Subgroups — C₆₁×#3375 #21443416

Order 7808.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#559	C₂	1	1	yes	maximal, characteristic
2	Q₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#489	C₂	1	2	yes	maximal, complemented
4	C₆₁×#486	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#474	C₂	1	2	yes	maximal, complemented
7	C₆₁×#464	C₂	1	2	yes	maximal, complemented
8	C₆₁×#460	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#440	C₂	1	2	yes	maximal, complemented
11	C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₆₁	—	2	1	no	—
15	C₄⋊Q₈×C₆₁	—	2	2	no	—
16	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	#127×C₆₁	C₂²	1	1	yes	characteristic
19	#124×C₆₁	—	2	1	no	—
20	#124×C₆₁	C₂²	1	2	yes	—
21	#123×C₆₁	—	2	2	no	—
22	C₄×Q₈×C₆₁	—	2	1	no	—
23	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
25	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₆₁	—	2	1	no	—
28	C₄×D₄×C₆₁	C₂²	1	2	yes	—
29	#107×C₆₁	C₂²	1	1	yes	characteristic
30	#107×C₆₁	C₂²	1	2	yes	—
31	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
33	#103×C₆₁	C₂²	1	1	yes	characteristic
34	#103×C₆₁	C₂²	1	1	yes	characteristic
35	#103×C₆₁	C₂²	1	2	yes	—
36	#102×C₆₁	C₂²	1	2	yes	—
37	#102×C₆₁	C₂²	1	2	yes	—
38	#102×C₆₁	C₂²	1	2	yes	—
39	#97×C₆₁	C₂²	1	2	yes	—
40	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₆₁	—	2	2	no	—
42	C₂×Q₈×C₆₁	—	2	1	no	—
43	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₆₁	—	2	1	no	—
47	C₂×D₄×C₆₁	—	2	1	no	—
48	C₂×D₄×C₆₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₆₁	—	2	2	no	—
50	Q₁₆×C₆₁	—	2	2	no	—
51	SD₁₆×C₆₁	—	2	4	no	—
52	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂³	1	2	yes	—
55	C₄⋊C₄×C₆₁	—	2	2	no	—
56	C₄⋊C₄×C₆₁	—	2	1	no	—
57	C₄⋊C₄×C₆₁	—	2	1	no	—
58	C₄⋊C₄×C₆₁	—	2	2	no	—
59	C₄⋊C₄×C₆₁	—	2	1	no	—
60	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
62	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₆₁	—	2	2	no	—
66	C₂²⋊C₄×C₆₁	—	2	2	no	—
67	C₂²⋊C₄×C₆₁	—	2	2	no	—
68	C₄²×C₆₁	—	2	1	no	—
69	C₄²×C₆₁	—	2	2	no	—
70	C₄²×C₆₁	—	2	1	no	—
71	C₄²×C₆₁	C₂³	1	1	yes	characteristic
72	C₄²×C₆₁	C₂³	1	2	yes	—
73	C₂³×C₆₁	—	2	2	no	—
74	Q₈×C₆₁	—	2	2	no	—
75	Q₈×C₆₁	—	2	1	no	—
76	Q₈×C₆₁	—	2	1	no	—
77	Q₈×C₆₁	—	2	1	no	—
78	Q₈×C₆₁	—	8	2	no	—
79	Q₈×C₆₁	—	8	1	no	—
80	Q₈×C₆₁	#41	1	2	yes	—
81	D₄×C₆₁	—	2	2	no	—
82	D₄×C₆₁	—	2	2	no	—
83	D₄×C₆₁	—	4	2	no	—
84	D₄×C₆₁	—	8	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	8	2	no	—
91	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₈×C₆₁	—	2	2	no	—
97	C₈×C₆₁	—	2	1	no	—
98	C₈×C₆₁	—	2	1	no	—
99	C₂²×C₆₁	—	4	2	no	—
100	C₂²×C₆₁	—	4	2	no	—
101	C₂²×C₆₁	—	4	2	no	—
102	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₆₁	—	2	2	no	—
104	C₄×C₆₁	—	2	2	no	—
105	C₄×C₆₁	—	2	2	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	—	4	1	no	—
108	C₄×C₆₁	—	4	1	no	—
109	C₄×C₆₁	—	4	1	no	—
110	C₄×C₆₁	—	4	2	no	—
111	C₄×C₆₁	#137	1	2	yes	—
112	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
113	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
114	#3375	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₁	—	8	2	no	—
116	C₂×C₆₁	#548	1	1	yes	characteristic
117	C₂×C₆₁	C₂×#137	1	1	yes	characteristic
118	C₂×C₆₁	#576	1	1	yes	characteristic
119	#559	C₂×C₆₁	1	1	yes	characteristic
120	Q₈²	C₂×C₆₁	1	1	yes	characteristic, complemented
121	#489	C₂×C₆₁	1	2	yes	complemented
