Subgroups — C₆₁×#34232 #29537665

Order 15616.

2606 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2366	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2365	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2363	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2353	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2341	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2339	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2333	C₂	1	4	yes	maximal, complemented
9	C₆₁×#2319	C₂	1	2	yes	maximal, complemented
10	C₄×#124×C₆₁	C₂	1	4	yes	maximal, complemented
11	C₆₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#2275	C₂	1	2	yes	maximal, complemented
13	C₄×#117×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#532	C₄	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₁	C₂²	1	2	yes	—
19	C₂×#121×C₆₁	C₂²	1	2	yes	—
20	C₂×#121×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₁	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
26	C₂×#117×C₆₁	C₂²	1	2	yes	—
27	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#391	C₂²	1	4	yes	—
29	C₆₁×#390	C₂²	1	4	yes	—
30	C₆₁×#390	C₂²	1	4	yes	—
31	C₆₁×#390	C₂²	1	8	yes	—
32	C₆₁×#389	C₂²	1	4	yes	—
33	C₆₁×#388	C₂²	1	8	yes	—
34	C₆₁×#388	C₂²	1	8	yes	—
35	C₆₁×#387	C₂²	1	8	yes	—
36	C₆₁×#386	C₂²	1	8	yes	—
37	C₆₁×#386	C₂²	1	8	yes	—
38	C₆₁×#383	C₂²	1	2	yes	—
39	C₆₁×#382	C₂²	1	2	yes	—
40	C₆₁×#381	C₂²	1	2	yes	—
41	C₆₁×#381	C₂²	1	4	yes	—
42	C₆₁×#380	C₂²	1	1	yes	characteristic
43	C₆₁×#380	C₂²	1	1	yes	characteristic
44	C₆₁×#380	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
50	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
52	C₄×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	—
53	C₄×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
55	C₄×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	—
56	C₆₁×#376	C₂²	1	2	yes	—
57	C₆₁×#376	C₂²	1	2	yes	—
58	C₂×#95×C₆₁	C₂²	1	4	yes	—
59	C₄³×C₆₁	C₂²	1	2	yes	—
60	C₂×#41×C₆₁	—	4	8	no	—
61	C₂×#41×C₆₁	C₂×C₄	1	16	yes	complemented
62	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
63	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
67	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₁	C₂³	1	8	yes	—
70	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₁	C₂×C₄	1	8	yes	complemented
73	#127×C₆₁	C₂×C₄	1	8	yes	—
74	#124×C₆₁	C₂×C₄	1	16	yes	complemented
75	#121×C₆₁	C₂×C₄	1	16	yes	complemented
76	#121×C₆₁	C₂×C₄	1	16	yes	complemented
77	C₄×Q₈×C₆₁	—	4	8	no	—
78	C₄×D₄×C₆₁	—	2	16	no	—
79	C₄×D₄×C₆₁	—	4	8	no	—
80	C₄×D₄×C₆₁	C₂×C₄	1	16	yes	complemented
81	C₄×D₄×C₆₁	C₂×C₄	1	16	yes	complemented
82	#117×C₆₁	—	2	8	no	—
83	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₆₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₁	C₂³	1	2	yes	—
94	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
98	C₂×C₄²×C₆₁	C₂×C₄	1	8	yes	complemented
99	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
100	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
101	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
102	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
104	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
105	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
109	#95×C₆₁	C₂³	1	2	yes	—
110	#95×C₆₁	C₂³	1	2	yes	—
111	#95×C₆₁	C₂³	1	8	yes	—
112	#95×C₆₁	C₂³	1	2	yes	—
113	#95×C₆₁	C₂³	1	4	yes	—
114	#95×C₆₁	C₂³	1	2	yes	—
115	#95×C₆₁	C₂³	1	4	yes	—
116	#95×C₆₁	C₂³	1	4	yes	—
117	#95×C₆₁	C₂³	1	8	yes	—
118	#95×C₆₁	C₂³	1	4	yes	—
