Subgroups — C₂²×C₆₁×#349 #29536768

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#349	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
5	C₂²×M₁₆×C₆₁	—	2	6	no	—
6	C₂²×M₁₆×C₆₁	—	2	2	no	—
7	C₂²×M₁₆×C₆₁	C₄	1	6	yes	—
8	C₂²×M₁₆×C₆₁	C₂²	1	6	yes	—
9	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₆₁	—	2	6	no	—
12	C₂×M₃₂×C₆₁	C₂²	1	12	yes	complemented
13	C₂²×C₁₆×C₆₁	—	2	2	no	—
14	C₆₁×#349	C₂²	1	16	yes	direct product factor
15	C₂⁵×C₆₁	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₆₁	—	2	3	no	—
17	C₂³×C₄×C₆₁	—	2	1	no	—
18	C₂³×C₄×C₆₁	—	2	3	no	—
19	C₂³×C₄×C₆₁	—	2	1	no	—
20	C₂³×C₄×C₆₁	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	C₈	1	6	yes	—
22	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₆₁	—	2	12	no	—
25	C₂×M₁₆×C₆₁	—	2	12	no	—
26	C₂×M₁₆×C₆₁	—	2	12	no	—
27	C₂×M₁₆×C₆₁	—	2	12	no	—
28	C₂×M₁₆×C₆₁	C₂×C₄	1	12	yes	—
29	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
30	C₂²×C₈×C₆₁	—	2	3	no	—
31	C₂²×C₈×C₆₁	—	2	1	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
34	C₂²×C₈×C₆₁	D₄	1	6	yes	—
35	C₂²×C₈×C₆₁	D₄	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
37	M₃₂×C₆₁	—	2	12	no	—
38	M₃₂×C₆₁	C₂³	1	8	yes	complemented
39	C₂×C₁₆×C₆₁	—	2	12	no	—
40	C₂⁴×C₆₁	—	2	3	no	—
41	C₂⁴×C₆₁	—	2	1	no	—
42	C₂⁴×C₆₁	—	4	4	no	—
43	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
44	C₂⁴×C₆₁	C₂×C₈	1	6	yes	—
45	C₂²×C₄×C₆₁	—	2	3	no	—
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	—	2	12	no	—
48	C₂²×C₄×C₆₁	—	2	6	no	—
49	C₂²×C₄×C₆₁	—	2	6	no	—
50	C₂²×C₄×C₆₁	—	2	6	no	—
51	C₂²×C₄×C₆₁	—	2	6	no	—
52	C₂²×C₄×C₆₁	—	2	6	no	—
53	C₂²×C₄×C₆₁	—	2	6	no	—
54	C₂²×C₄×C₆₁	C₂²⋊C₄	1	3	yes	—
55	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×C₈	1	6	yes	—
57	C₂²×C₄×C₆₁	C₂×C₈	1	12	yes	—
58	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
59	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	M₁₆	1	3	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
63	M₁₆×C₆₁	—	2	24	no	—
64	M₁₆×C₆₁	—	2	8	no	—
65	C₂×C₈×C₆₁	—	2	6	no	—
66	C₂×C₈×C₆₁	—	2	6	no	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	6	yes	—
68	C₂×C₈×C₆₁	C₂²⋊C₄	1	6	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
71	C₂×C₈×C₆₁	C₂×D₄	1	6	yes	—
72	C₂×C₈×C₆₁	C₂×D₄	1	6	yes	—
73	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
74	C₁₆×C₆₁	—	2	8	no	—
75	C₂³×C₆₁	—	2	4	no	—
76	C₂³×C₆₁	—	2	6	no	—
77	C₂³×C₆₁	—	2	6	no	—
78	C₂³×C₆₁	—	4	12	no	—
79	C₂³×C₆₁	—	4	12	no	—
80	C₂³×C₆₁	—	4	4	no	—
81	C₂³×C₆₁	C₂²⋊C₈	1	3	yes	—
82	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
83	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
84	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₆₁	—	2	6	no	—
86	C₂×C₄×C₆₁	—	2	6	no	—
87	C₂×C₄×C₆₁	—	2	4	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	12	no	—
90	C₂×C₄×C₆₁	—	2	12	no	—
91	C₂×C₄×C₆₁	C₂²⋊C₈	1	6	yes	—
92	C₂×C₄×C₆₁	C₂²⋊C₈	1	3	yes	—
