Subgroups — C₂×C₆₁×#3193 #29564169

Order 15616.

2666 conjugacy classes of subgroups.

The subgroup lattice has 576 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	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3193	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₆₁×#484	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₁×#480	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#468	C₂	1	4	yes	maximal, complemented
7	C₂×C₆₁×#448	C₂	1	2	yes	maximal, complemented
8	C₂×C₆₁×#450	C₂	1	2	yes	maximal, complemented
9	C₂×C₆₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₄×C₆₁	—	2	1	no	—
12	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₁	C₂²	1	4	yes	complemented
15	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₆₁	—	2	4	no	—
19	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	complemented
21	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	8	yes	complemented
22	C₂×#121×C₆₁	—	2	1	no	—
23	C₂×#121×C₆₁	—	2	4	no	—
24	C₂×#121×C₆₁	—	2	1	no	—
25	C₂×#121×C₆₁	C₂²	1	2	yes	—
26	C₂×#121×C₆₁	C₂²	1	2	yes	—
27	C₂×#121×C₆₁	C₂²	1	8	yes	—
28	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₂²⋊C₄×C₆₁	—	2	1	no	—
32	C₂²×C₂²⋊C₄×C₆₁	—	2	1	no	—
33	C₆₁×#484	C₂²	1	16	yes	complemented
34	C₆₁×#480	C₂²	1	16	yes	complemented
35	C₆₁×#468	C₂²	1	32	yes	complemented
36	C₆₁×#450	C₂²	1	16	yes	complemented
37	C₆₁×#448	C₂²	1	16	yes	complemented
38	C₂×#107×C₆₁	C₂²	1	4	yes	complemented
39	C₂×#103×C₆₁	C₂²	1	4	yes	complemented
40	C₂×#102×C₆₁	C₂²	1	4	yes	complemented
41	C₆₁×#407	C₂²	1	8	yes	complemented
42	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	complemented
44	C₂⁵×C₆₁	—	2	1	no	—
45	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₆₁	—	2	2	no	—
47	C₂²×Q₈×C₆₁	—	2	1	no	—
48	C₂²×Q₈×C₆₁	—	4	4	no	—
49	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
50	C₂²×Q₈×C₆₁	C₂³	1	4	yes	complemented
51	C₂²×D₄×C₆₁	—	2	8	no	—
52	C₂²×D₄×C₆₁	—	2	2	no	—
53	C₂²×D₄×C₆₁	—	2	2	no	—
54	C₂²×D₄×C₆₁	—	2	2	no	—
55	C₂²×D₄×C₆₁	—	2	1	no	—
56	C₂²×D₄×C₆₁	—	4	4	no	—
57	C₂²×D₄×C₆₁	—	4	8	no	—
58	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
60	C₂³×C₄×C₆₁	—	2	1	no	—
61	C₂³×C₄×C₆₁	—	2	1	no	—
62	C₂³×C₄×C₆₁	—	2	1	no	—
63	C₂³×C₄×C₆₁	—	4	4	no	—
64	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₆₁	D₄	1	4	yes	—
67	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
72	C₂×SD₁₆×C₆₁	—	4	8	no	—
73	C₂×SD₁₆×C₆₁	C₂³	1	16	yes	complemented
74	C₂×M₁₆×C₆₁	—	4	4	no	—
75	C₂×M₁₆×C₆₁	C₂³	1	8	yes	complemented
76	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
77	C₂²⋊Q₈×C₆₁	—	2	4	no	—
78	C₂²⋊Q₈×C₆₁	—	2	8	no	—
79	C₂²⋊Q₈×C₆₁	—	2	8	no	—
80	C₂²⋊Q₈×C₆₁	—	2	4	no	—
81	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
82	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	complemented
83	#121×C₆₁	—	2	4	no	—
84	#121×C₆₁	—	2	4	no	—
85	#121×C₆₁	—	2	8	no	—
86	#121×C₆₁	—	2	8	no	—
87	#121×C₆₁	C₂³	1	8	yes	—
88	#121×C₆₁	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