119	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
120	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
121	#41×C₆₁	—	4	32	no	—
122	C₂×Q₈×C₆₁	—	2	8	no	—
123	C₂×Q₈×C₆₁	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₆₁	—	2	8	no	—
125	C₂×D₄×C₆₁	—	2	8	no	—
126	C₂×D₄×C₆₁	—	4	2	no	—
127	C₂×D₄×C₆₁	—	4	2	no	—
128	C₂×D₄×C₆₁	—	4	8	no	—
129	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
130	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
131	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
132	C₂×D₄×C₆₁	C₂²×C₄	1	16	yes	—
133	C₂²×C₄×C₆₁	—	2	4	no	—
134	C₂²×C₄×C₆₁	—	2	4	no	—
135	C₂²×C₄×C₆₁	—	2	16	no	—
136	C₂²×C₄×C₆₁	—	4	1	no	—
137	C₂²×C₄×C₆₁	—	4	1	no	—
138	C₂²×C₄×C₆₁	—	4	8	no	—
139	C₂²×C₄×C₆₁	—	4	8	no	—
140	C₂²×C₄×C₆₁	—	4	8	no	—
141	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
143	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₆₁	C₂²×C₄	1	16	yes	—
145	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
156	C₄⋊C₄×C₆₁	—	2	4	no	—
157	C₄⋊C₄×C₆₁	—	2	4	no	—
158	C₄⋊C₄×C₆₁	—	2	4	no	—
159	C₄⋊C₄×C₆₁	—	4	8	no	—
160	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
161	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
162	C₄⋊C₄×C₆₁	#41	1	8	yes	complemented
163	C₂²⋊C₄×C₆₁	—	2	8	no	—
164	C₂²⋊C₄×C₆₁	—	2	16	no	—
165	C₂²⋊C₄×C₆₁	—	4	8	no	—
166	C₂²⋊C₄×C₆₁	—	4	2	no	—
167	C₂²⋊C₄×C₆₁	—	4	2	no	—
168	C₂²⋊C₄×C₆₁	—	4	8	no	—
169	C₂²⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
170	C₂²⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
171	C₄²×C₆₁	—	2	8	no	—
172	C₄²×C₆₁	—	2	8	no	—
173	C₄²×C₆₁	—	2	8	no	—
174	C₄²×C₆₁	—	4	2	no	—
175	C₄²×C₆₁	—	4	2	no	—
176	C₄²×C₆₁	—	4	8	no	—
177	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
178	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
179	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
180	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
181	C₂³×C₆₁	—	2	2	no	—
182	C₂³×C₆₁	—	2	8	no	—
183	C₂³×C₆₁	—	2	8	no	—
184	C₂³×C₆₁	—	2	2	no	—
185	C₂³×C₆₁	—	2	1	no	—
186	C₂³×C₆₁	—	2	1	no	—
187	C₂³×C₆₁	—	2	4	no	—
188	C₂³×C₆₁	—	2	4	no	—
189	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
190	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
191	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
192	Q₈×C₆₁	—	2	16	no	—
193	D₄×C₆₁	—	2	16	no	—
194	D₄×C₆₁	—	2	16	no	—
195	D₄×C₆₁	—	4	8	no	—
196	D₄×C₆₁	—	4	16	no	—
197	C₂×C₄×C₆₁	—	2	2	no	—
198	C₂×C₄×C₆₁	—	2	4	no	—
199	C₂×C₄×C₆₁	—	2	8	no	—
200	C₂×C₄×C₆₁	—	2	2	no	—
201	C₂×C₄×C₆₁	—	2	1	no	—
202	C₂×C₄×C₆₁	—	2	1	no	—
203	C₂×C₄×C₆₁	—	2	4	no	—
204	C₂×C₄×C₆₁	—	2	1	no	—
205	C₂×C₄×C₆₁	—	2	1	no	—
206	C₂×C₄×C₆₁	—	2	8	no	—
207	C₂×C₄×C₆₁	—	2	8	no	—
208	C₂×C₄×C₆₁	—	2	4	no	—
209	C₂×C₄×C₆₁	—	2	4	no	—
210	C₂×C₄×C₆₁	—	2	4	no	—
211	C₂×C₄×C₆₁	—	2	4	no	—
212	C₂×C₄×C₆₁	—	2	4	no	—
213	C₂×C₄×C₆₁	—	2	4	no	—
214	C₂×C₄×C₆₁	—	2	4	no	—
215	C₂×C₄×C₆₁	—	2	4	no	—
216	C₂×C₄×C₆₁	—	2	4	no	—
217	C₂×C₄×C₆₁	—	4	4	no	—
218	C₂×C₄×C₆₁	—	4	16	no	—
219	C₂×C₄×C₆₁	—	8	16	no	—
220	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
223	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
224	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	—
225	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
227	C₂×C₄×C₆₁	C₂×#41	1	8	yes	—
228	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
229	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
230	#34232	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂²×C₆₁	—	2	2	no	—
232	C₂²×C₆₁	—	2	8	no	—
233	C₂²×C₆₁	—	2	2	no	—
234	C₂²×C₆₁	—	2	8	no	—
