Subgroups — Q₈×C₂₉×M₃₂ #20328377

Order 7424.

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×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#2183	C₂	1	6	yes	maximal, complemented
3	Q₈×C₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
4	C₂₉×#2150	C₂	1	3	yes	maximal
5	C₄×C₂₉×M₃₂	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₂₉	C₄	1	1	yes	characteristic
7	C₂×C₂₉×M₃₂	C₂²	1	3	yes	—
8	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₂₉	C₄	1	3	yes	—
10	C₈×Q₈×C₂₉	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₂₉	C₂²	1	1	yes	characteristic
12	C₈×Q₈×C₂₉	C₂²	1	3	yes	—
13	C₂×C₂₉×C₄⋊C₈	C₄	1	3	yes	—
14	C₂×C₄×C₈×C₂₉	C₂²	1	3	yes	—
15	C₂₉×C₄⋊C₁₆	C₂²	1	6	yes	—
16	C₂₉×C₄⋊C₁₆	C₂²	1	6	yes	—
17	C₂₉×#346	C₂²	1	6	yes	—
18	C₄×C₁₆×C₂₉	C₂²	1	6	yes	—
19	C₂²×Q₈×C₂₉	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₂₉	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₂₉	C₈	1	3	yes	—
23	C₄×Q₈×C₂₉	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₂₉	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₂₉	C₈	1	3	yes	—
27	C₂×C₄²×C₂₉	C₂×C₄	1	3	yes	—
28	C₂₉×M₃₂	—	2	6	no	—
29	C₂₉×M₃₂	Q₈	1	4	yes	direct product factor
30	C₂×C₁₆×C₂₉	C₂³	1	6	yes	—
31	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
32	C₂₉×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₂₉×C₄⋊C₈	C₂×C₄	1	6	yes	—
34	C₂₉×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₂₉	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₂₉	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₂₉	C₂³	1	3	yes	—
38	C₄×C₈×C₂₉	C₂³	1	3	yes	—
39	C₂×Q₈×C₂₉	—	2	4	no	—
40	C₂×Q₈×C₂₉	C₂×C₈	1	3	yes	—
41	C₂×Q₈×C₂₉	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₂₉	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₉	—	2	6	no	—
46	C₂×C₈×C₂₉	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₂₉	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₂₉	#41	1	2	yes	—
51	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₂₉	C₂×C₈	1	6	yes	—
53	C₄⋊C₄×C₂₉	C₂×C₈	1	3	yes	—
54	C₄⋊C₄×C₂₉	C₂×C₈	1	3	yes	—
55	C₄²×C₂₉	C₂×C₈	1	3	yes	—
56	C₄²×C₂₉	C₂×C₈	1	3	yes	—
57	C₄²×C₂₉	C₂²×C₄	1	3	yes	—
58	C₄²×C₂₉	C₂²×C₄	1	3	yes	—
59	C₁₆×C₂₉	—	2	6	no	—
60	C₁₆×C₂₉	C₂×Q₈	1	4	yes	complemented
61	Q₈×M₃₂	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₉	C₂²×C₈	1	1	yes	characteristic
63	Q₈×C₂₉	—	2	6	no	—
64	Q₈×C₂₉	M₃₂	1	4	yes	direct product factor
65	C₂×C₄×C₂₉	—	2	6	no	—
66	C₂×C₄×C₂₉	—	4	3	no	—
67	C₂×C₄×C₂₉	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₂₉	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₂₉	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₂₉	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₂₉	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₂₉	#131	1	2	yes	—
74	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₂₉	—	2	3	no	—
76	C₈×C₂₉	—	2	3	no	—
77	C₈×C₂₉	C₄×Q₈	1	2	yes	—
78	C₈×C₂₉	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₂₉	C₂×#41	1	1	yes	characteristic
80	C₂×C₈×Q₈	C₂×C₂₉	1	1	yes	characteristic
81	#2183	C₂×C₂₉	1	6	yes	complemented
82	Q₈×C₁₆	C₂×C₂₉	1	2	yes	complemented
83	#2150	C₂×C₂₉	1	3	yes	—
84	C₄×M₃₂	C₂×C₂₉	1	3	yes	—
85	C₂²×C₂₉	—	2	1	no	—
86	C₂²×C₂₉	—	2	1	no	—
87	C₂²×C₂₉	C₈×Q₈	1	2	yes	—
88	C₂²×C₂₉	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₂₉	—	2	3	no	—
90	C₄×C₂₉	—	2	3	no	—
91	C₄×C₂₉	—	2	3	no	—
92	C₄×C₂₉	C₈×Q₈	1	2	yes	—
93	C₄×C₂₉	C₂×M₃₂	1	6	yes	—
94	C₄×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₂₉	C₂×#131	1	1	yes	characteristic
96	C₂×C₄×Q₈	C₄×C₂₉	1	1	yes	characteristic
97	C₂×M₃₂	C₂²×C₂₉	1	3	yes	—
98	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