91	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	complemented
100	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
102	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
104	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
106	#107×C₆₁	C₂³	1	16	yes	complemented
107	#103×C₆₁	C₂³	1	16	yes	complemented
108	#102×C₆₁	C₂³	1	16	yes	complemented
109	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
110	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	complemented
111	C₂⁴×C₆₁	—	2	4	no	—
112	C₂⁴×C₆₁	—	2	4	no	—
113	C₂⁴×C₆₁	—	2	4	no	—
114	C₂⁴×C₆₁	—	2	2	no	—
115	C₂⁴×C₆₁	—	4	2	no	—
116	C₂⁴×C₆₁	—	4	4	no	—
117	C₂⁴×C₆₁	—	4	4	no	—
118	C₂⁴×C₆₁	—	4	4	no	—
119	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
120	C₂⁴×C₆₁	C₂×D₄	1	4	yes	—
121	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₆₁	—	2	2	no	—
124	C₂×Q₈×C₆₁	—	2	4	no	—
125	C₂×Q₈×C₆₁	—	2	8	no	—
126	C₂×Q₈×C₆₁	—	4	8	no	—
127	C₂×Q₈×C₆₁	—	4	4	no	—
128	C₂×Q₈×C₆₁	—	4	8	no	—
129	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
130	C₂×D₄×C₆₁	—	2	8	no	—
131	C₂×D₄×C₆₁	—	2	8	no	—
132	C₂×D₄×C₆₁	—	2	4	no	—
133	C₂×D₄×C₆₁	—	2	2	no	—
134	C₂×D₄×C₆₁	—	2	8	no	—
135	C₂×D₄×C₆₁	—	4	8	no	—
136	C₂×D₄×C₆₁	—	4	4	no	—
137	C₂×D₄×C₆₁	—	4	8	no	—
138	C₂×D₄×C₆₁	—	4	32	no	—
139	C₂×D₄×C₆₁	—	4	8	no	—
140	C₂×D₄×C₆₁	—	4	8	no	—
141	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₁	—	2	4	no	—
143	C₂²×C₄×C₆₁	—	2	4	no	—
144	C₂²×C₄×C₆₁	—	2	4	no	—
145	C₂²×C₄×C₆₁	—	2	4	no	—
146	C₂²×C₄×C₆₁	—	2	4	no	—
147	C₂²×C₄×C₆₁	—	2	4	no	—
148	C₂²×C₄×C₆₁	—	2	2	no	—
149	C₂²×C₄×C₆₁	—	2	2	no	—
150	C₂²×C₄×C₆₁	—	2	2	no	—
151	C₂²×C₄×C₆₁	—	4	8	no	—
152	C₂²×C₄×C₆₁	—	4	8	no	—
153	C₂²×C₄×C₆₁	—	4	2	no	—
154	C₂²×C₄×C₆₁	—	4	2	no	—
155	C₂²×C₄×C₆₁	—	4	2	no	—
156	C₂²×C₄×C₆₁	—	4	4	no	—
157	C₂²×C₄×C₆₁	—	4	4	no	—
158	C₂²×C₄×C₆₁	—	4	4	no	—
159	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₆₁	—	4	16	no	—
173	M₁₆×C₆₁	—	4	8	no	—
174	C₂×C₈×C₆₁	—	2	8	no	—
175	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
176	C₄⋊C₄×C₆₁	—	2	8	no	—
177	C₄⋊C₄×C₆₁	—	2	8	no	—
178	C₄⋊C₄×C₆₁	—	2	4	no	—
179	C₄⋊C₄×C₆₁	—	2	2	no	—
180	C₄⋊C₄×C₆₁	—	2	2	no	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
184	C₂²⋊C₄×C₆₁	—	2	8	no	—
185	C₂²⋊C₄×C₆₁	—	2	8	no	—
186	C₂²⋊C₄×C₆₁	—	2	8	no	—
187	C₂²⋊C₄×C₆₁	—	2	8	no	—
188	C₂³×C₆₁	—	2	2	no	—
189	C₂³×C₆₁	—	2	2	no	—
190	C₂³×C₆₁	—	2	2	no	—
191	C₂³×C₆₁	—	2	2	no	—
192	C₂³×C₆₁	—	2	2	no	—
193	C₂³×C₆₁	—	2	8	no	—
194	C₂³×C₆₁	—	2	2	no	—
195	C₂³×C₆₁	—	2	8	no	—
196	C₂³×C₆₁	—	2	2	no	—
197	C₂³×C₆₁	—	2	4	no	—
198	C₂³×C₆₁	—	4	4	no	—
199	C₂³×C₆₁	—	4	16	no	—
200	C₂³×C₆₁	—	4	8	no	—
201	C₂³×C₆₁	—	4	8	no	—
202	C₂³×C₆₁	—	4	4	no	—
203	C₂³×C₆₁	—	4	4	no	—
204	C₂³×C₆₁	—	4	8	no	—
205	C₂³×C₆₁	—	4	4	no	—
206	C₂³×C₆₁	—	8	16	no	—
207	C₂³×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
208	C₂³×C₆₁	#137	1	4	yes	—
209	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
210	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
211	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