235	C₂²×C₆₁	—	2	2	no	—
236	C₂²×C₆₁	—	2	8	no	—
237	C₂²×C₆₁	—	4	8	no	—
238	C₂²×C₆₁	—	4	2	no	—
239	C₂²×C₆₁	—	4	2	no	—
240	C₂²×C₆₁	—	4	8	no	—
241	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
242	C₂²×C₆₁	C₂²×#41	1	2	yes	—
243	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
244	C₂²×C₆₁	#585	1	1	yes	characteristic
245	C₂²×C₆₁	#585	1	1	yes	characteristic
246	C₄×C₆₁	—	2	2	no	—
247	C₄×C₆₁	—	2	2	no	—
248	C₄×C₆₁	—	2	8	no	—
249	C₄×C₆₁	—	2	2	no	—
250	C₄×C₆₁	—	2	2	no	—
251	C₄×C₆₁	—	2	8	no	—
252	C₄×C₆₁	—	4	4	no	—
253	C₄×C₆₁	—	4	4	no	—
254	C₄×C₆₁	—	4	4	no	—
255	C₄×C₆₁	C₄×#41	1	4	yes	—
256	C₄×C₆₁	#518	1	4	yes	—
257	C₂×#532	C₂×C₆₁	1	1	yes	characteristic
258	#2366	C₂×C₆₁	1	2	yes	complemented
259	#2365	C₂×C₆₁	1	2	yes	complemented
260	#2363	C₂×C₆₁	1	4	yes	complemented
261	#2353	C₂×C₆₁	1	4	yes	complemented
262	#2341	C₂×C₆₁	1	2	yes	complemented
263	#2339	C₂×C₆₁	1	2	yes	complemented
264	#2333	C₂×C₆₁	1	4	yes	complemented
265	#2319	C₂×C₆₁	1	2	yes	complemented
266	C₄×#124	C₂×C₆₁	1	4	yes	complemented
267	#2283	C₂×C₆₁	1	1	yes	characteristic, complemented
268	#2275	C₂×C₆₁	1	2	yes	complemented
269	C₄×#117	C₂×C₆₁	1	1	yes	characteristic, complemented
270	C₂×C₆₁	—	4	2	no	—
271	C₂×C₆₁	—	4	8	no	—
272	C₂×C₆₁	C₂×C₄×#41	1	1	yes	characteristic
273	C₂×C₆₁	C₂×#518	1	1	yes	characteristic
274	C₂×C₆₁	#3428	1	1	yes	characteristic
275	C₂×C₆₁	#3514	1	2	yes	—
276	C₂×C₆₁	#3517	1	2	yes	—
277	C₂²×#41	C₂²×C₆₁	1	2	yes	—
278	#532	C₄×C₆₁	1	16	yes	complemented
279	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
280	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
281	C₂×#124	C₂²×C₆₁	1	2	yes	—
282	C₂×#121	C₂²×C₆₁	1	2	yes	—
283	C₂×#121	C₂²×C₆₁	1	2	yes	—
284	C₂×C₄×Q₈	C₂²×C₆₁	1	4	yes	—
285	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
286	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
287	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
288	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
289	C₂×#117	C₂²×C₆₁	1	2	yes	—
290	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
291	#391	C₂²×C₆₁	1	4	yes	—
292	#390	C₂²×C₆₁	1	4	yes	—
293	#390	C₂²×C₆₁	1	8	yes	—
294	#390	C₂²×C₆₁	1	4	yes	—
295	#389	C₂²×C₆₁	1	4	yes	—
296	#388	C₂²×C₆₁	1	8	yes	—
297	#388	C₂²×C₆₁	1	8	yes	—
298	#387	C₂²×C₆₁	1	8	yes	—
299	#386	C₂²×C₆₁	1	8	yes	—
300	#386	C₂²×C₆₁	1	8	yes	—
301	#383	C₂²×C₆₁	1	2	yes	—
302	#382	C₂²×C₆₁	1	2	yes	—
303	#381	C₂²×C₆₁	1	4	yes	—
304	#381	C₂²×C₆₁	1	2	yes	—
305	#380	C₂²×C₆₁	1	4	yes	—
306	#380	C₂²×C₆₁	1	1	yes	characteristic
307	#380	C₂²×C₆₁	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
310	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
312	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
313	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₆₁	1	8	yes	—
315	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
316	C₄×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
317	C₄×C₂²⋊C₄	C₂²×C₆₁	1	8	yes	—
318	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
319	#376	C₂²×C₆₁	1	2	yes	—
320	#376	C₂²×C₆₁	1	2	yes	—
321	C₂×#95	C₂²×C₆₁	1	4	yes	—
322	C₄³	C₂²×C₆₁	1	2	yes	—
323	C₆₁	#34232	1	1	yes	characteristic, direct product factor, Sylow
324	C₂×#41	—	4	8	no	—
325	C₂×#41	C₂×C₄×C₆₁	1	16	yes	complemented
326	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
327	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
328	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
329	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
330	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
