Subgroups — C₂²×C₆₁×#553 #29566138

Order 15616.

8798 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#553	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#126×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₂²×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#123×C₆₁	C₂	1	2	yes	maximal, complemented
5	C₂²×#121×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#121×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#121×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₄×C₆₁	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#117×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₁	C₂²	1	2	yes	—
14	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₆₁	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
20	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#553	C₂²	1	256	yes	direct product factor
22	C₂×#126×C₆₁	C₂²	1	48	yes	complemented
23	C₂×#125×C₆₁	C₂²	1	24	yes	complemented
24	C₂×#123×C₆₁	C₂²	1	48	yes	complemented
25	C₂×#121×C₆₁	C₂²	1	48	yes	complemented
26	C₂×#121×C₆₁	C₂²	1	24	yes	complemented
27	C₂×#121×C₆₁	C₂²	1	24	yes	complemented
28	C₂×C₄×D₄×C₆₁	C₂²	1	48	yes	complemented
29	C₂×C₄×D₄×C₆₁	C₂²	1	24	yes	complemented
30	C₂×C₄×D₄×C₆₁	C₂²	1	24	yes	complemented
31	C₂×#117×C₆₁	C₂²	1	24	yes	complemented
32	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
38	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
39	C₂²×C₄⋊C₄×C₆₁	C₂²	1	24	yes	complemented
40	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
41	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
42	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
43	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
44	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
45	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
46	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
47	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
48	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
49	C₂⁵×C₆₁	C₂³	1	2	yes	—
50	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₆₁	—	2	16	no	—
52	C₂²×D₄×C₆₁	—	2	8	no	—
53	C₂²×D₄×C₆₁	—	2	8	no	—
54	C₂²×D₄×C₆₁	—	4	8	no	—
55	C₂²×D₄×C₆₁	C₂³	1	24	yes	—
56	C₂²×D₄×C₆₁	C₂³	1	12	yes	—
57	C₂²×D₄×C₆₁	C₂³	1	12	yes	—
58	C₂²×D₄×C₆₁	C₂³	1	24	yes	—
59	C₂³×C₄×C₆₁	—	2	8	no	—
60	C₂³×C₄×C₆₁	—	2	8	no	—
61	C₂³×C₄×C₆₁	—	4	8	no	—
62	C₂³×C₄×C₆₁	—	4	8	no	—
63	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
64	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
65	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₁	C₂³	1	24	yes	—
73	C₂³×C₄×C₆₁	C₂³	1	24	yes	—
74	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
75	#126×C₆₁	C₂³	1	128	yes	complemented
76	#125×C₆₁	C₂³	1	64	yes	complemented
77	#123×C₆₁	C₂³	1	128	yes	complemented
78	#121×C₆₁	C₂³	1	128	yes	complemented
79	#121×C₆₁	C₂³	1	64	yes	complemented
80	#121×C₆₁	C₂³	1	64	yes	complemented
81	C₄×D₄×C₆₁	C₂³	1	128	yes	complemented
82	C₄×D₄×C₆₁	C₂³	1	64	yes	complemented
83	C₄×D₄×C₆₁	C₂³	1	64	yes	complemented
84	#117×C₆₁	C₂³	1	64	yes	complemented
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
86	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
87	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
89	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
90	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
