Subgroups — Q₈×#13133291 #28328438

Order 13568.

530 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	#20291130	C₂	1	6	yes	maximal, complemented
2	Q₈×#11719910	C₂	1	2	yes	maximal, complemented
3	#20291047	C₂	1	3	yes	maximal
4	C₄×#13133291	C₂	1	3	yes	maximal
5	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	3	yes	—
7	C₂×#13133291	C₂²	1	1	yes	characteristic
8	C₄⋊#11719910	C₂²	1	6	yes	—
9	C₄⋊#11719910	C₂²	1	6	yes	—
10	#16811511	C₂²	1	6	yes	—
11	C₄×#11719910	C₂²	1	6	yes	—
12	Q₈×#88322	C₄	1	3	yes	—
13	Q₈×#88322	C₄	1	1	yes	characteristic
14	Q₈×#88322	C₂²	1	1	yes	characteristic
15	Q₈×#88322	C₂²	1	3	yes	—
16	C₂×C₄⋊#88322	C₄	1	3	yes	—
17	C₂×C₄×#88322	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	#13133291	—	2	6	no	—
20	#13133291	Q₈	1	4	yes	direct product factor
21	C₂×#11719910	C₂³	1	6	yes	—
22	C₂×#11719910	C₂³	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	3	yes	—
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	C₄⋊#88322	C₂×C₄	1	3	yes	—
26	C₄⋊#88322	C₂×C₄	1	6	yes	—
27	C₄⋊#88322	C₂×C₄	1	3	yes	—
28	C₄×#88322	C₂×C₄	1	3	yes	—
29	C₄×#88322	C₂×C₄	1	3	yes	—
30	C₄×#88322	C₂³	1	3	yes	—
31	C₄×#88322	C₂³	1	3	yes	—
32	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₈	1	3	yes	—
34	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
36	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	C₈	1	3	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	C₂×C₈	1	3	yes	—
41	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
46	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
48	C₄²×C₅₃	C₂×C₈	1	3	yes	—
49	C₄²×C₅₃	C₂×C₈	1	3	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
52	C₂×#88322	—	2	6	no	—
53	C₂×#88322	C₂²×C₄	1	3	yes	—
54	C₂×#88322	C₂²×C₄	1	3	yes	—
55	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₂×Q₈	1	2	yes	—
57	C₂×#88322	#41	1	2	yes	—
58	C₂×#88322	C₂⁴	1	1	yes	characteristic
59	#11719910	—	2	6	no	—
60	#11719910	C₂×Q₈	1	4	yes	complemented
61	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₅₃	—	2	6	no	—
63	Q₈×C₅₃	M₃₂	1	4	yes	complemented
64	C₂×C₄×C₅₃	—	2	6	no	—
65	C₂×C₄×C₅₃	—	4	3	no	—
66	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₅₃	#131	1	2	yes	—
73	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
74	#88322	—	2	3	no	—
75	#88322	—	2	3	no	—
76	#88322	C₄×Q₈	1	2	yes	—
77	#88322	C₂²×Q₈	1	1	yes	characteristic
78	#88322	C₂×#41	1	1	yes	characteristic
79	Q₈×M₃₂	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
83	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₅₃	—	2	3	no	—
85	C₄×C₅₃	—	2	3	no	—
86	C₄×C₅₃	—	2	3	no	—
87	C₄×C₅₃	C₈×Q₈	1	2	yes	—
88	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
89	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
91	C₂×C₈×Q₈	—	53	1	no	—
92	#2183	—	53	6	no	—
93	Q₈×C₁₆	—	53	2	no	—
94	#2150	—	53	3	no	—
95	C₄×M₃₂	—	53	3	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
98	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
99	C₂×C₅₃	#3407	1	1	yes	characteristic
100	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
