Subgroups — Q₈×#107×C₆₁ #29526159

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal
3	C₆₁×#1947	C₂	1	6	yes	maximal
4	C₆₁×#1845	C₂	1	3	yes	maximal
5	C₄×#107×C₆₁	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₆₁	—	2	2	no	—
7	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₆₁	C₄	1	8	yes	complemented
9	C₂×#107×C₆₁	C₂²	1	3	yes	—
10	C₂×#107×C₆₁	C₂²	1	6	yes	—
11	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
14	C₆₁×#389	—	2	6	no	—
15	C₆₁×#384	C₂²	1	6	yes	—
16	C₄×C₄⋊C₄×C₆₁	C₂²	1	6	yes	—
17	C₆₁×#340	C₂²	1	6	yes	—
18	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
20	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₆₁	—	4	8	no	—
22	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₆₁	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
25	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
26	C₂×C₄⋊C₄×C₆₁	D₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₆₁	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
29	C₂×C₄²×C₆₁	—	2	6	no	—
30	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
31	#107×C₆₁	—	2	6	no	—
32	#107×C₆₁	—	4	6	no	—
33	#107×C₆₁	Q₈	1	4	yes	direct product factor
34	C₄⋊C₈×C₆₁	C₂×C₄	1	12	yes	—
35	C₄×C₈×C₆₁	C₂×C₄	1	12	yes	—
36	#95×C₆₁	—	2	6	no	—
37	C₂×Q₈×C₆₁	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₆₁	D₈	1	4	yes	—
39	C₂×Q₈×C₆₁	Q₁₆	1	4	yes	—
40	C₂²×C₄×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	—	2	6	no	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₆₁	—	2	3	no	—
46	C₂×C₈×C₆₁	—	2	3	no	—
47	C₂×C₈×C₆₁	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₆₁	#41	1	2	yes	—
51	C₄⋊C₄×C₆₁	—	2	2	no	—
52	C₄⋊C₄×C₆₁	—	2	6	no	—
53	C₄⋊C₄×C₆₁	—	2	6	no	—
54	C₄⋊C₄×C₆₁	—	4	12	no	—
55	C₄⋊C₄×C₆₁	—	8	6	no	—
56	C₄⋊C₄×C₆₁	C₄⋊C₄	1	12	yes	—
57	C₄⋊C₄×C₆₁	C₂×Q₈	1	4	yes	—
58	C₄²×C₆₁	—	4	12	no	—
59	C₄²×C₆₁	C₂²×C₄	1	6	yes	—
60	C₄²×C₆₁	C₂×Q₈	1	6	yes	—
61	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₆₁	#107	1	16	yes	direct product factor
63	C₂×C₄×C₆₁	—	2	3	no	—
64	C₂×C₄×C₆₁	—	2	3	no	—
65	C₂×C₄×C₆₁	—	2	4	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	—	4	2	no	—
68	C₂×C₄×C₆₁	—	4	6	no	—
69	C₂×C₄×C₆₁	—	4	6	no	—
70	C₂×C₄×C₆₁	—	4	6	no	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
73	C₂×C₄×C₆₁	C₂×D₈	1	6	yes	—
74	C₂×C₄×C₆₁	C₂×Q₁₆	1	6	yes	—
75	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
79	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
81	C₈×C₆₁	—	2	6	no	—
82	C₈×C₆₁	C₄×Q₈	1	4	yes	complemented
83	Q₈×#107	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
86	C₂²×C₆₁	#550	1	1	yes	characteristic
87	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
88	C₂²×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
89	C₂²×C₆₁	#576	1	1	yes	characteristic
90	C₂²×C₆₁	#578	1	1	yes	characteristic
91	C₄×C₆₁	—	2	6	no	—
92	C₄×C₆₁	—	4	4	no	—
93	C₄×C₆₁	—	8	6	no	—
94	C₄×C₆₁	C₂×#107	1	12	yes	—
95	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₆₁	#520	1	1	yes	characteristic
97	C₄×C₆₁	#557	1	1	yes	characteristic
98	C₄×C₆₁	Q₈²	1	1	yes	characteristic
99	C₂×C₈×Q₈	C₂×C₆₁	1	1	yes	characteristic
100	Q₈×C₄⋊C₄	C₂×C₆₁	1	2	yes	—
101	#1947	C₂×C₆₁	1	6	yes	—
102	#1845	C₂×C₆₁	1	3	yes	—
103	C₄×#107	C₂×C₆₁	1	3	yes	—
104	C₂×C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
106	C₂×C₆₁	#2913	1	1	yes	characteristic
107	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
108	C₂×C₆₁	Q₈×Q₁₆	1	1	yes	characteristic
109	C₂×C₆₁	#3391	1	1	yes	characteristic
110	C₂×C₆₁	#3395	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	2	2	no	—
112	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
113	C₈×Q₈	C₄×C₆₁	1	8	yes	complemented
114	C₂×#107	C₂²×C₆₁	1	3	yes	—
115	C₂×#107	C₂²×C₆₁	1	6	yes	—
116	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