91	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
92	C₂×C₄⋊C₄×C₆₁	C₂³	1	64	yes	complemented
93	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
94	C₂×C₂²⋊C₄×C₆₁	C₂³	1	12	yes	—
95	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
96	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
97	C₂×C₂²⋊C₄×C₆₁	C₂³	1	12	yes	—
98	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
99	C₂×C₄²×C₆₁	C₂³	1	24	yes	—
100	C₂×C₄²×C₆₁	C₂³	1	12	yes	—
101	C₂⁴×C₆₁	—	2	4	no	—
102	C₂⁴×C₆₁	—	2	4	no	—
103	C₂⁴×C₆₁	—	2	4	no	—
104	C₂⁴×C₆₁	—	2	4	no	—
105	C₂⁴×C₆₁	—	2	8	no	—
106	C₂⁴×C₆₁	—	2	8	no	—
107	C₂⁴×C₆₁	—	2	8	no	—
108	C₂⁴×C₆₁	—	2	4	no	—
109	C₂⁴×C₆₁	#41	1	8	yes	—
110	C₂⁴×C₆₁	C₂⁴	1	6	yes	—
111	C₂⁴×C₆₁	C₂⁴	1	12	yes	—
112	C₂⁴×C₆₁	C₂⁴	1	6	yes	—
113	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
114	C₂×D₄×C₆₁	—	2	96	no	—
115	C₂×D₄×C₆₁	—	2	48	no	—
116	C₂×D₄×C₆₁	—	2	48	no	—
117	C₂×D₄×C₆₁	—	4	48	no	—
118	C₂×D₄×C₆₁	C₂⁴	1	32	yes	—
119	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
120	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
121	C₂×D₄×C₆₁	C₂⁴	1	32	yes	—
122	C₂²×C₄×C₆₁	—	2	4	no	—
123	C₂²×C₄×C₆₁	—	2	4	no	—
124	C₂²×C₄×C₆₁	—	2	8	no	—
125	C₂²×C₄×C₆₁	—	2	4	no	—
126	C₂²×C₄×C₆₁	—	2	4	no	—
127	C₂²×C₄×C₆₁	—	2	8	no	—
128	C₂²×C₄×C₆₁	—	2	8	no	—
129	C₂²×C₄×C₆₁	—	2	48	no	—
130	C₂²×C₄×C₆₁	—	2	48	no	—
131	C₂²×C₄×C₆₁	—	4	48	no	—
132	C₂²×C₄×C₆₁	—	4	48	no	—
133	C₂²×C₄×C₆₁	#41	1	8	yes	—
134	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
135	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
136	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
137	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
138	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
139	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
140	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
141	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
142	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
143	C₂²×C₄×C₆₁	C₂⁴	1	32	yes	—
144	C₂²×C₄×C₆₁	C₂⁴	1	32	yes	—
145	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
146	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
147	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
148	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
149	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
150	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
151	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
152	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
153	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
154	C₂²⋊C₄×C₆₁	C₂⁴	1	16	yes	—
155	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
156	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
157	C₂²⋊C₄×C₆₁	C₂⁴	1	16	yes	—
158	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
159	C₄²×C₆₁	C₂⁴	1	32	yes	—
160	C₄²×C₆₁	C₂⁴	1	16	yes	—
161	C₂³×C₆₁	—	2	32	no	—
162	C₂³×C₆₁	—	2	12	no	—
163	C₂³×C₆₁	—	2	12	no	—
164	C₂³×C₆₁	—	2	12	no	—
165	C₂³×C₆₁	—	2	12	no	—
166	C₂³×C₆₁	—	2	24	no	—
167	C₂³×C₆₁	—	2	24	no	—
168	C₂³×C₆₁	—	2	24	no	—
169	C₂³×C₆₁	—	2	12	no	—
170	C₂³×C₆₁	—	4	16	no	—
171	C₂³×C₆₁	—	4	32	no	—
172	C₂³×C₆₁	C₂×#41	1	24	yes	—
173	C₂³×C₆₁	C₂×#41	1	4	yes	—
174	C₂³×C₆₁	C₂×#41	1	4	yes	—
175	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
176	C₂³×C₆₁	C₂⁵	1	4	yes	—
177	C₂³×C₆₁	C₂⁵	1	8	yes	—
178	C₂³×C₆₁	C₂⁵	1	4	yes	—
