Subgroups — Q₈×C₁₃×M₃₂ #15719816

Order 3328.

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	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2183	C₂	1	6	yes	maximal, complemented
3	Q₈×C₁₃×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2150	C₂	1	3	yes	maximal
5	C₄×C₁₃×M₃₂	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	3	yes	—
8	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₁₃	C₄	1	3	yes	—
10	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₈×Q₈×C₁₃	C₂²	1	3	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₄	1	3	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
15	C₁₃×C₄⋊C₁₆	C₂²	1	6	yes	—
16	C₁₃×C₄⋊C₁₆	C₂²	1	6	yes	—
17	C₁₃×#346	C₂²	1	6	yes	—
18	C₄×C₁₃×C₁₆	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	C₈	1	3	yes	—
23	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×C₄⋊C₄	C₈	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
28	C₁₃×M₃₂	—	2	6	no	—
29	C₁₃×M₃₂	Q₈	1	4	yes	direct product factor
30	C₂×C₁₃×C₁₆	C₂³	1	6	yes	—
31	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
32	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
34	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₁₃	C₂³	1	3	yes	—
38	C₄×C₈×C₁₃	C₂³	1	3	yes	—
39	Q₈×M₃₂	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	C₂×C₈	1	3	yes	—
42	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	—	2	6	no	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₃	#41	1	2	yes	—
52	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
54	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
55	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
56	C₄²×C₁₃	C₂×C₈	1	3	yes	—
57	C₄²×C₁₃	C₂×C₈	1	3	yes	—
58	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
60	C₁₃×C₁₆	—	2	6	no	—
61	C₁₃×C₁₆	C₂×Q₈	1	4	yes	complemented
62	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic
63	#2183	C₂×C₁₃	1	6	yes	complemented
64	Q₈×C₁₆	C₂×C₁₃	1	2	yes	complemented
65	#2150	C₂×C₁₃	1	3	yes	—
66	C₄×M₃₂	C₂×C₁₃	1	3	yes	—
67	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₁₃	—	2	6	no	—
69	Q₈×C₁₃	M₃₂	1	4	yes	direct product factor
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	—	4	3	no	—
72	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₁₃	—	2	3	no	—
81	C₈×C₁₃	—	2	3	no	—
82	C₈×C₁₃	C₄×Q₈	1	2	yes	—
83	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×Q₈	C₄×C₁₃	1	1	yes	characteristic
86	C₂×M₃₂	C₂²×C₁₃	1	3	yes	—
87	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
88	C₈×Q₈	C₄×C₁₃	1	3	yes	—
89	C₈×Q₈	C₄×C₁₃	1	1	yes	characteristic
90	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
91	C₈×Q₈	C₂²×C₁₃	1	3	yes	—
92	C₂×C₄⋊C₈	C₄×C₁₃	1	3	yes	—
93	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
94	C₄⋊C₁₆	C₂²×C₁₃	1	6	yes	—
95	C₄⋊C₁₆	C₂²×C₁₃	1	6	yes	—
96	#346	C₂²×C₁₃	1	6	yes	—
97	C₄×C₁₆	C₂²×C₁₃	1	6	yes	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
101	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₁₃	—	2	3	no	—
103	C₄×C₁₃	—	2	3	no	—
104	C₄×C₁₃	—	2	3	no	—
105	C₄×C₁₃	C₈×Q₈	1	2	yes	—
106	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
107	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
109	C₂²×Q₈	C₈×C₁₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×C₄×C₁₃	1	3	yes	—
111	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
112	C₄×Q₈	C₈×C₁₃	1	3	yes	—
113	C₄×Q₈	C₈×C₁₃	1	1	yes	characteristic
114	C₄×Q₈	C₂×C₄×C₁₃	1	3	yes	—
115	C₄×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₈×C₁₃	1	3	yes	—
117	C₂×C₄²	C₂×C₄×C₁₃	1	3	yes	—
118	M₃₂	—	2	6	no	—
119	M₃₂	Q₈×C₁₃	1	4	yes	direct product factor
120	C₂×C₁₆	C₂³×C₁₃	1	6	yes	—
121	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
122	C₄⋊C₈	C₂×C₄×C₁₃	1	3	yes	—
123	C₄⋊C₈	C₂×C₄×C₁₃	1	6	yes	—
124	C₄⋊C₈	C₂×C₄×C₁₃	1	3	yes	—
125	C₄×C₈	C₂×C₄×C₁₃	1	3	yes	—
126	C₄×C₈	C₂×C₄×C₁₃	1	3	yes	—
127	C₄×C₈	C₂³×C₁₃	1	3	yes	—
128	C₄×C₈	C₂³×C₁₃	1	3	yes	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
131	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
132	C₂×C₁₃	#3407	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	C₂×C₈×C₁₃	1	3	yes	—
135	C₂×Q₈	C₂×C₈×C₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₈×C₁₃	1	3	yes	—
137	C₂²×C₄	C₂×C₈×C₁₃	1	3	yes	—
138	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
139	C₂×C₈	—	2	6	no	—
140	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
141	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
142	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
144	C₂×C₈	C₁₃×#41	1	2	yes	—
145	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₈×C₁₃	1	6	yes	—
147	C₄⋊C₄	C₂×C₈×C₁₃	1	3	yes	—
148	C₄⋊C₄	C₂×C₈×C₁₃	1	3	yes	—
149	C₄²	C₂×C₈×C₁₃	1	3	yes	—
150	C₄²	C₂×C₈×C₁₃	1	3	yes	—
151	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
152	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
153	C₁₆	—	2	6	no	—
154	C₁₆	C₂×Q₈×C₁₃	1	4	yes	complemented
155	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	C₁₃×M₃₂	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₁₃	1	3	yes	—
163	C₂×C₄	C₂²×C₈×C₁₃	1	3	yes	—
164	C₂×C₄	C₂²×C₈×C₁₃	1	3	yes	—
165	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₈×C₁₃	1	3	yes	—
167	C₂×C₄	C₁₃×#131	1	2	yes	—
168	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	C₄×Q₈×C₁₃	1	2	yes	—
172	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
173	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₈×Q₈×C₁₃	1	2	yes	—
177	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₈×Q₈×C₁₃	1	2	yes	—
182	C₄	C₂×C₁₃×M₃₂	1	6	yes	—
183	C₄	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
184	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×C₈×Q₈×C₁₃	1	1	yes	characteristic
187	C₂	C₂²×C₁₃×M₃₂	1	1	yes	characteristic
188	C₂	C₁₃×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂³×C₁₃	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2183	C₂	semidirect
3. Q₈×C₁₃×C₁₆	C₂	semidirect
29. C₁₃×M₃₂	Q₈	direct (N × H)
39. Q₈×M₃₂	C₁₃	direct (N × H)
61. C₁₃×C₁₆	C₂×Q₈	semidirect
63. #2183	C₂×C₁₃	semidirect
64. Q₈×C₁₆	C₂×C₁₃	semidirect
69. Q₈×C₁₃	M₃₂	direct (N × H)
119. M₃₂	Q₈×C₁₃	direct (N × H)
154. C₁₆	C₂×Q₈×C₁₃	semidirect
155. C₁₃	Q₈×M₃₂	direct (N × H)
158. Q₈	C₁₃×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₁₃	2
C₁₃×#2183	2
Q₈×C₁₃×C₁₆	2
C₁₃×#2150	2
C₄×C₁₃×M₃₂	2
Q₈×M₃₂	13

Subgroups — Q8×C13×M32 #15719816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₃×M₃₂ #15719816