93	C₂×C₄×C₆₁	C₂²⋊C₈	1	3	yes	—
94	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
95	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
96	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
98	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
99	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
100	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
101	C₈×C₆₁	—	2	4	no	—
102	C₈×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
103	C₈×C₆₁	C₂²×D₄	1	2	yes	—
104	C₂²×#349	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₆₁	—	2	6	no	—
106	C₂²×C₆₁	—	2	6	no	—
107	C₂²×C₆₁	—	2	4	no	—
108	C₂²×C₆₁	—	4	4	no	—
109	C₂²×C₆₁	—	4	12	no	—
110	C₂²×C₆₁	—	4	12	no	—
111	C₂²×C₆₁	#349	1	4	yes	direct product factor
112	C₂²×C₆₁	C₂×C₂²⋊C₈	1	3	yes	—
113	C₂²×C₆₁	C₂×C₂²⋊C₈	1	3	yes	—
114	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₆₁	—	2	4	no	—
116	C₄×C₆₁	C₂×C₂²⋊C₈	1	3	yes	—
117	C₄×C₆₁	C₂×C₂²⋊C₈	1	3	yes	—
118	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
120	C₂³×M₁₆	C₂×C₆₁	1	1	yes	characteristic
121	C₂²×M₃₂	C₂×C₆₁	1	2	yes	complemented
122	C₂×#349	C₂×C₆₁	1	12	yes	direct product factor
123	C₂×C₆₁	—	2	1	no	—
124	C₂×C₆₁	—	2	3	no	—
125	C₂×C₆₁	—	4	4	no	—
126	C₂×C₆₁	C₂×#349	1	6	yes	direct product factor
127	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
128	C₂⁴×C₄	C₄×C₆₁	1	1	yes	characteristic
129	C₂²×M₁₆	—	2	2	no	—
130	C₂²×M₁₆	—	2	6	no	—
131	C₂²×M₁₆	C₄×C₆₁	1	6	yes	—
132	C₂²×M₁₆	C₂²×C₆₁	1	6	yes	—
133	C₂³×C₈	C₄×C₆₁	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
135	C₂×M₃₂	—	2	6	no	—
136	C₂×M₃₂	C₂²×C₆₁	1	12	yes	complemented
137	C₂²×C₁₆	—	2	2	no	—
138	#349	C₂²×C₆₁	1	16	yes	direct product factor
139	C₆₁	C₂²×#349	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁵	C₈×C₆₁	1	1	yes	characteristic
141	C₂³×C₄	—	2	3	no	—
142	C₂³×C₄	—	2	1	no	—
143	C₂³×C₄	—	2	3	no	—
144	C₂³×C₄	—	2	1	no	—
145	C₂³×C₄	C₈×C₆₁	1	1	yes	characteristic
146	C₂³×C₄	C₈×C₆₁	1	6	yes	—
147	C₂³×C₄	C₂×C₄×C₆₁	1	6	yes	—
148	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
149	C₂×M₁₆	—	2	12	no	—
150	C₂×M₁₆	—	2	12	no	—
151	C₂×M₁₆	—	2	12	no	—
152	C₂×M₁₆	—	2	12	no	—
153	C₂×M₁₆	C₂×C₄×C₆₁	1	12	yes	—
154	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
155	C₂²×C₈	—	2	3	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
158	C₂²×C₈	C₂×C₄×C₆₁	1	3	yes	—
159	C₂²×C₈	D₄×C₆₁	1	2	yes	—
160	C₂²×C₈	D₄×C₆₁	1	6	yes	—
161	C₂²×C₈	C₂³×C₆₁	1	3	yes	—
162	M₃₂	—	2	12	no	—
163	M₃₂	C₂³×C₆₁	1	8	yes	complemented
164	C₂×C₁₆	—	2	12	no	—
165	C₂⁴	—	2	3	no	—
166	C₂⁴	—	2	1	no	—
167	C₂⁴	—	4	4	no	—
168	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
169	C₂⁴	C₂×C₈×C₆₁	1	6	yes	—
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	12	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	6	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	C₂²⋊C₄×C₆₁	1	3	yes	—
180	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₈×C₆₁	1	6	yes	—