179	C₂³×C₆₁	C₂⁵	1	3	yes	—
180	D₄×C₆₁	—	2	64	no	—
181	D₄×C₆₁	—	2	32	no	—
182	D₄×C₆₁	—	2	32	no	—
183	D₄×C₆₁	—	4	32	no	—
184	C₂×C₄×C₆₁	—	2	12	no	—
185	C₂×C₄×C₆₁	—	2	12	no	—
186	C₂×C₄×C₆₁	—	2	24	no	—
187	C₂×C₄×C₆₁	—	2	12	no	—
188	C₂×C₄×C₆₁	—	2	12	no	—
189	C₂×C₄×C₆₁	—	2	24	no	—
190	C₂×C₄×C₆₁	—	2	24	no	—
191	C₂×C₄×C₆₁	—	2	32	no	—
192	C₂×C₄×C₆₁	—	2	32	no	—
193	C₂×C₄×C₆₁	—	4	32	no	—
194	C₂×C₄×C₆₁	—	4	32	no	—
195	C₂×C₄×C₆₁	C₂×#41	1	24	yes	—
196	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
197	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
198	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
199	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
200	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
201	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
202	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
203	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
204	C₂²×#553	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂²×C₆₁	—	2	4	no	—
206	C₂²×C₆₁	—	2	4	no	—
207	C₂²×C₆₁	—	2	48	no	—
208	C₂²×C₆₁	—	2	4	no	—
209	C₂²×C₆₁	—	2	4	no	—
210	C₂²×C₆₁	—	2	8	no	—
211	C₂²×C₆₁	—	2	8	no	—
212	C₂²×C₆₁	—	2	8	no	—
213	C₂²×C₆₁	—	2	4	no	—
214	C₂²×C₆₁	—	4	24	no	—
215	C₂²×C₆₁	—	4	48	no	—
216	C₂²×C₆₁	#553	1	16	yes	direct product factor
217	C₂²×C₆₁	C₂²×#41	1	8	yes	—
218	C₂²×C₆₁	C₂²×#41	1	6	yes	—
219	C₂²×C₆₁	C₂²×#41	1	6	yes	—
220	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	6	yes	—
221	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
222	C₄×C₆₁	—	2	4	no	—
223	C₄×C₆₁	—	2	4	no	—
224	C₄×C₆₁	—	2	8	no	—
225	C₄×C₆₁	—	2	4	no	—
226	C₄×C₆₁	—	2	4	no	—
227	C₄×C₆₁	—	2	8	no	—
228	C₄×C₆₁	—	2	8	no	—
229	C₄×C₆₁	C₂²×#41	1	8	yes	—
230	C₂×#553	C₂×C₆₁	1	48	yes	direct product factor
231	C₂²×#126	C₂×C₆₁	1	2	yes	complemented
232	C₂²×#125	C₂×C₆₁	1	1	yes	characteristic, complemented
233	C₂²×#123	C₂×C₆₁	1	2	yes	complemented
234	C₂²×#121	C₂×C₆₁	1	2	yes	complemented
235	C₂²×#121	C₂×C₆₁	1	1	yes	characteristic, complemented
236	C₂²×#121	C₂×C₆₁	1	1	yes	characteristic, complemented
237	C₂²×C₄×D₄	C₂×C₆₁	1	2	yes	complemented
238	C₂²×C₄×D₄	C₂×C₆₁	1	1	yes	characteristic, complemented
239	C₂²×C₄×D₄	C₂×C₆₁	1	1	yes	characteristic, complemented
240	C₂²×#117	C₂×C₆₁	1	1	yes	characteristic, complemented
241	C₂³×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
242	C₂×C₆₁	—	2	8	no	—
243	C₂×C₆₁	—	4	4	no	—
244	C₂×C₆₁	—	4	8	no	—
245	C₂×C₆₁	C₂×#553	1	12	yes	direct product factor
246	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
247	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
248	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
249	C₂³×D₄	C₂²×C₆₁	1	2	yes	—
250	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
251	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
252	C₂³×D₄	C₂²×C₆₁	1	2	yes	—
253	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
254	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
255	C₂⁴×C₄	C₂²×C₆₁	1	2	yes	—
256	C₂⁴×C₄	C₂²×C₆₁	1	2	yes	—
257	#553	C₂²×C₆₁	1	256	yes	direct product factor
258	C₂×#126	C₂²×C₆₁	1	48	yes	complemented
259	C₂×#125	C₂²×C₆₁	1	24	yes	complemented
260	C₂×#123	C₂²×C₆₁	1	48	yes	complemented
261	C₂×#121	C₂²×C₆₁	1	48	yes	complemented
262	C₂×#121	C₂²×C₆₁	1	24	yes	complemented
263	C₂×#121	C₂²×C₆₁	1	24	yes	complemented
264	C₂×C₄×D₄	C₂²×C₆₁	1	48	yes	complemented