101	C₂×M₃₂	—	53	1	no	—
102	C₂×M₃₂	—	53	3	no	—
103	C₈×Q₈	—	53	1	no	—
104	C₈×Q₈	—	53	3	no	—
105	C₈×Q₈	—	53	3	no	—
106	C₈×Q₈	—	53	1	no	—
107	C₂×C₄⋊C₈	—	53	3	no	—
108	C₂×C₄×C₈	—	53	3	no	—
109	C₄⋊C₁₆	—	53	6	no	—
110	C₄⋊C₁₆	—	53	6	no	—
111	#346	—	53	6	no	—
112	C₄×C₁₆	—	53	6	no	—
113	C₅₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	#88324	1	1	yes	characteristic
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	—	53	3	no	—
117	C₄×Q₈	#88324	1	3	yes	—
118	C₄×Q₈	#88324	1	1	yes	characteristic
119	C₄×Q₈	C₂×#6157	1	3	yes	—
120	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
121	C₂×C₄⋊C₄	#88324	1	3	yes	—
122	C₂×C₄²	C₂×#6157	1	3	yes	—
123	M₃₂	—	53	4	no	—
124	M₃₂	—	106	6	no	—
125	C₂×C₁₆	—	53	2	no	—
126	C₂×C₁₆	—	53	6	no	—
127	C₄⋊C₈	—	53	3	no	—
128	C₄⋊C₈	—	53	6	no	—
129	C₄⋊C₈	—	53	3	no	—
130	C₄×C₈	—	53	3	no	—
131	C₄×C₈	—	53	3	no	—
132	C₄×C₈	—	53	3	no	—
133	C₄×C₈	—	53	3	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₂×#88324	1	3	yes	—
136	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
137	C₂²×C₄	C₂×#88324	1	3	yes	—
138	C₂²×C₄	C₂×#88324	1	3	yes	—
139	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	3	no	—
145	C₂×C₈	—	53	3	no	—
146	C₂×C₈	—	106	6	no	—
147	C₄⋊C₄	C₂×#88324	1	6	yes	—
148	C₄⋊C₄	C₂×#88324	1	3	yes	—
149	C₄⋊C₄	C₂×#88324	1	3	yes	—
150	C₄²	C₂×#88324	1	3	yes	—
151	C₄²	C₂×#88324	1	3	yes	—
152	C₄²	C₂²×#6157	1	3	yes	—
153	C₄²	C₂²×#6157	1	3	yes	—
154	C₁₆	—	53	4	no	—
155	C₁₆	—	106	6	no	—
156	C₂³	C₂²×#88324	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#13133291	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	Q₈×#6157	1	2	yes	—
162	C₂×C₄	C₂²×#88324	1	3	yes	—
163	C₂×C₄	C₂²×#88324	1	3	yes	—
164	C₂×C₄	C₂²×#88324	1	3	yes	—
165	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
166	C₂×C₄	C₂²×#88324	1	3	yes	—
167	C₂×C₄	#13133305	1	2	yes	—
168	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
169	C₈	—	53	2	no	—
170	C₈	—	53	1	no	—
171	C₈	—	53	1	no	—
172	C₈	—	106	3	no	—
173	C₈	—	106	3	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×#88324	1	2	yes	—
177	C₂²	C₂³×#88324	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×#88324	1	2	yes	—
182	C₄	C₂×#13133291	1	6	yes	—
183	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
184	C₄	C₂×#13133305	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×#88324	1	1	yes	characteristic
187	C₂	C₂²×#13133291	1	1	yes	characteristic
188	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₅₃	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×Q₈	#6157
hypocenter	C₅₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20291130	C₂	semidirect
2. Q₈×#11719910	C₂	semidirect
20. #13133291	Q₈	direct (N × H)
60. #11719910	C₂×Q₈	semidirect
63. Q₈×C₅₃	M₃₂	semidirect
113. C₅₃	Q₈×M₃₂	semidirect
158. Q₈	#13133291	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291130	2
Q₈×#11719910	2
#20291047	2
C₄×#13133291	2
C₂×Q₈×#88322	2
Q₈×M₃₂	53

Subgroups — Q8×#13133291 #28328438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13133291 #28328438