54	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
55	C₄²×C₁₃	C₂×D₄	1	3	yes	—
56	C₄²×C₁₃	C₂×D₄	1	3	yes	—
57	#6099	—	2	2	no	—
58	#6099	—	2	6	no	—
59	#6099	—	2	6	no	—
60	#6099	#41	1	4	yes	—
61	C₂×#1055	—	2	2	no	—
62	C₂×#1055	—	2	6	no	—
63	Q₈×M₁₆	—	13	2	no	—
64	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
65	#1950	—	13	6	no	—
66	#1837	—	13	3	no	—
67	C₄×#108	—	13	3	no	—
68	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
69	Q₈×C₁₃	—	2	6	no	—
70	Q₈×C₁₃	#108	1	4	yes	complemented
71	C₂×C₄×C₁₃	—	2	6	no	—
72	C₂×C₄×C₁₃	—	4	3	no	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
80	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
81	C₈×C₁₃	—	2	6	no	—
82	C₈×C₁₃	—	2	2	no	—
83	#1055	—	2	4	no	—
84	#1055	—	4	6	no	—
85	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
86	#446	—	26	6	no	—
87	C₈×Q₈	—	26	2	no	—
88	C₈×Q₈	Dic₁₃	1	6	yes	—
89	C₈×Q₈	Dic₁₃	1	2	yes	—
90	C₂×#108	—	13	6	no	—
91	#423	—	13	6	no	—
92	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
93	C₄×M₁₆	—	13	6	no	—
94	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
95	#341	—	13	3	no	—
96	#341	—	13	6	no	—
97	#341	—	13	1	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₃	#557	1	2	yes	—
102	C₄×C₁₃	—	2	3	no	—
103	C₄×C₁₃	—	2	3	no	—
104	C₄×C₁₃	—	2	3	no	—
105	C₄×C₁₃	C₂×#108	1	6	yes	—
106	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
107	C₄×C₁₃	#519	1	1	yes	characteristic
108	C₄×C₁₃	#548	1	2	yes	—
109	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
110	C₂×M₁₆	—	13	2	no	—
111	C₂×M₁₆	—	13	6	no	—
112	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
114	C₄×Q₈	D₅₂	1	1	yes	characteristic
115	C₄×Q₈	D₅₂	1	3	yes	—
116	C₄×Q₈	C₂×Dic₁₃	1	3	yes	—
117	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	Dic₂₆	1	3	yes	—
119	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
120	#108	—	26	12	no	—
121	C₄⋊C₈	—	26	6	no	—
122	C₄⋊C₈	—	26	6	no	—
123	C₄⋊C₈	C₂×Dic₁₃	1	6	yes	—
124	C₄⋊C₈	C₂×Dic₁₃	1	6	yes	—
125	#97	—	26	6	no	—
126	C₄×C₈	—	26	6	no	—
127	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
128	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	#2351	1	1	yes	characteristic
131	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
132	C₂×C₁₃	#2916	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	C₄⋊Dic₁₃	1	3	yes	—
135	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
137	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
138	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
139	M₁₆	—	13	4	no	—
140	M₁₆	—	26	2	no	—
141	M₁₆	—	26	6	no	—
142	M₁₆	—	26	6	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	6	no	—
145	C₂×C₈	—	26	2	no	—
146	C₂×C₈	—	26	6	no	—
147	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
148	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
149	C₄⋊C₄	C₄⋊Dic₁₃	1	6	yes	—
150	C₄⋊C₄	C₄⋊Dic₁₃	1	3	yes	—
151	C₄⋊C₄	C₄⋊Dic₁₃	1	3	yes	—
152	C₄²	C₂×D₅₂	1	3	yes	—
153	C₄²	C₂×D₅₂	1	3	yes	—
154	C₄²	C₂²×Dic₁₃	1	3	yes	—
155	C₄²	C₂²×Dic₁₃	1	3	yes	—
156	C₁₃	#22746	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
158	Q₈	—	2	6	no	—
159	Q₈	#88059	1	4	yes	complemented
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	4	3	no	—
162	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
166	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
167	C₂×C₄	#88254	1	2	yes	—
168	C₂×C₄	#41×D₁₃	1	2	yes	—
169	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	6	no	—
172	C₈	—	26	4	no	—
173	C₈	—	52	6	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#11719274	1	2	yes	—
177	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#88059	1	6	yes	—
182	C₄	#11719281	1	2	yes	—
183	C₄	#11719517	1	1	yes	characteristic
184	C₄	#41×Dic₁₃	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13117757	1	1	yes	characteristic
187	C₂	C₂²×#88059	1	1	yes	characteristic
188	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#22746
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
70. Q₈×C₁₃	#108	semidirect
156. C₁₃	#22746	semidirect
159. Q₈	#88059	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6099	2
#13115612	2
#13114974	2
C₄×#88059	2
C₂×C₈×Q₈×C₁₃	2
#22746	13