Subgroups — C₂²×C₆₁×#539 #29566105

Order 15616.

8062 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#539	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#126×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₂²×#125×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₂²×#123×C₆₁	C₂	1	2	yes	maximal, complemented
5	C₂²×#123×C₆₁	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₂²⋊Q₈×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₂²×#121×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄×D₄×C₆₁	C₂	1	2	yes	maximal, complemented
10	C₂³×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₁	C₂²	1	2	yes	—
13	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#539	C₂²	1	256	yes	direct product factor
19	C₂×#126×C₆₁	C₂²	1	48	yes	complemented
20	C₂×#125×C₆₁	C₂²	1	48	yes	complemented
21	C₂×#123×C₆₁	C₂²	1	48	yes	complemented
22	C₂×#123×C₆₁	C₂²	1	48	yes	complemented
23	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	24	yes	complemented
24	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	48	yes	complemented
25	C₂×#121×C₆₁	C₂²	1	24	yes	—
26	C₂×C₄×D₄×C₆₁	C₂²	1	48	yes	complemented
27	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
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	24	yes	complemented
36	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
38	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
39	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
40	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
41	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
42	C₂⁵×C₆₁	C₂³	1	2	yes	—
43	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₆₁	—	4	4	no	—
45	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
46	C₂²×D₄×C₆₁	—	2	16	no	—
47	C₂²×D₄×C₆₁	—	4	4	no	—
48	C₂²×D₄×C₆₁	C₂³	1	24	yes	—
49	C₂²×D₄×C₆₁	C₂³	1	12	yes	—
50	C₂³×C₄×C₆₁	—	2	8	no	—
51	C₂³×C₄×C₆₁	—	4	4	no	—
52	C₂³×C₄×C₆₁	—	4	8	no	—
53	C₂³×C₄×C₆₁	—	4	4	no	—
54	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
55	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
63	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
64	C₂³×C₄×C₆₁	C₂³	1	24	yes	—
65	#126×C₆₁	C₂³	1	128	yes	complemented
66	#125×C₆₁	C₂³	1	128	yes	complemented
67	#123×C₆₁	C₂³	1	128	yes	complemented
68	#123×C₆₁	C₂³	1	128	yes	complemented
69	C₂²⋊Q₈×C₆₁	C₂³	1	64	yes	complemented
70	C₂²⋊Q₈×C₆₁	C₂³	1	128	yes	complemented
71	#121×C₆₁	C₂³	1	64	yes	—
72	C₄×D₄×C₆₁	C₂³	1	128	yes	complemented
73	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
74	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
75	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
76	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
77	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
78	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
79	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
80	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
81	C₂×C₄⋊C₄×C₆₁	C₂³	1	64	yes	complemented
82	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
83	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
84	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
85	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
86	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
87	C₂×C₄²×C₆₁	C₂³	1	24	yes	—
88	C₂⁴×C₆₁	—	2	4	no	—
89	C₂⁴×C₆₁	—	2	4	no	—
90	C₂⁴×C₆₁	—	2	8	no	—
91	C₂⁴×C₆₁	—	2	8	no	—
92	C₂⁴×C₆₁	—	2	8	no	—
93	C₂⁴×C₆₁	#41	1	8	yes	—
94	C₂⁴×C₆₁	C₂⁴	1	12	yes	—
95	C₂⁴×C₆₁	C₂⁴	1	6	yes	—
96	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