182	C₂²×C₄	C₂×C₈×C₆₁	1	12	yes	—
183	C₂²×C₄	C₂×C₈×C₆₁	1	3	yes	—
184	C₂²×C₄	M₁₆×C₆₁	1	3	yes	—
185	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
188	M₁₆	—	2	24	no	—
189	M₁₆	—	2	8	no	—
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	—	2	6	no	—
192	C₂×C₈	C₂²⋊C₄×C₆₁	1	6	yes	—
193	C₂×C₈	C₂²⋊C₄×C₆₁	1	6	yes	—
194	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
195	C₂×C₈	C₂²×C₄×C₆₁	1	3	yes	—
196	C₂×C₈	C₂×D₄×C₆₁	1	6	yes	—
197	C₂×C₈	C₂×D₄×C₆₁	1	6	yes	—
198	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
199	C₁₆	—	2	8	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	6	no	—
202	C₂³	—	2	6	no	—
203	C₂³	—	4	12	no	—
204	C₂³	—	4	12	no	—
205	C₂³	—	4	4	no	—
206	C₂³	C₂²⋊C₈×C₆₁	1	3	yes	—
207	C₂³	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
208	C₂³	C₂²×C₈×C₆₁	1	3	yes	—
209	C₂³	C₂²×C₈×C₆₁	1	4	yes	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	2	12	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₂²⋊C₈×C₆₁	1	6	yes	—
217	C₂×C₄	C₂²⋊C₈×C₆₁	1	3	yes	—
218	C₂×C₄	C₂²⋊C₈×C₆₁	1	3	yes	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
221	C₂×C₄	C₂²×C₈×C₆₁	1	4	yes	—
222	C₂×C₄	C₂²×C₈×C₆₁	1	3	yes	—
223	C₂×C₄	C₂×M₁₆×C₆₁	1	3	yes	—
224	C₂×C₄	C₂×M₁₆×C₆₁	1	3	yes	—
225	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
226	C₈	—	2	4	no	—
227	C₈	C₂×C₂²⋊C₄×C₆₁	1	6	yes	—
228	C₈	C₂²×D₄×C₆₁	1	2	yes	—
229	C₂²	—	2	6	no	—
230	C₂²	—	2	6	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	12	no	—
234	C₂²	—	4	12	no	—
235	C₂²	C₆₁×#349	1	4	yes	direct product factor
236	C₂²	C₂×C₂²⋊C₈×C₆₁	1	3	yes	—
237	C₂²	C₂×C₂²⋊C₈×C₆₁	1	3	yes	—
238	C₂²	C₂³×C₈×C₆₁	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	C₂×C₂²⋊C₈×C₆₁	1	3	yes	—
241	C₄	C₂×C₂²⋊C₈×C₆₁	1	3	yes	—
242	C₄	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
243	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	3	no	—
246	C₂	—	4	4	no	—
247	C₂	C₂×C₆₁×#349	1	6	yes	direct product factor
248	C₂	C₂²×C₂²⋊C₈×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₆₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂²⋊C₈
prime radical	C₂⁵×C₆₁	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₆₁	C₂	semidirect
3. C₂×C₆₁×#349	C₂	direct (N × H)
12. C₂×M₃₂×C₆₁	C₂²	semidirect
14. C₆₁×#349	C₂²	direct (N × H)
38. M₃₂×C₆₁	C₂³	semidirect
104. C₂²×#349	C₆₁	direct (N × H)
111. C₂²×C₆₁	#349	direct (N × H)
121. C₂²×M₃₂	C₂×C₆₁	semidirect
122. C₂×#349	C₂×C₆₁	direct (N × H)
126. C₂×C₆₁	C₂×#349	direct (N × H)
136. C₂×M₃₂	C₂²×C₆₁	semidirect
138. #349	C₂²×C₆₁	direct (N × H)
139. C₆₁	C₂²×#349	direct (N × H)
163. M₃₂	C₂³×C₆₁	semidirect
235. C₂²	C₆₁×#349	direct (N × H)
247. C₂	C₂×C₆₁×#349	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₁	2
C₂²×M₃₂×C₆₁	2
C₂×C₆₁×#349	2
C₂²×#349	61

Subgroups — C22×C61×#349 #29536768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#349 #29536768