265	C₂×C₄×D₄	C₂²×C₆₁	1	24	yes	complemented
266	C₂×C₄×D₄	C₂²×C₆₁	1	24	yes	complemented
267	C₂×#117	C₂²×C₆₁	1	24	yes	complemented
268	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
269	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
273	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
274	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₆₁	1	24	yes	complemented
276	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
277	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
278	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
279	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
280	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
281	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
282	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
283	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
284	C₆₁	C₂²×#553	1	1	yes	characteristic, direct product factor, Sylow
285	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
286	C₂⁵	C₂³×C₆₁	1	2	yes	—
287	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
288	C₂²×D₄	—	2	16	no	—
289	C₂²×D₄	—	2	8	no	—
290	C₂²×D₄	—	2	8	no	—
291	C₂²×D₄	—	4	8	no	—
292	C₂²×D₄	C₂³×C₆₁	1	24	yes	—
293	C₂²×D₄	C₂³×C₆₁	1	12	yes	—
294	C₂²×D₄	C₂³×C₆₁	1	12	yes	—
295	C₂²×D₄	C₂³×C₆₁	1	24	yes	—
296	C₂³×C₄	—	2	8	no	—
297	C₂³×C₄	—	2	8	no	—
298	C₂³×C₄	—	4	8	no	—
299	C₂³×C₄	—	4	8	no	—
300	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
302	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
303	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
305	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
307	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
308	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
309	C₂³×C₄	C₂³×C₆₁	1	24	yes	—
310	C₂³×C₄	C₂³×C₆₁	1	24	yes	—
311	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
312	#126	C₂³×C₆₁	1	128	yes	complemented
313	#125	C₂³×C₆₁	1	64	yes	complemented
314	#123	C₂³×C₆₁	1	128	yes	complemented
315	#121	C₂³×C₆₁	1	128	yes	complemented
316	#121	C₂³×C₆₁	1	64	yes	complemented
317	#121	C₂³×C₆₁	1	64	yes	complemented
318	C₄×D₄	C₂³×C₆₁	1	128	yes	complemented
319	C₄×D₄	C₂³×C₆₁	1	64	yes	complemented
320	C₄×D₄	C₂³×C₆₁	1	64	yes	complemented
321	#117	C₂³×C₆₁	1	64	yes	complemented
322	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
323	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
324	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
325	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
326	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
327	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
328	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
329	C₂×C₄⋊C₄	C₂³×C₆₁	1	64	yes	complemented
330	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₆₁	1	12	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₆₁	1	12	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
336	C₂×C₄²	C₂³×C₆₁	1	24	yes	—
337	C₂×C₄²	C₂³×C₆₁	1	12	yes	—
338	C₂⁴	—	2	4	no	—
339	C₂⁴	—	2	4	no	—
340	C₂⁴	—	2	4	no	—
341	C₂⁴	—	2	4	no	—
342	C₂⁴	—	2	8	no	—
343	C₂⁴	—	2	8	no	—
344	C₂⁴	—	2	8	no	—
345	C₂⁴	—	2	4	no	—
346	C₂⁴	#41×C₆₁	1	8	yes	—
347	C₂⁴	C₂⁴×C₆₁	1	6	yes	—
348	C₂⁴	C₂⁴×C₆₁	1	12	yes	—
349	C₂⁴	C₂⁴×C₆₁	1	6	yes	—
350	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
351	C₂×D₄	—	2	96	no	—
352	C₂×D₄	—	2	48	no	—