99	C₈×Q₈	C₄×C₂₉	1	3	yes	—
100	C₈×Q₈	C₄×C₂₉	1	1	yes	characteristic
101	C₈×Q₈	C₂²×C₂₉	1	1	yes	characteristic
102	C₈×Q₈	C₂²×C₂₉	1	3	yes	—
103	C₂×C₄⋊C₈	C₄×C₂₉	1	3	yes	—
104	C₂×C₄×C₈	C₂²×C₂₉	1	3	yes	—
105	C₄⋊C₁₆	C₂²×C₂₉	1	6	yes	—
106	C₄⋊C₁₆	C₂²×C₂₉	1	6	yes	—
107	#346	C₂²×C₂₉	1	6	yes	—
108	C₄×C₁₆	C₂²×C₂₉	1	6	yes	—
109	C₂×C₂₉	—	2	2	no	—
110	C₂×C₂₉	C₂×C₈×Q₈	1	1	yes	characteristic
111	C₂×C₂₉	C₂²×M₃₂	1	1	yes	characteristic
112	C₂×C₂₉	#3407	1	1	yes	characteristic
113	C₂²×Q₈	C₈×C₂₉	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₄×C₂₉	1	3	yes	—
115	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
116	C₄×Q₈	C₈×C₂₉	1	3	yes	—
117	C₄×Q₈	C₈×C₂₉	1	1	yes	characteristic
118	C₄×Q₈	C₂×C₄×C₂₉	1	3	yes	—
119	C₄×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₈×C₂₉	1	3	yes	—
121	C₂×C₄²	C₂×C₄×C₂₉	1	3	yes	—
122	M₃₂	—	2	6	no	—
123	M₃₂	Q₈×C₂₉	1	4	yes	direct product factor
124	C₂×C₁₆	C₂³×C₂₉	1	6	yes	—
125	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
126	C₄⋊C₈	C₂×C₄×C₂₉	1	3	yes	—
127	C₄⋊C₈	C₂×C₄×C₂₉	1	6	yes	—
128	C₄⋊C₈	C₂×C₄×C₂₉	1	3	yes	—
129	C₄×C₈	C₂×C₄×C₂₉	1	3	yes	—
130	C₄×C₈	C₂×C₄×C₂₉	1	3	yes	—
131	C₄×C₈	C₂³×C₂₉	1	3	yes	—
132	C₄×C₈	C₂³×C₂₉	1	3	yes	—
133	C₂₉	Q₈×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₂×C₈×C₂₉	1	3	yes	—
136	C₂×Q₈	C₂×C₈×C₂₉	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₂₉	1	3	yes	—
138	C₂²×C₄	C₂×C₈×C₂₉	1	3	yes	—
139	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
140	C₂×C₈	—	2	6	no	—
141	C₂×C₈	C₂²×C₄×C₂₉	1	3	yes	—
142	C₂×C₈	C₂²×C₄×C₂₉	1	3	yes	—
143	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
145	C₂×C₈	#41×C₂₉	1	2	yes	—
146	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₈×C₂₉	1	6	yes	—
148	C₄⋊C₄	C₂×C₈×C₂₉	1	3	yes	—
149	C₄⋊C₄	C₂×C₈×C₂₉	1	3	yes	—
150	C₄²	C₂×C₈×C₂₉	1	3	yes	—
151	C₄²	C₂×C₈×C₂₉	1	3	yes	—
152	C₄²	C₂²×C₄×C₂₉	1	3	yes	—
153	C₄²	C₂²×C₄×C₂₉	1	3	yes	—
154	C₁₆	—	2	6	no	—
155	C₁₆	C₂×Q₈×C₂₉	1	4	yes	complemented
156	C₂³	C₂²×C₈×C₂₉	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	C₂₉×M₃₂	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₄×Q₈×C₂₉	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₂₉	1	3	yes	—
163	C₂×C₄	C₂²×C₈×C₂₉	1	3	yes	—
164	C₂×C₄	C₂²×C₈×C₂₉	1	3	yes	—
165	C₂×C₄	C₂²×C₈×C₂₉	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₈×C₂₉	1	3	yes	—
167	C₂×C₄	C₂₉×#131	1	2	yes	—
168	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	C₄×Q₈×C₂₉	1	2	yes	—
172	C₈	C₂²×Q₈×C₂₉	1	1	yes	characteristic
173	C₈	C₂×#41×C₂₉	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₈×Q₈×C₂₉	1	2	yes	—
177	C₂²	C₂³×C₈×C₂₉	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₈×Q₈×C₂₉	1	2	yes	—
182	C₄	C₂×C₂₉×M₃₂	1	6	yes	—
183	C₄	C₂×C₄×Q₈×C₂₉	1	1	yes	characteristic
184	C₄	C₂×C₂₉×#131	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×C₈×Q₈×C₂₉	1	1	yes	characteristic
187	C₂	C₂²×C₂₉×M₃₂	1	1	yes	characteristic
188	C₂	C₂₉×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂³×C₈
prime radical	C₂³×C₂₉	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₉×#2183	C₂	semidirect
3. Q₈×C₁₆×C₂₉	C₂	semidirect
29. C₂₉×M₃₂	Q₈	direct (N × H)
60. C₁₆×C₂₉	C₂×Q₈	semidirect
61. Q₈×M₃₂	C₂₉	direct (N × H)
64. Q₈×C₂₉	M₃₂	direct (N × H)
81. #2183	C₂×C₂₉	semidirect
82. Q₈×C₁₆	C₂×C₂₉	semidirect
123. M₃₂	Q₈×C₂₉	direct (N × H)
133. C₂₉	Q₈×M₃₂	direct (N × H)
155. C₁₆	C₂×Q₈×C₂₉	semidirect
158. Q₈	C₂₉×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×C₂₉	2
C₂₉×#2183	2
Q₈×C₁₆×C₂₉	2
C₂₉×#2150	2
C₄×C₂₉×M₃₂	2
Q₈×M₃₂	29

Subgroups — Q8×C29×M32 #20328377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₉×M₃₂ #20328377