Subgroups — Q₈×M₃₂×C₇₉ #34145166

Order 20224.

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×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#2183	C₂	1	6	yes	maximal, complemented
3	Q₈×C₁₆×C₇₉	C₂	1	2	yes	maximal, complemented
4	C₇₉×#2150	C₂	1	3	yes	maximal
5	C₄×M₃₂×C₇₉	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₇₉	C₂²	1	3	yes	—
8	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₇₉	C₄	1	3	yes	—
10	C₈×Q₈×C₇₉	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₈×Q₈×C₇₉	C₂²	1	3	yes	—
13	C₂×C₄⋊C₈×C₇₉	C₄	1	3	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
15	C₄⋊C₁₆×C₇₉	C₂²	1	6	yes	—
16	C₄⋊C₁₆×C₇₉	C₂²	1	6	yes	—
17	#346×C₇₉	C₂²	1	6	yes	—
18	C₄×C₁₆×C₇₉	C₂²	1	6	yes	—
19	C₂²×Q₈×C₇₉	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₇₉	C₈	1	3	yes	—
23	C₄×Q₈×C₇₉	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₇₉	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₇₉	C₈	1	3	yes	—
27	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
28	M₃₂×C₇₉	—	2	6	no	—
29	M₃₂×C₇₉	Q₈	1	4	yes	direct product factor
30	C₂×C₁₆×C₇₉	C₂³	1	6	yes	—
31	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
32	C₄⋊C₈×C₇₉	C₂×C₄	1	3	yes	—
33	C₄⋊C₈×C₇₉	C₂×C₄	1	6	yes	—
34	C₄⋊C₈×C₇₉	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₇₉	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₇₉	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₇₉	C₂³	1	3	yes	—
38	C₄×C₈×C₇₉	C₂³	1	3	yes	—
39	C₂×Q₈×C₇₉	—	2	4	no	—
40	C₂×Q₈×C₇₉	C₂×C₈	1	3	yes	—
41	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	6	no	—
46	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₇₉	#41	1	2	yes	—
51	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₇₉	C₂×C₈	1	6	yes	—
53	C₄⋊C₄×C₇₉	C₂×C₈	1	3	yes	—
54	C₄⋊C₄×C₇₉	C₂×C₈	1	3	yes	—
55	C₄²×C₇₉	C₂×C₈	1	3	yes	—
56	C₄²×C₇₉	C₂×C₈	1	3	yes	—
57	C₄²×C₇₉	C₂²×C₄	1	3	yes	—
58	C₄²×C₇₉	C₂²×C₄	1	3	yes	—
59	C₁₆×C₇₉	—	2	6	no	—
60	C₁₆×C₇₉	C₂×Q₈	1	4	yes	complemented
61	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₇₉	—	2	6	no	—
63	Q₈×C₇₉	M₃₂	1	4	yes	direct product factor
64	C₂×C₄×C₇₉	—	2	6	no	—
65	C₂×C₄×C₇₉	—	4	3	no	—
66	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₇₉	#131	1	2	yes	—
73	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
74	C₈×C₇₉	—	2	3	no	—
75	C₈×C₇₉	—	2	3	no	—
76	C₈×C₇₉	C₄×Q₈	1	2	yes	—
77	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	—	2	1	no	—
81	C₂²×C₇₉	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₇₉	—	2	3	no	—
84	C₄×C₇₉	—	2	3	no	—
85	C₄×C₇₉	—	2	3	no	—
86	C₄×C₇₉	C₈×Q₈	1	2	yes	—
87	C₄×C₇₉	C₂×M₃₂	1	6	yes	—
88	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
90	Q₈×M₃₂	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₇₉	—	2	2	no	—
92	C₂×C₇₉	C₂×C₈×Q₈	1	1	yes	characteristic
93	C₂×C₇₉	C₂²×M₃₂	1	1	yes	characteristic
94	C₂×C₇₉	#3407	1	1	yes	characteristic
95	C₂×C₈×Q₈	C₂×C₇₉	1	1	yes	characteristic
96	#2183	C₂×C₇₉	1	6	yes	complemented
97	Q₈×C₁₆	C₂×C₇₉	1	2	yes	complemented
98	#2150	C₂×C₇₉	1	3	yes	—
99	C₄×M₃₂	C₂×C₇₉	1	3	yes	—
100	C₇₉	Q₈×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×C₄×Q₈	C₄×C₇₉	1	1	yes	characteristic
102	C₂×M₃₂	C₂²×C₇₉	1	3	yes	—
103	C₂×M₃₂	C₂²×C₇₉	1	1	yes	characteristic
104	C₈×Q₈	C₄×C₇₉	1	3	yes	—
105	C₈×Q₈	C₄×C₇₉	1	1	yes	characteristic
106	C₈×Q₈	C₂²×C₇₉	1	1	yes	characteristic
107	C₈×Q₈	C₂²×C₇₉	1	3	yes	—
108	C₂×C₄⋊C₈	C₄×C₇₉	1	3	yes	—
109	C₂×C₄×C₈	C₂²×C₇₉	1	3	yes	—
110	C₄⋊C₁₆	C₂²×C₇₉	1	6	yes	—