117	C₂×C₄⋊C₈	C₂²×C₆₁	1	3	yes	—
118	C₂×C₄×C₈	C₂²×C₆₁	1	3	yes	—
119	#389	—	2	6	no	—
120	#384	C₂²×C₆₁	1	6	yes	—
121	C₄×C₄⋊C₄	C₂²×C₆₁	1	6	yes	—
122	#340	C₂²×C₆₁	1	6	yes	—
123	C₆₁	Q₈×#107	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₆₁	1	3	yes	—
126	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
127	C₄×Q₈	—	4	8	no	—
128	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	—
129	C₄×Q₈	Q₈×C₆₁	1	4	yes	—
130	C₂×C₄⋊C₄	—	2	6	no	—
131	C₂×C₄⋊C₄	—	2	6	no	—
132	C₂×C₄⋊C₄	D₄×C₆₁	1	3	yes	—
133	C₂×C₄⋊C₄	C₂³×C₆₁	1	6	yes	—
134	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
135	C₂×C₄²	—	2	6	no	—
136	C₂×C₄²	C₂³×C₆₁	1	3	yes	—
137	#107	—	2	6	no	—
138	#107	—	4	6	no	—
139	#107	Q₈×C₆₁	1	4	yes	direct product factor
140	C₄⋊C₈	C₂×C₄×C₆₁	1	12	yes	—
141	C₄×C₈	C₂×C₄×C₆₁	1	12	yes	—
142	#95	—	2	6	no	—
143	C₂×Q₈	C₄⋊C₄×C₆₁	1	4	yes	—
144	C₂×Q₈	D₈×C₆₁	1	4	yes	—
145	C₂×Q₈	Q₁₆×C₆₁	1	4	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	6	no	—
148	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
149	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
150	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	3	no	—
153	C₂×C₈	C₂²×C₄×C₆₁	1	6	yes	—
154	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
155	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
156	C₂×C₈	#41×C₆₁	1	2	yes	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	6	no	—
159	C₄⋊C₄	—	2	6	no	—
160	C₄⋊C₄	—	4	12	no	—
161	C₄⋊C₄	—	8	6	no	—
162	C₄⋊C₄	C₄⋊C₄×C₆₁	1	12	yes	—
163	C₄⋊C₄	C₂×Q₈×C₆₁	1	4	yes	—
164	C₄²	—	4	12	no	—
165	C₄²	C₂²×C₄×C₆₁	1	6	yes	—
166	C₄²	C₂×Q₈×C₆₁	1	6	yes	—
167	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
168	Q₈	#107×C₆₁	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	6	no	—
175	C₂×C₄	—	4	6	no	—
176	C₂×C₄	—	4	6	no	—
177	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	6	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	6	yes	—
179	C₂×C₄	C₂×D₈×C₆₁	1	6	yes	—
180	C₂×C₄	C₂×Q₁₆×C₆₁	1	6	yes	—
181	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
184	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
185	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
186	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	C₄×Q₈×C₆₁	1	4	yes	complemented
189	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
190	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
191	C₂²	C₆₁×#550	1	1	yes	characteristic
192	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
193	C₂²	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
194	C₂²	C₆₁×#576	1	1	yes	characteristic
195	C₂²	C₆₁×#578	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	6	no	—
199	C₄	C₂×#107×C₆₁	1	12	yes	—
200	C₄	C₂×C₄×Q₈×C₆₁	1	1	yes	characteristic
201	C₄	C₆₁×#520	1	1	yes	characteristic
202	C₄	C₆₁×#557	1	1	yes	characteristic
203	C₄	Q₈²×C₆₁	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊C₄×C₆₁	1	1	yes	characteristic
205	C₂	C₂²×#107×C₆₁	1	1	yes	characteristic
206	C₂	C₆₁×#2913	1	1	yes	characteristic
207	C₂	Q₈×D₈×C₆₁	1	1	yes	characteristic
208	C₂	Q₈×Q₁₆×C₆₁	1	1	yes	characteristic
209	C₂	C₆₁×#3391	1	1	yes	characteristic
210	C₂	C₆₁×#3395	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₆₁	C₄	semidirect
33. #107×C₆₁	Q₈	direct (N × H)
62. Q₈×C₆₁	#107	direct (N × H)
82. C₈×C₆₁	C₄×Q₈	semidirect
83. Q₈×#107	C₆₁	direct (N × H)
113. C₈×Q₈	C₄×C₆₁	semidirect
123. C₆₁	Q₈×#107	direct (N × H)
139. #107	Q₈×C₆₁	direct (N × H)
168. Q₈	#107×C₆₁	direct (N × H)
188. C₈	C₄×Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₆₁	2
Q₈×C₄⋊C₄×C₆₁	2
C₆₁×#1947	2
C₆₁×#1845	2
C₄×#107×C₆₁	2
Q₈×#107	61

Subgroups — Q8×#107×C61 #29526159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#107×C₆₁ #29526159