97	C₂×Q₈×C₆₁	—	4	24	no	—
98	C₂×Q₈×C₆₁	C₂⁴	1	16	yes	—
99	C₂×D₄×C₆₁	—	2	96	no	—
100	C₂×D₄×C₆₁	—	4	24	no	—
101	C₂×D₄×C₆₁	C₂⁴	1	32	yes	—
102	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
103	C₂²×C₄×C₆₁	—	2	48	no	—
104	C₂²×C₄×C₆₁	—	2	8	no	—
105	C₂²×C₄×C₆₁	—	2	8	no	—
106	C₂²×C₄×C₆₁	—	2	8	no	—
107	C₂²×C₄×C₆₁	—	2	4	no	—
108	C₂²×C₄×C₆₁	—	2	8	no	—
109	C₂²×C₄×C₆₁	—	2	4	no	—
110	C₂²×C₄×C₆₁	—	2	8	no	—
111	C₂²×C₄×C₆₁	—	4	24	no	—
112	C₂²×C₄×C₆₁	—	4	48	no	—
113	C₂²×C₄×C₆₁	—	4	24	no	—
114	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
115	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
116	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
117	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
118	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
119	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
120	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
121	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
122	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
123	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
124	C₂²×C₄×C₆₁	C₂⁴	1	32	yes	—
125	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
126	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
127	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
128	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
129	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
130	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
131	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
132	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
133	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
134	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
135	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
136	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
137	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
138	C₄²×C₆₁	C₂⁴	1	32	yes	—
139	C₂³×C₆₁	—	2	32	no	—
140	C₂³×C₆₁	—	2	12	no	—
141	C₂³×C₆₁	—	2	12	no	—
142	C₂³×C₆₁	—	2	24	no	—
143	C₂³×C₆₁	—	2	24	no	—
144	C₂³×C₆₁	—	2	24	no	—
145	C₂³×C₆₁	—	4	32	no	—
146	C₂³×C₆₁	C₂×#41	1	24	yes	—
147	C₂³×C₆₁	C₂×#41	1	4	yes	—
148	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
149	C₂³×C₆₁	2₋¹⁺⁴	1	4	yes	—
150	C₂³×C₆₁	C₂⁵	1	8	yes	—
151	C₂³×C₆₁	C₂⁵	1	4	yes	—
152	C₂³×C₆₁	C₂⁵	1	3	yes	—
153	Q₈×C₆₁	—	4	16	no	—
154	D₄×C₆₁	—	2	64	no	—
155	D₄×C₆₁	—	4	16	no	—
156	C₂×C₄×C₆₁	—	2	32	no	—
157	C₂×C₄×C₆₁	—	2	24	no	—
158	C₂×C₄×C₆₁	—	2	24	no	—
159	C₂×C₄×C₆₁	—	2	24	no	—
160	C₂×C₄×C₆₁	—	2	12	no	—
161	C₂×C₄×C₆₁	—	2	24	no	—
162	C₂×C₄×C₆₁	—	2	12	no	—
163	C₂×C₄×C₆₁	—	2	24	no	—
164	C₂×C₄×C₆₁	—	4	16	no	—
165	C₂×C₄×C₆₁	—	4	32	no	—
166	C₂×C₄×C₆₁	—	4	16	no	—
167	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
168	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
169	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
170	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
171	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
172	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
173	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
174	C₂²×#539	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂²×C₆₁	—	2	4	no	—
176	C₂²×C₆₁	—	2	4	no	—
177	C₂²×C₆₁	—	2	48	no	—
178	C₂²×C₆₁	—	2	8	no	—
179	C₂²×C₆₁	—	2	8	no	—
180	C₂²×C₆₁	—	2	8	no	—
181	C₂²×C₆₁	—	4	48	no	—
182	C₂²×C₆₁	#539	1	16	yes	direct product factor
183	C₂²×C₆₁	C₂²×#41	1	8	yes	—
184	C₂²×C₆₁	C₂²×#41	1	6	yes	—
185	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	6	yes	—