111	C₄⋊C₁₆	C₂²×C₇₉	1	6	yes	—
112	#346	C₂²×C₇₉	1	6	yes	—
113	C₄×C₁₆	C₂²×C₇₉	1	6	yes	—
114	C₂²×Q₈	C₈×C₇₉	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₇₉	1	3	yes	—
116	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
117	C₄×Q₈	C₈×C₇₉	1	3	yes	—
118	C₄×Q₈	C₈×C₇₉	1	1	yes	characteristic
119	C₄×Q₈	C₂×C₄×C₇₉	1	3	yes	—
120	C₄×Q₈	C₂×C₄×C₇₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₈×C₇₉	1	3	yes	—
122	C₂×C₄²	C₂×C₄×C₇₉	1	3	yes	—
123	M₃₂	—	2	6	no	—
124	M₃₂	Q₈×C₇₉	1	4	yes	direct product factor
125	C₂×C₁₆	C₂³×C₇₉	1	6	yes	—
126	C₂×C₁₆	C₂³×C₇₉	1	2	yes	—
127	C₄⋊C₈	C₂×C₄×C₇₉	1	3	yes	—
128	C₄⋊C₈	C₂×C₄×C₇₉	1	6	yes	—
129	C₄⋊C₈	C₂×C₄×C₇₉	1	3	yes	—
130	C₄×C₈	C₂×C₄×C₇₉	1	3	yes	—
131	C₄×C₈	C₂×C₄×C₇₉	1	3	yes	—
132	C₄×C₈	C₂³×C₇₉	1	3	yes	—
133	C₄×C₈	C₂³×C₇₉	1	3	yes	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₂×C₈×C₇₉	1	3	yes	—
136	C₂×Q₈	C₂×C₈×C₇₉	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₇₉	1	3	yes	—
138	C₂²×C₄	C₂×C₈×C₇₉	1	3	yes	—
139	C₂²×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
140	C₂×C₈	—	2	6	no	—
141	C₂×C₈	C₂²×C₄×C₇₉	1	3	yes	—
142	C₂×C₈	C₂²×C₄×C₇₉	1	3	yes	—
143	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₇₉	1	2	yes	—
145	C₂×C₈	#41×C₇₉	1	2	yes	—
146	C₂×C₈	C₂⁴×C₇₉	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₈×C₇₉	1	6	yes	—
148	C₄⋊C₄	C₂×C₈×C₇₉	1	3	yes	—
149	C₄⋊C₄	C₂×C₈×C₇₉	1	3	yes	—
150	C₄²	C₂×C₈×C₇₉	1	3	yes	—
151	C₄²	C₂×C₈×C₇₉	1	3	yes	—
152	C₄²	C₂²×C₄×C₇₉	1	3	yes	—
153	C₄²	C₂²×C₄×C₇₉	1	3	yes	—
154	C₁₆	—	2	6	no	—
155	C₁₆	C₂×Q₈×C₇₉	1	4	yes	complemented
156	C₂³	C₂²×C₈×C₇₉	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	M₃₂×C₇₉	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₄×Q₈×C₇₉	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
163	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
164	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
165	C₂×C₄	C₂²×C₈×C₇₉	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
167	C₂×C₄	#131×C₇₉	1	2	yes	—
168	C₂×C₄	C₂³×C₄×C₇₉	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	C₄×Q₈×C₇₉	1	2	yes	—
172	C₈	C₂²×Q₈×C₇₉	1	1	yes	characteristic
173	C₈	C₂×#41×C₇₉	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₈×Q₈×C₇₉	1	2	yes	—
177	C₂²	C₂³×C₈×C₇₉	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₈×Q₈×C₇₉	1	2	yes	—
182	C₄	C₂×M₃₂×C₇₉	1	6	yes	—
183	C₄	C₂×C₄×Q₈×C₇₉	1	1	yes	characteristic
184	C₄	C₂×#131×C₇₉	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×C₈×Q₈×C₇₉	1	1	yes	characteristic
187	C₂	C₂²×M₃₂×C₇₉	1	1	yes	characteristic
188	C₂	C₇₉×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₇₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₇₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₇₉
socle	C₂²×C₇₉	C₂³×C₈
prime radical	C₂³×C₇₉	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₉×#2183	C₂	semidirect
3. Q₈×C₁₆×C₇₉	C₂	semidirect
29. M₃₂×C₇₉	Q₈	direct (N × H)
60. C₁₆×C₇₉	C₂×Q₈	semidirect
63. Q₈×C₇₉	M₃₂	direct (N × H)
90. Q₈×M₃₂	C₇₉	direct (N × H)
96. #2183	C₂×C₇₉	semidirect
97. Q₈×C₁₆	C₂×C₇₉	semidirect
100. C₇₉	Q₈×M₃₂	direct (N × H)
124. M₃₂	Q₈×C₇₉	direct (N × H)
155. C₁₆	C₂×Q₈×C₇₉	semidirect
158. Q₈	M₃₂×C₇₉	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×C₇₉	2
C₇₉×#2183	2
Q₈×C₁₆×C₇₉	2
C₇₉×#2150	2
C₄×M₃₂×C₇₉	2
Q₈×M₃₂	79

Subgroups — Q8×M32×C79 #34145166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×M₃₂×C₇₉ #34145166