353	C₂×D₄	—	2	48	no	—
354	C₂×D₄	—	4	48	no	—
355	C₂×D₄	C₂⁴×C₆₁	1	32	yes	—
356	C₂×D₄	C₂⁴×C₆₁	1	16	yes	—
357	C₂×D₄	C₂⁴×C₆₁	1	16	yes	—
358	C₂×D₄	C₂⁴×C₆₁	1	32	yes	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	8	no	—
365	C₂²×C₄	—	2	8	no	—
366	C₂²×C₄	—	2	48	no	—
367	C₂²×C₄	—	2	48	no	—
368	C₂²×C₄	—	4	48	no	—
369	C₂²×C₄	—	4	48	no	—
370	C₂²×C₄	#41×C₆₁	1	8	yes	—
371	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
372	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
373	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
374	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
375	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
376	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
377	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
378	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
379	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
380	C₂²×C₄	C₂⁴×C₆₁	1	32	yes	—
381	C₂²×C₄	C₂⁴×C₆₁	1	32	yes	—
382	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
383	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
384	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
385	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
386	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
387	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
388	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
389	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
390	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
391	C₂²⋊C₄	C₂⁴×C₆₁	1	16	yes	—
392	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
393	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
394	C₂²⋊C₄	C₂⁴×C₆₁	1	16	yes	—
395	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
396	C₄²	C₂⁴×C₆₁	1	32	yes	—
397	C₄²	C₂⁴×C₆₁	1	16	yes	—
398	C₂³	—	2	32	no	—
399	C₂³	—	2	12	no	—
400	C₂³	—	2	12	no	—
401	C₂³	—	2	12	no	—
402	C₂³	—	2	12	no	—
403	C₂³	—	2	24	no	—
404	C₂³	—	2	24	no	—
405	C₂³	—	2	24	no	—
406	C₂³	—	2	12	no	—
407	C₂³	—	4	16	no	—
408	C₂³	—	4	32	no	—
409	C₂³	C₂×#41×C₆₁	1	24	yes	—
410	C₂³	C₂×#41×C₆₁	1	4	yes	—
411	C₂³	C₂×#41×C₆₁	1	4	yes	—
412	C₂³	H(5,𝔽₂)×C₆₁	1	4	yes	—
413	C₂³	C₂⁵×C₆₁	1	4	yes	—
414	C₂³	C₂⁵×C₆₁	1	8	yes	—
415	C₂³	C₂⁵×C₆₁	1	4	yes	—
416	C₂³	C₂⁵×C₆₁	1	3	yes	—
417	D₄	—	2	64	no	—
418	D₄	—	2	32	no	—
419	D₄	—	2	32	no	—
420	D₄	—	4	32	no	—
421	C₂×C₄	—	2	12	no	—
422	C₂×C₄	—	2	12	no	—
423	C₂×C₄	—	2	24	no	—
424	C₂×C₄	—	2	12	no	—
425	C₂×C₄	—	2	12	no	—
426	C₂×C₄	—	2	24	no	—
427	C₂×C₄	—	2	24	no	—
428	C₂×C₄	—	2	32	no	—
429	C₂×C₄	—	2	32	no	—
430	C₂×C₄	—	4	32	no	—
431	C₂×C₄	—	4	32	no	—
432	C₂×C₄	C₂×#41×C₆₁	1	24	yes	—
433	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
434	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
435	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
436	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
437	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
438	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
439	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
440	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
441	C₂²	—	2	4	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	48	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	8	no	—
447	C₂²	—	2	8	no	—
448	C₂²	—	2	8	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	4	24	no	—
451	C₂²	—	4	48	no	—
