Subgroups — C₂×C₆₁×#2817 #29540082

Order 15616.

2478 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₆₁	C₂	1	2	yes	maximal
2	C₆₁×#2817	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₆₁×#400	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#397	C₂	1	4	yes	maximal
5	C₂×C₆₁×#395	C₂	1	4	yes	maximal, complemented
6	C₂×C₆₁×#381	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Q₈×C₆₁	—	2	16	no	—
10	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	16	yes	—
11	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#400	C₂²	1	32	yes	complemented
17	C₆₁×#397	C₂²	1	32	yes	—
18	C₆₁×#395	C₂²	1	32	yes	complemented
19	C₆₁×#381	C₂²	1	8	yes	—
20	C₂×#95×C₆₁	C₂²	1	4	yes	—
21	C₂×#95×C₆₁	C₂²	1	8	yes	—
22	C₂×#95×C₆₁	C₂²	1	4	yes	—
23	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Q₈×C₆₁	—	2	8	no	—
25	C₂²×Q₈×C₆₁	—	4	8	no	—
26	C₂²×Q₈×C₆₁	C₂³	1	8	yes	—
27	C₂³×C₄×C₆₁	—	4	4	no	—
28	C₂³×C₄×C₆₁	D₄	1	8	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
31	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
32	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
34	C₂²⋊Q₈×C₆₁	—	2	64	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	16	no	—
37	C₂×C₄⋊C₄×C₆₁	—	4	8	no	—
38	C₂×C₄⋊C₄×C₆₁	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₂³	1	16	yes	—
41	C₂×C₂²⋊C₄×C₆₁	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₆₁	—	4	4	no	—
43	C₂×C₂²⋊C₄×C₆₁	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
45	#95×C₆₁	C₂³	1	16	yes	—
46	#95×C₆₁	C₂³	1	32	yes	—
47	#95×C₆₁	C₂³	1	16	yes	—
48	C₂⁴×C₆₁	—	2	4	no	—
49	C₂⁴×C₆₁	—	2	8	no	—
50	C₂⁴×C₆₁	C₂×D₄	1	4	yes	—
51	C₂⁴×C₆₁	C₂⁴	1	2	yes	—
52	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
53	C₂×Q₈×C₆₁	—	2	16	no	—
54	C₂×Q₈×C₆₁	—	4	16	no	—
55	C₂×Q₈×C₆₁	—	4	32	no	—
56	C₂²×C₄×C₆₁	—	2	8	no	—
57	C₂²×C₄×C₆₁	—	2	8	no	—
58	C₂²×C₄×C₆₁	—	2	8	no	—
59	C₂²×C₄×C₆₁	—	2	8	no	—
60	C₂²×C₄×C₆₁	—	2	16	no	—
61	C₂²×C₄×C₆₁	—	2	16	no	—
62	C₂²×C₄×C₆₁	—	2	16	no	—
63	C₂²×C₄×C₆₁	—	4	16	no	—
64	C₂²×C₄×C₆₁	—	4	8	no	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	16	yes	—
67	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
71	C₄⋊C₄×C₆₁	—	2	16	no	—
72	C₄⋊C₄×C₆₁	—	2	32	no	—
73	C₄⋊C₄×C₆₁	—	4	16	no	—
74	C₂²⋊C₄×C₆₁	—	2	32	no	—
75	C₂²⋊C₄×C₆₁	—	4	8	no	—
76	C₂³×C₆₁	—	2	8	no	—
77	C₂³×C₆₁	—	2	16	no	—
78	C₂³×C₆₁	—	2	4	no	—
79	C₂³×C₆₁	—	2	8	no	—
80	C₂³×C₆₁	—	4	16	no	—
81	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
82	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
83	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
84	C₂³×C₆₁	2₋¹⁺⁴	1	8	yes	—
85	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
86	Q₈×C₆₁	—	4	32	no	—
87	C₂×C₄×C₆₁	—	2	16	no	—
88	C₂×C₄×C₆₁	—	2	8	no	—
89	C₂×C₄×C₆₁	—	2	8	no	—
90	C₂×C₄×C₆₁	—	2	8	no	—
91	C₂×C₄×C₆₁	—	2	8	no	—
92	C₂×C₄×C₆₁	—	2	16	no	—
93	C₂×C₄×C₆₁	—	2	16	no	—
94	C₂×C₄×C₆₁	—	2	16	no	—
95	C₂×C₄×C₆₁	—	4	16	no	—
96	C₂×C₄×C₆₁	—	4	16	no	—
97	C₂×C₄×C₆₁	—	4	16	no	—
98	C₂×C₄×C₆₁	—	4	16	no	—
99	C₂×C₄×C₆₁	C₂²×D₄	1	8	yes	—
100	C₂×#2817	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₆₁	—	2	4	no	—
102	C₂²×C₆₁	—	2	8	no	—
103	C₂²×C₆₁	—	4	16	no	—
104	C₂²×C₆₁	—	4	8	no	—
105	C₂²×C₆₁	#534	1	4	yes	—
106	C₂²×C₆₁	#536	1	8	yes	—
107	C₂²×C₆₁	#563	1	16	yes	—
108	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
109	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	4	yes	—
111	C₄×C₆₁	—	2	8	no	—
112	C₄×C₆₁	—	4	8	no	—
113	C₄×C₆₁	—	4	8	no	—
114	C₄×C₆₁	—	4	8	no	—
115	C₂²×C₂²⋊Q₈	C₂×C₆₁	1	2	yes	—
116	#2817	C₂×C₆₁	1	16	yes	direct product factor
117	C₂×#400	C₂×C₆₁	1	4	yes	complemented
118	C₂×#397	C₂×C₆₁	1	4	yes	—
119	C₂×#395	C₂×C₆₁	1	4	yes	complemented
120	C₂×#381	C₂×C₆₁	1	1	yes	characteristic
121	C₂×C₆₁	—	4	4	no	—
122	C₂×C₆₁	#2817	1	8	yes	direct product factor
123	C₂×C₆₁	C₂×#534	1	1	yes	characteristic
124	C₂×C₆₁	C₂×#536	1	2	yes	—
125	C₂×C₆₁	C₂×#563	1	4	yes	—
126	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
127	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
128	C₂×C₂²⋊Q₈	—	2	16	no	—
129	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	16	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