331	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₆₁	1	8	yes	—
334	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
335	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
336	C₄⋊Q₈	C₂×C₄×C₆₁	1	8	yes	complemented
337	#127	C₂×C₄×C₆₁	1	8	yes	—
338	#124	C₂×C₄×C₆₁	1	16	yes	complemented
339	#121	C₂×C₄×C₆₁	1	16	yes	complemented
340	#121	C₂×C₄×C₆₁	1	16	yes	complemented
341	C₄×Q₈	—	4	8	no	—
342	C₄×D₄	—	2	16	no	—
343	C₄×D₄	—	4	8	no	—
344	C₄×D₄	C₂×C₄×C₆₁	1	16	yes	complemented
345	C₄×D₄	C₂×C₄×C₆₁	1	16	yes	complemented
346	#117	—	2	8	no	—
347	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
348	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
351	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₆₁	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₆₁	1	2	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
362	C₂×C₄²	C₂×C₄×C₆₁	1	8	yes	complemented
363	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
364	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
365	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
366	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
367	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
368	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
369	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
372	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
373	#95	C₂³×C₆₁	1	2	yes	—
374	#95	C₂³×C₆₁	1	2	yes	—
375	#95	C₂³×C₆₁	1	8	yes	—
376	#95	C₂³×C₆₁	1	2	yes	—
377	#95	C₂³×C₆₁	1	4	yes	—
378	#95	C₂³×C₆₁	1	2	yes	—
379	#95	C₂³×C₆₁	1	4	yes	—
380	#95	C₂³×C₆₁	1	4	yes	—
381	#95	C₂³×C₆₁	1	8	yes	—
382	#95	C₂³×C₆₁	1	4	yes	—
383	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
384	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
385	#41	—	4	32	no	—
386	C₂×Q₈	—	2	8	no	—
387	C₂×Q₈	C₂²×C₄×C₆₁	1	8	yes	—
388	C₂×D₄	—	2	8	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	4	8	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
394	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
395	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
396	C₂×D₄	C₂²×C₄×C₆₁	1	16	yes	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	16	no	—
400	C₂²×C₄	—	4	8	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	—	4	1	no	—
403	C₂²×C₄	—	4	8	no	—
404	C₂²×C₄	—	4	8	no	—
405	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
406	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
407	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
408	C₂²×C₄	C₂²×C₄×C₆₁	1	16	yes	—
409	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
418	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
419	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	4	8	no	—
424	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
425	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
426	C₄⋊C₄	#41×C₆₁	1	8	yes	complemented
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	16	no	—
429	C₂²⋊C₄	—	4	8	no	—
430	C₂²⋊C₄	—	4	8	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	—	4	2	no	—
433	C₂²⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
434	C₂²⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
435	C₄²	—	2	8	no	—
436	C₄²	—	2	8	no	—
437	C₄²	—	2	8	no	—
438	C₄²	—	4	8	no	—
439	C₄²	—	4	2	no	—
440	C₄²	—	4	2	no	—
441	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
442	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
443	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