122	#486	C₂×C₆₁	1	1	yes	characteristic
123	#477	C₂×C₆₁	1	1	yes	characteristic, complemented
124	#474	C₂×C₆₁	1	2	yes	complemented
125	#464	C₂×C₆₁	1	2	yes	complemented
126	#460	C₂×C₆₁	1	1	yes	characteristic
127	#446	C₂×C₆₁	1	1	yes	characteristic, complemented
128	#440	C₂×C₆₁	1	2	yes	complemented
129	C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
130	C₆₁	#3375	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
132	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	—
133	C₄⋊Q₈	—	2	1	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
137	#127	C₂²×C₆₁	1	1	yes	characteristic
138	#124	—	2	1	no	—
139	#124	C₂²×C₆₁	1	2	yes	—
140	#123	—	2	2	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
143	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
144	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
145	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	C₂²×C₆₁	1	2	yes	—
148	#107	C₂²×C₆₁	1	1	yes	characteristic
149	#107	C₂²×C₆₁	1	2	yes	—
150	C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
152	#103	C₂²×C₆₁	1	1	yes	characteristic
153	#103	C₂²×C₆₁	1	1	yes	characteristic
154	#103	C₂²×C₆₁	1	2	yes	—
155	#102	C₂²×C₆₁	1	2	yes	—
156	#102	C₂²×C₆₁	1	2	yes	—
157	#102	C₂²×C₆₁	1	2	yes	—
158	#97	C₂²×C₆₁	1	2	yes	—
159	C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	D₄×C₆₁	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂³×C₆₁	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	Q₁₆	—	2	2	no	—
170	SD₁₆	—	2	4	no	—
171	C₂×C₈	C₂³×C₆₁	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₆₁	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₆₁	1	2	yes	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
180	C₄⋊C₄	D₄×C₆₁	1	2	yes	—
181	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
182	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	C₂³×C₆₁	1	1	yes	characteristic
191	C₄²	C₂³×C₆₁	1	2	yes	—
192	C₂³	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	1	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	8	2	no	—
198	Q₈	—	8	1	no	—
199	Q₈	#41×C₆₁	1	2	yes	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	8	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	#137×C₆₁	1	2	yes	—
231	C₄	C₂×#41×C₆₁	1	1	yes	characteristic
232	C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
233	C₂	—	8	2	no	—
234	C₂	C₆₁×#548	1	1	yes	characteristic
235	C₂	C₂×#137×C₆₁	1	1	yes	characteristic
236	C₂	C₆₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
prime radical	#127×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈²×C₆₁	C₂	semidirect
3. C₆₁×#489	C₂	semidirect
5. C₆₁×#477	C₂	semidirect
6. C₆₁×#474	C₂	semidirect
7. C₆₁×#464	C₂	semidirect
9. C₆₁×#446	C₂	semidirect
10. C₆₁×#440	C₂	semidirect
11. C₄×Q₁₆×C₆₁	C₂	semidirect
114. #3375	C₆₁	direct (N × H)
120. Q₈²	C₂×C₆₁	semidirect
121. #489	C₂×C₆₁	semidirect
123. #477	C₂×C₆₁	semidirect
124. #474	C₂×C₆₁	semidirect
125. #464	C₂×C₆₁	semidirect
127. #446	C₂×C₆₁	semidirect
128. #440	C₂×C₆₁	semidirect
129. C₄×Q₁₆	C₂×C₆₁	semidirect
130. C₆₁	#3375	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#559	2
Q₈²×C₆₁	2
C₆₁×#489	2
C₆₁×#486	2
C₆₁×#477	2
C₆₁×#474	2
C₆₁×#464	2
C₆₁×#460	2
C₆₁×#446	2
C₆₁×#440	2
C₄×Q₁₆×C₆₁	2
#3375	61

Subgroups — C61×#3375 #21443416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3375 #21443416