452	C₂²	C₆₁×#553	1	16	yes	direct product factor
453	C₂²	C₂²×#41×C₆₁	1	8	yes	—
454	C₂²	C₂²×#41×C₆₁	1	6	yes	—
455	C₂²	C₂²×#41×C₆₁	1	6	yes	—
456	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	6	yes	—
457	C₂²	C₂⁶×C₆₁	1	1	yes	characteristic
458	C₄	—	2	4	no	—
459	C₄	—	2	4	no	—
460	C₄	—	2	8	no	—
461	C₄	—	2	4	no	—
462	C₄	—	2	4	no	—
463	C₄	—	2	8	no	—
464	C₄	—	2	8	no	—
465	C₄	C₂²×#41×C₆₁	1	8	yes	—
466	C₂	—	2	8	no	—
467	C₂	—	4	4	no	—
468	C₂	—	4	8	no	—
469	C₂	C₂×C₆₁×#553	1	12	yes	direct product factor
470	C₂	C₂³×#41×C₆₁	1	1	yes	characteristic
471	C₂	C₂³×#41×C₆₁	1	1	yes	characteristic
472	C₂	C₂²×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#553	C₂	direct (N × H)
2. C₂²×#126×C₆₁	C₂	semidirect
3. C₂²×#125×C₆₁	C₂	semidirect
4. C₂²×#123×C₆₁	C₂	semidirect
5. C₂²×#121×C₆₁	C₂	semidirect
6. C₂²×#121×C₆₁	C₂	semidirect
7. C₂²×#121×C₆₁	C₂	semidirect
8. C₂²×C₄×D₄×C₆₁	C₂	semidirect
9. C₂²×C₄×D₄×C₆₁	C₂	semidirect
10. C₂²×C₄×D₄×C₆₁	C₂	semidirect
11. C₂²×#117×C₆₁	C₂	semidirect
12. C₂³×C₄⋊C₄×C₆₁	C₂	semidirect
21. C₆₁×#553	C₂²	direct (N × H)
22. C₂×#126×C₆₁	C₂²	semidirect
23. C₂×#125×C₆₁	C₂²	semidirect
24. C₂×#123×C₆₁	C₂²	semidirect
25. C₂×#121×C₆₁	C₂²	semidirect
26. C₂×#121×C₆₁	C₂²	semidirect
27. C₂×#121×C₆₁	C₂²	semidirect
28. C₂×C₄×D₄×C₆₁	C₂²	semidirect
29. C₂×C₄×D₄×C₆₁	C₂²	semidirect
30. C₂×C₄×D₄×C₆₁	C₂²	semidirect
31. C₂×#117×C₆₁	C₂²	semidirect
39. C₂²×C₄⋊C₄×C₆₁	C₂²	semidirect
75. #126×C₆₁	C₂³	semidirect
76. #125×C₆₁	C₂³	semidirect
77. #123×C₆₁	C₂³	semidirect
78. #121×C₆₁	C₂³	semidirect
79. #121×C₆₁	C₂³	semidirect
80. #121×C₆₁	C₂³	semidirect
81. C₄×D₄×C₆₁	C₂³	semidirect
82. C₄×D₄×C₆₁	C₂³	semidirect
83. C₄×D₄×C₆₁	C₂³	semidirect
84. #117×C₆₁	C₂³	semidirect
92. C₂×C₄⋊C₄×C₆₁	C₂³	semidirect
204. C₂²×#553	C₆₁	direct (N × H)
216. C₂²×C₆₁	#553	direct (N × H)
230. C₂×#553	C₂×C₆₁	direct (N × H)
231. C₂²×#126	C₂×C₆₁	semidirect
232. C₂²×#125	C₂×C₆₁	semidirect
233. C₂²×#123	C₂×C₆₁	semidirect
234. C₂²×#121	C₂×C₆₁	semidirect
235. C₂²×#121	C₂×C₆₁	semidirect
236. C₂²×#121	C₂×C₆₁	semidirect
237. C₂²×C₄×D₄	C₂×C₆₁	semidirect
238. C₂²×C₄×D₄	C₂×C₆₁	semidirect
239. C₂²×C₄×D₄	C₂×C₆₁	semidirect
240. C₂²×#117	C₂×C₆₁	semidirect
241. C₂³×C₄⋊C₄	C₂×C₆₁	semidirect
245. C₂×C₆₁	C₂×#553	direct (N × H)
257. #553	C₂²×C₆₁	direct (N × H)
258. C₂×#126	C₂²×C₆₁	semidirect
259. C₂×#125	C₂²×C₆₁	semidirect
260. C₂×#123	C₂²×C₆₁	semidirect
261. C₂×#121	C₂²×C₆₁	semidirect
262. C₂×#121	C₂²×C₆₁	semidirect
263. C₂×#121	C₂²×C₆₁	semidirect
264. C₂×C₄×D₄	C₂²×C₆₁	semidirect
265. C₂×C₄×D₄	C₂²×C₆₁	semidirect
266. C₂×C₄×D₄	C₂²×C₆₁	semidirect
267. C₂×#117	C₂²×C₆₁	semidirect
275. C₂²×C₄⋊C₄	C₂²×C₆₁	semidirect
284. C₆₁	C₂²×#553	direct (N × H)
312. #126	C₂³×C₆₁	semidirect
313. #125	C₂³×C₆₁	semidirect
314. #123	C₂³×C₆₁	semidirect
315. #121	C₂³×C₆₁	semidirect
316. #121	C₂³×C₆₁	semidirect
317. #121	C₂³×C₆₁	semidirect
318. C₄×D₄	C₂³×C₆₁	semidirect
319. C₄×D₄	C₂³×C₆₁	semidirect
320. C₄×D₄	C₂³×C₆₁	semidirect
321. #117	C₂³×C₆₁	semidirect
329. C₂×C₄⋊C₄	C₂³×C₆₁	semidirect
452. C₂²	C₆₁×#553	direct (N × H)
469. C₂	C₂×C₆₁×#553	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#553	2
C₂²×#126×C₆₁	2
C₂²×#125×C₆₁	2
C₂²×#123×C₆₁	2
C₂²×#121×C₆₁	2
C₂²×C₄×D₄×C₆₁	2
C₂²×#117×C₆₁	2
C₂³×C₄⋊C₄×C₆₁	2
C₂²×#553	61

Subgroups — C22×C61×#553 #29566138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#553 #29566138