186	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	6	yes	—
187	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
188	C₄×C₆₁	—	2	8	no	—
189	C₄×C₆₁	—	2	8	no	—
190	C₄×C₆₁	—	2	8	no	—
191	C₄×C₆₁	—	2	4	no	—
192	C₄×C₆₁	—	2	8	no	—
193	C₄×C₆₁	—	2	4	no	—
194	C₄×C₆₁	—	2	8	no	—
195	C₂×#539	C₂×C₆₁	1	48	yes	direct product factor
196	C₂²×#126	C₂×C₆₁	1	2	yes	complemented
197	C₂²×#125	C₂×C₆₁	1	2	yes	complemented
198	C₂²×#123	C₂×C₆₁	1	2	yes	complemented
199	C₂²×#123	C₂×C₆₁	1	2	yes	complemented
200	C₂²×C₂²⋊Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
201	C₂²×C₂²⋊Q₈	C₂×C₆₁	1	2	yes	complemented
202	C₂²×#121	C₂×C₆₁	1	1	yes	characteristic
203	C₂²×C₄×D₄	C₂×C₆₁	1	2	yes	complemented
204	C₂³×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
205	C₂×C₆₁	—	2	8	no	—
206	C₂×C₆₁	—	4	8	no	—
207	C₂×C₆₁	C₂×#539	1	12	yes	direct product factor
208	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
209	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₆₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
212	C₂³×D₄	C₂²×C₆₁	1	2	yes	—
213	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
214	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
215	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
216	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
217	C₂⁴×C₄	C₂²×C₆₁	1	2	yes	—
218	#539	C₂²×C₆₁	1	256	yes	direct product factor
219	C₂×#126	C₂²×C₆₁	1	48	yes	complemented
220	C₂×#125	C₂²×C₆₁	1	48	yes	complemented
221	C₂×#123	C₂²×C₆₁	1	48	yes	complemented
222	C₂×#123	C₂²×C₆₁	1	48	yes	complemented
223	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	24	yes	complemented
224	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	48	yes	complemented
225	C₂×#121	C₂²×C₆₁	1	24	yes	—
226	C₂×C₄×D₄	C₂²×C₆₁	1	48	yes	complemented
227	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
228	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
229	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
230	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
231	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
232	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
233	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
234	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
235	C₂²×C₄⋊C₄	C₂²×C₆₁	1	24	yes	complemented
236	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
237	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
238	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
239	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
240	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
241	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
242	C₆₁	C₂²×#539	1	1	yes	characteristic, direct product factor, Sylow
243	C₂⁵	C₂³×C₆₁	1	2	yes	—
244	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
245	C₂²×Q₈	—	4	4	no	—
246	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
247	C₂²×D₄	—	2	16	no	—
248	C₂²×D₄	—	4	4	no	—
249	C₂²×D₄	C₂³×C₆₁	1	24	yes	—
250	C₂²×D₄	C₂³×C₆₁	1	12	yes	—
251	C₂³×C₄	—	2	8	no	—
252	C₂³×C₄	—	4	4	no	—
253	C₂³×C₄	—	4	8	no	—
254	C₂³×C₄	—	4	4	no	—
255	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
256	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
257	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
258	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
259	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
260	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
261	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