212	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₆₁	—	2	8	no	—
214	Q₈×C₆₁	—	4	8	no	—
215	Q₈×C₆₁	—	4	4	no	—
216	D₄×C₆₁	—	2	8	no	—
217	D₄×C₆₁	—	4	8	no	—
218	D₄×C₆₁	—	4	16	no	—
219	D₄×C₆₁	—	4	16	no	—
220	D₄×C₆₁	—	4	4	no	—
221	C₂×C₄×C₆₁	—	2	2	no	—
222	C₂×C₄×C₆₁	—	2	2	no	—
223	C₂×C₄×C₆₁	—	2	2	no	—
224	C₂×C₄×C₆₁	—	2	4	no	—
225	C₂×C₄×C₆₁	—	2	4	no	—
226	C₂×C₄×C₆₁	—	2	4	no	—
227	C₂×C₄×C₆₁	—	2	4	no	—
228	C₂×C₄×C₆₁	—	2	4	no	—
229	C₂×C₄×C₆₁	—	2	4	no	—
230	C₂×C₄×C₆₁	—	2	4	no	—
231	C₂×C₄×C₆₁	—	2	4	no	—
232	C₂×C₄×C₆₁	—	2	4	no	—
233	C₂×C₄×C₆₁	—	4	4	no	—
234	C₂×C₄×C₆₁	—	4	4	no	—
235	C₂×C₄×C₆₁	—	4	4	no	—
236	C₂×C₄×C₆₁	—	4	8	no	—
237	C₂×C₄×C₆₁	—	4	8	no	—
238	C₂×C₄×C₆₁	—	4	4	no	—
239	C₂×C₄×C₆₁	—	4	8	no	—
240	C₂×C₄×C₆₁	—	4	4	no	—
241	C₂×C₄×C₆₁	—	4	4	no	—
242	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
247	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
248	C₈×C₆₁	—	2	8	no	—
249	C₂×#3193	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂²×C₆₁	—	2	2	no	—
251	C₂²×C₆₁	—	2	2	no	—
252	C₂²×C₆₁	—	2	8	no	—
253	C₂²×C₆₁	—	2	8	no	—
254	C₂²×C₆₁	—	2	2	no	—
255	C₂²×C₆₁	—	2	2	no	—
256	C₂²×C₆₁	—	2	2	no	—
257	C₂²×C₆₁	—	2	2	no	—
258	C₂²×C₆₁	—	2	2	no	—
259	C₂²×C₆₁	—	4	16	no	—
260	C₂²×C₆₁	—	4	4	no	—
261	C₂²×C₆₁	—	4	4	no	—
262	C₂²×C₆₁	—	4	4	no	—
263	C₂²×C₆₁	—	4	4	no	—
264	C₂²×C₆₁	—	8	16	no	—
265	C₂²×C₆₁	—	8	8	no	—
266	C₂²×C₆₁	#534	1	2	yes	—
267	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
268	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
269	C₂²×C₆₁	C₂×#137	1	4	yes	—
270	C₂²×C₆₁	C₂×#137	1	2	yes	—
271	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
272	C₄×C₆₁	—	2	2	no	—
273	C₄×C₆₁	—	2	2	no	—
274	C₄×C₆₁	—	2	2	no	—
275	C₄×C₆₁	—	4	4	no	—
276	C₄×C₆₁	—	4	4	no	—
277	C₄×C₆₁	—	4	4	no	—
278	C₄×C₆₁	#534	1	2	yes	—
279	C₄×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
280	C₂²×C₂²⋊Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
281	C₂²×#121	C₂×C₆₁	1	1	yes	characteristic
282	#3193	C₂×C₆₁	1	16	yes	direct product factor
283	C₂×#484	C₂×C₆₁	1	2	yes	complemented
284	C₂×#480	C₂×C₆₁	1	2	yes	complemented
285	C₂×#468	C₂×C₆₁	1	4	yes	complemented
286	C₂×#448	C₂×C₆₁	1	2	yes	complemented
287	C₂×#450	C₂×C₆₁	1	2	yes	complemented
288	C₂×#407	C₂×C₆₁	1	1	yes	characteristic, complemented
289	C₂×C₆₁	—	2	4	no	—
290	C₂×C₆₁	—	2	4	no	—
291	C₂×C₆₁	—	4	2	no	—
292	C₂×C₆₁	—	8	4	no	—
293	C₂×C₆₁	#3193	1	4	yes	direct product factor
294	C₂×C₆₁	C₂×#534	1	1	yes	characteristic
295	C₂×C₆₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
297	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
298	C₂³×D₄	—	2	1	no	—
299	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
300	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
301	C₂²×SD₁₆	C₂²×C₆₁	1	4	yes	complemented
302	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	complemented
303	C₂×C₂²⋊Q₈	—	2	1	no	—
304	C₂×C₂²⋊Q₈	—	2	1	no	—
305	C₂×C₂²⋊Q₈	—	2	4	no	—
306	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
307	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	complemented