444	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	4	no	—
453	C₂³	C₂³×C₄×C₆₁	1	8	yes	—
454	C₂³	C₂³×C₄×C₆₁	1	2	yes	—
455	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
456	Q₈	—	2	16	no	—
457	D₄	—	2	16	no	—
458	D₄	—	2	16	no	—
459	D₄	—	4	8	no	—
460	D₄	—	4	16	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	—	4	16	no	—
483	C₂×C₄	—	8	16	no	—
484	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
485	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
486	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
487	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
488	C₂×C₄	C₂³×C₄×C₆₁	1	8	yes	—
489	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
491	C₂×C₄	C₂×#41×C₆₁	1	8	yes	—
492	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
493	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	8	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	4	8	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	8	no	—
504	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
505	C₂²	C₂²×#41×C₆₁	1	2	yes	—
506	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
507	C₂²	C₆₁×#585	1	1	yes	characteristic
508	C₂²	C₆₁×#585	1	1	yes	characteristic
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	8	no	—
512	C₄	—	2	2	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	8	no	—
515	C₄	—	4	4	no	—
516	C₄	—	4	4	no	—
517	C₄	—	4	4	no	—
518	C₄	C₄×#41×C₆₁	1	4	yes	—
519	C₄	C₆₁×#518	1	4	yes	—
520	C₂	—	4	2	no	—
521	C₂	—	4	8	no	—
522	C₂	C₂×C₄×#41×C₆₁	1	1	yes	characteristic
523	C₂	C₂×C₆₁×#518	1	1	yes	characteristic
524	C₂	C₆₁×#3428	1	1	yes	characteristic
525	C₂	C₆₁×#3514	1	2	yes	—
526	C₂	C₆₁×#3517	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₁
Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂×C₆₁×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2366	C₂	semidirect
3. C₆₁×#2365	C₂	semidirect
4. C₆₁×#2363	C₂	semidirect
5. C₆₁×#2353	C₂	semidirect
6. C₆₁×#2341	C₂	semidirect
7. C₆₁×#2339	C₂	semidirect
8. C₆₁×#2333	C₂	semidirect
9. C₆₁×#2319	C₂	semidirect
10. C₄×#124×C₆₁	C₂	semidirect
11. C₆₁×#2283	C₂	semidirect
12. C₆₁×#2275	C₂	semidirect
13. C₄×#117×C₆₁	C₂	semidirect
15. C₆₁×#532	C₄	semidirect
61. C₂×#41×C₆₁	C₂×C₄	semidirect
72. C₄⋊Q₈×C₆₁	C₂×C₄	semidirect
74. #124×C₆₁	C₂×C₄	semidirect
75. #121×C₆₁	C₂×C₄	semidirect
76. #121×C₆₁	C₂×C₄	semidirect
80. C₄×D₄×C₆₁	C₂×C₄	semidirect
81. C₄×D₄×C₆₁	C₂×C₄	semidirect
98. C₂×C₄²×C₆₁	C₂×C₄	semidirect
162. C₄⋊C₄×C₆₁	#41	semidirect
230. #34232	C₆₁	direct (N × H)
258. #2366	C₂×C₆₁	semidirect
259. #2365	C₂×C₆₁	semidirect
260. #2363	C₂×C₆₁	semidirect
261. #2353	C₂×C₆₁	semidirect
262. #2341	C₂×C₆₁	semidirect
263. #2339	C₂×C₆₁	semidirect
264. #2333	C₂×C₆₁	semidirect
265. #2319	C₂×C₆₁	semidirect
266. C₄×#124	C₂×C₆₁	semidirect
267. #2283	C₂×C₆₁	semidirect
268. #2275	C₂×C₆₁	semidirect
269. C₄×#117	C₂×C₆₁	semidirect
278. #532	C₄×C₆₁	semidirect
323. C₆₁	#34232	direct (N × H)
325. C₂×#41	C₂×C₄×C₆₁	semidirect
336. C₄⋊Q₈	C₂×C₄×C₆₁	semidirect
338. #124	C₂×C₄×C₆₁	semidirect
339. #121	C₂×C₄×C₆₁	semidirect
340. #121	C₂×C₄×C₆₁	semidirect
344. C₄×D₄	C₂×C₄×C₆₁	semidirect
345. C₄×D₄	C₂×C₄×C₆₁	semidirect
362. C₂×C₄²	C₂×C₄×C₆₁	semidirect
426. C₄⋊C₄	#41×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#532	2
C₆₁×#2366	2
C₆₁×#2365	2
C₆₁×#2363	2
C₆₁×#2353	2
C₆₁×#2341	2
C₆₁×#2339	2
C₆₁×#2333	2
C₆₁×#2319	2
C₄×#124×C₆₁	2
C₆₁×#2283	2
C₆₁×#2275	2
C₄×#117×C₆₁	2
#34232	61

Subgroups — C61×#34232 #29537665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#34232 #29537665