262	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
263	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
264	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
265	C₂³×C₄	C₂³×C₆₁	1	24	yes	—
266	#126	C₂³×C₆₁	1	128	yes	complemented
267	#125	C₂³×C₆₁	1	128	yes	complemented
268	#123	C₂³×C₆₁	1	128	yes	complemented
269	#123	C₂³×C₆₁	1	128	yes	complemented
270	C₂²⋊Q₈	C₂³×C₆₁	1	64	yes	complemented
271	C₂²⋊Q₈	C₂³×C₆₁	1	128	yes	complemented
272	#121	C₂³×C₆₁	1	64	yes	—
273	C₄×D₄	C₂³×C₆₁	1	128	yes	complemented
274	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
275	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
276	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
277	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₁	1	64	yes	complemented
283	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
288	C₂×C₄²	C₂³×C₆₁	1	24	yes	—
289	C₂⁴	—	2	4	no	—
290	C₂⁴	—	2	4	no	—
291	C₂⁴	—	2	8	no	—
292	C₂⁴	—	2	8	no	—
293	C₂⁴	—	2	8	no	—
294	C₂⁴	#41×C₆₁	1	8	yes	—
295	C₂⁴	C₂⁴×C₆₁	1	12	yes	—
296	C₂⁴	C₂⁴×C₆₁	1	6	yes	—
297	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
298	C₂×Q₈	—	4	24	no	—
299	C₂×Q₈	C₂⁴×C₆₁	1	16	yes	—
300	C₂×D₄	—	2	96	no	—
301	C₂×D₄	—	4	24	no	—
302	C₂×D₄	C₂⁴×C₆₁	1	32	yes	—
303	C₂×D₄	C₂⁴×C₆₁	1	16	yes	—
304	C₂²×C₄	—	2	48	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	8	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	8	no	—
312	C₂²×C₄	—	4	24	no	—
313	C₂²×C₄	—	4	48	no	—
314	C₂²×C₄	—	4	24	no	—
315	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
316	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
317	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
318	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
319	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
320	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
321	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
322	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
323	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
324	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
325	C₂²×C₄	C₂⁴×C₆₁	1	32	yes	—
326	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
327	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
328	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
329	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
330	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
331	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
332	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
333	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
334	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
335	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
336	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
337	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
338	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
339	C₄²	C₂⁴×C₆₁	1	32	yes	—
340	C₂³	—	2	32	no	—
341	C₂³	—	2	12	no	—
342	C₂³	—	2	12	no	—
343	C₂³	—	2	24	no	—
344	C₂³	—	2	24	no	—
345	C₂³	—	2	24	no	—
346	C₂³	—	4	32	no	—
347	C₂³	C₂×#41×C₆₁	1	24	yes	—
348	C₂³	C₂×#41×C₆₁	1	4	yes	—
349	C₂³	H(5,𝔽₂)×C₆₁	1	4	yes	—
350	C₂³	2₋¹⁺⁴×C₆₁	1	4	yes	—
351	C₂³	C₂⁵×C₆₁	1	8	yes	—
352	C₂³	C₂⁵×C₆₁	1	4	yes	—
353	C₂³	C₂⁵×C₆₁	1	3	yes	—
354	Q₈	—	4	16	no	—
355	D₄	—	2	64	no	—
356	D₄	—	4	16	no	—
357	C₂×C₄	—	2	32	no	—
358	C₂×C₄	—	2	24	no	—
359	C₂×C₄	—	2	24	no	—
360	C₂×C₄	—	2	24	no	—
361	C₂×C₄	—	2	12	no	—
362	C₂×C₄	—	2	24	no	—
363	C₂×C₄	—	2	12	no	—
364	C₂×C₄	—	2	24	no	—
365	C₂×C₄	—	4	16	no	—
366	C₂×C₄	—	4	32	no	—