133	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
134	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
135	#400	C₂²×C₆₁	1	32	yes	complemented
136	#397	C₂²×C₆₁	1	32	yes	—
137	#395	C₂²×C₆₁	1	32	yes	complemented
138	#381	C₂²×C₆₁	1	8	yes	—
139	C₂×#95	C₂²×C₆₁	1	8	yes	—
140	C₂×#95	C₂²×C₆₁	1	4	yes	—
141	C₂×#95	C₂²×C₆₁	1	4	yes	—
142	C₆₁	C₂×#2817	1	1	yes	characteristic, direct product factor, Sylow
143	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
144	C₂²×Q₈	—	2	8	no	—
145	C₂²×Q₈	—	4	8	no	—
146	C₂²×Q₈	C₂³×C₆₁	1	8	yes	—
147	C₂³×C₄	—	4	4	no	—
148	C₂³×C₄	D₄×C₆₁	1	8	yes	—
149	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
150	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
151	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
152	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
153	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
154	C₂²⋊Q₈	—	2	64	no	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	16	no	—
157	C₂×C₄⋊C₄	—	4	8	no	—
158	C₂×C₄⋊C₄	C₂³×C₆₁	1	16	yes	—
159	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
160	C₂×C₄⋊C₄	C₂³×C₆₁	1	16	yes	—
161	C₂×C₂²⋊C₄	—	2	16	no	—
162	C₂×C₂²⋊C₄	—	4	4	no	—
163	C₂×C₂²⋊C₄	C₂³×C₆₁	1	16	yes	—
164	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
165	#95	C₂³×C₆₁	1	16	yes	—
166	#95	C₂³×C₆₁	1	32	yes	—
167	#95	C₂³×C₆₁	1	16	yes	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	C₂×D₄×C₆₁	1	4	yes	—
171	C₂⁴	C₂⁴×C₆₁	1	2	yes	—
172	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
173	C₂×Q₈	—	2	16	no	—
174	C₂×Q₈	—	4	16	no	—
175	C₂×Q₈	—	4	32	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	16	no	—
181	C₂²×C₄	—	2	16	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	4	16	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	C₂×D₄×C₆₁	1	8	yes	—
186	C₂²×C₄	C₂×D₄×C₆₁	1	16	yes	—
187	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
188	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
189	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
190	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	2	32	no	—
193	C₄⋊C₄	—	4	16	no	—
194	C₂²⋊C₄	—	2	32	no	—
195	C₂²⋊C₄	—	4	8	no	—
196	C₂³	—	2	8	no	—
197	C₂³	—	2	16	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	8	no	—
200	C₂³	—	4	16	no	—
201	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
202	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
203	C₂³	H(5,𝔽₂)×C₆₁	1	4	yes	—
204	C₂³	2₋¹⁺⁴×C₆₁	1	8	yes	—
205	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
206	Q₈	—	4	32	no	—
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	16	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	4	16	no	—
216	C₂×C₄	—	4	16	no	—
217	C₂×C₄	—	4	16	no	—
218	C₂×C₄	—	4	16	no	—
219	C₂×C₄	C₂²×D₄×C₆₁	1	8	yes	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	4	16	no	—
223	C₂²	—	4	8	no	—
224	C₂²	C₆₁×#534	1	4	yes	—
225	C₂²	C₆₁×#536	1	8	yes	—
226	C₂²	C₆₁×#563	1	16	yes	—
227	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
228	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
229	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	4	yes	—
230	C₄	—	2	8	no	—
231	C₄	—	4	8	no	—
232	C₄	—	4	8	no	—
233	C₄	—	4	8	no	—
234	C₂	—	4	4	no	—
235	C₂	C₆₁×#2817	1	8	yes	direct product factor
236	C₂	C₂×C₆₁×#534	1	1	yes	characteristic
237	C₂	C₂×C₆₁×#536	1	2	yes	—
238	C₂	C₂×C₆₁×#563	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂⁵×C₆₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2817	C₂	direct (N × H)
3. C₂×C₆₁×#400	C₂	semidirect
5. C₂×C₆₁×#395	C₂	semidirect
16. C₆₁×#400	C₂²	semidirect
18. C₆₁×#395	C₂²	semidirect
100. C₂×#2817	C₆₁	direct (N × H)
116. #2817	C₂×C₆₁	direct (N × H)
117. C₂×#400	C₂×C₆₁	semidirect
119. C₂×#395	C₂×C₆₁	semidirect
122. C₂×C₆₁	#2817	direct (N × H)
135. #400	C₂²×C₆₁	semidirect
137. #395	C₂²×C₆₁	semidirect
142. C₆₁	C₂×#2817	direct (N × H)
235. C₂	C₆₁×#2817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊Q₈×C₆₁	2
C₆₁×#2817	2
C₂×C₆₁×#400	2
C₂×C₆₁×#397	2
C₂×C₆₁×#395	2
C₂×C₆₁×#381	2
C₂×#2817	61

Subgroups — C2×C61×#2817 #29540082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2817 #29540082