367	C₂×C₄	—	4	16	no	—
368	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
369	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
370	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
371	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
372	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
373	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
374	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
375	C₂²	—	2	4	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	2	48	no	—
378	C₂²	—	2	8	no	—
379	C₂²	—	2	8	no	—
380	C₂²	—	2	8	no	—
381	C₂²	—	4	48	no	—
382	C₂²	C₆₁×#539	1	16	yes	direct product factor
383	C₂²	C₂²×#41×C₆₁	1	8	yes	—
384	C₂²	C₂²×#41×C₆₁	1	6	yes	—
385	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	6	yes	—
386	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	6	yes	—
387	C₂²	C₂⁶×C₆₁	1	1	yes	characteristic
388	C₄	—	2	8	no	—
389	C₄	—	2	8	no	—
390	C₄	—	2	8	no	—
391	C₄	—	2	4	no	—
392	C₄	—	2	8	no	—
393	C₄	—	2	4	no	—
394	C₄	—	2	8	no	—
395	C₂	—	2	8	no	—
396	C₂	—	4	8	no	—
397	C₂	C₂×C₆₁×#539	1	12	yes	direct product factor
398	C₂	C₂³×#41×C₆₁	1	1	yes	characteristic
399	C₂	C₂²×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
400	C₂	C₂²×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₁
prime radical	C₂²×#121×C₆₁	C₂
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×#539	C₂	direct (N × H)
2. C₂²×#126×C₆₁	C₂	semidirect
3. C₂²×#125×C₆₁	C₂	semidirect
4. C₂²×#123×C₆₁	C₂	semidirect
5. C₂²×#123×C₆₁	C₂	semidirect
6. C₂²×C₂²⋊Q₈×C₆₁	C₂	semidirect
7. C₂²×C₂²⋊Q₈×C₆₁	C₂	semidirect
9. C₂²×C₄×D₄×C₆₁	C₂	semidirect
10. C₂³×C₄⋊C₄×C₆₁	C₂	semidirect
18. C₆₁×#539	C₂²	direct (N × H)
19. C₂×#126×C₆₁	C₂²	semidirect
20. C₂×#125×C₆₁	C₂²	semidirect
21. C₂×#123×C₆₁	C₂²	semidirect
22. C₂×#123×C₆₁	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₆₁	C₂²	semidirect
24. C₂×C₂²⋊Q₈×C₆₁	C₂²	semidirect
26. C₂×C₄×D₄×C₆₁	C₂²	semidirect
35. C₂²×C₄⋊C₄×C₆₁	C₂²	semidirect
65. #126×C₆₁	C₂³	semidirect
66. #125×C₆₁	C₂³	semidirect
67. #123×C₆₁	C₂³	semidirect
68. #123×C₆₁	C₂³	semidirect
69. C₂²⋊Q₈×C₆₁	C₂³	semidirect
70. C₂²⋊Q₈×C₆₁	C₂³	semidirect
72. C₄×D₄×C₆₁	C₂³	semidirect
81. C₂×C₄⋊C₄×C₆₁	C₂³	semidirect
174. C₂²×#539	C₆₁	direct (N × H)
182. C₂²×C₆₁	#539	direct (N × H)
195. C₂×#539	C₂×C₆₁	direct (N × H)
196. C₂²×#126	C₂×C₆₁	semidirect
197. C₂²×#125	C₂×C₆₁	semidirect
198. C₂²×#123	C₂×C₆₁	semidirect
199. C₂²×#123	C₂×C₆₁	semidirect
200. C₂²×C₂²⋊Q₈	C₂×C₆₁	semidirect
201. C₂²×C₂²⋊Q₈	C₂×C₆₁	semidirect
203. C₂²×C₄×D₄	C₂×C₆₁	semidirect
204. C₂³×C₄⋊C₄	C₂×C₆₁	semidirect
207. C₂×C₆₁	C₂×#539	direct (N × H)
218. #539	C₂²×C₆₁	direct (N × H)
219. C₂×#126	C₂²×C₆₁	semidirect
220. C₂×#125	C₂²×C₆₁	semidirect
221. C₂×#123	C₂²×C₆₁	semidirect
222. C₂×#123	C₂²×C₆₁	semidirect
223. C₂×C₂²⋊Q₈	C₂²×C₆₁	semidirect
224. C₂×C₂²⋊Q₈	C₂²×C₆₁	semidirect
226. C₂×C₄×D₄	C₂²×C₆₁	semidirect
235. C₂²×C₄⋊C₄	C₂²×C₆₁	semidirect
242. C₆₁	C₂²×#539	direct (N × H)
266. #126	C₂³×C₆₁	semidirect
267. #125	C₂³×C₆₁	semidirect
268. #123	C₂³×C₆₁	semidirect
269. #123	C₂³×C₆₁	semidirect
270. C₂²⋊Q₈	C₂³×C₆₁	semidirect
271. C₂²⋊Q₈	C₂³×C₆₁	semidirect
273. C₄×D₄	C₂³×C₆₁	semidirect
282. C₂×C₄⋊C₄	C₂³×C₆₁	semidirect
382. C₂²	C₆₁×#539	direct (N × H)
397. C₂	C₂×C₆₁×#539	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×#539	2
C₂²×#126×C₆₁	2
C₂²×#125×C₆₁	2
C₂²×#123×C₆₁	2
C₂²×C₂²⋊Q₈×C₆₁	2
C₂²×#121×C₆₁	2
C₂²×C₄×D₄×C₆₁	2
C₂³×C₄⋊C₄×C₆₁	2
C₂²×#539	61

Subgroups — C22×C61×#539 #29566105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#539 #29566105