308	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	8	yes	complemented
309	C₂×#121	—	2	1	no	—
310	C₂×#121	—	2	1	no	—
311	C₂×#121	—	2	4	no	—
312	C₂×#121	C₂²×C₆₁	1	2	yes	—
313	C₂×#121	C₂²×C₆₁	1	8	yes	—
314	C₂×#121	C₂²×C₆₁	1	2	yes	—
315	C₂²×C₄⋊C₄	—	2	1	no	—
316	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
317	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic, complemented
318	C₂²×C₂²⋊C₄	—	2	1	no	—
319	C₂²×C₂²⋊C₄	—	2	1	no	—
320	#484	C₂²×C₆₁	1	16	yes	complemented
321	#480	C₂²×C₆₁	1	16	yes	complemented
322	#468	C₂²×C₆₁	1	32	yes	complemented
323	#450	C₂²×C₆₁	1	16	yes	complemented
324	#448	C₂²×C₆₁	1	16	yes	complemented
325	C₂×#107	C₂²×C₆₁	1	4	yes	complemented
326	C₂×#103	C₂²×C₆₁	1	4	yes	complemented
327	C₂×#102	C₂²×C₆₁	1	4	yes	complemented
328	#407	C₂²×C₆₁	1	8	yes	complemented
329	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
330	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	complemented
331	C₆₁	C₂×#3193	1	1	yes	characteristic, direct product factor, Sylow
332	C₂⁵	—	2	1	no	—
333	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
334	C₂²×Q₈	—	2	1	no	—
335	C₂²×Q₈	—	2	2	no	—
336	C₂²×Q₈	—	4	4	no	—
337	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
338	C₂²×Q₈	C₂³×C₆₁	1	4	yes	complemented
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	—	2	8	no	—
341	C₂²×D₄	—	2	2	no	—
342	C₂²×D₄	—	2	1	no	—
343	C₂²×D₄	—	2	2	no	—
344	C₂²×D₄	—	4	4	no	—
345	C₂²×D₄	—	4	8	no	—
346	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
347	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	—	4	4	no	—
352	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
353	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
354	C₂³×C₄	D₄×C₆₁	1	4	yes	—
355	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
360	C₂×SD₁₆	—	4	8	no	—
361	C₂×SD₁₆	C₂³×C₆₁	1	16	yes	complemented
362	C₂×M₁₆	—	4	4	no	—
363	C₂×M₁₆	C₂³×C₆₁	1	8	yes	complemented
364	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	8	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
370	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	complemented
371	#121	—	2	4	no	—
372	#121	—	2	4	no	—
373	#121	—	2	8	no	—
374	#121	—	2	8	no	—
375	#121	C₂³×C₆₁	1	8	yes	—
376	#121	C₂³×C₆₁	1	8	yes	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	8	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	complemented
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	8	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	C₂×C₂²⋊C₄	—	2	8	no	—
394	#107	C₂³×C₆₁	1	16	yes	complemented
395	#103	C₂³×C₆₁	1	16	yes	complemented
396	#102	C₂³×C₆₁	1	16	yes	complemented
397	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
398	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	complemented
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	2	4	no	—
402	C₂⁴	—	2	2	no	—
403	C₂⁴	—	4	2	no	—
404	C₂⁴	—	4	4	no	—
405	C₂⁴	—	4	4	no	—
406	C₂⁴	—	4	4	no	—
407	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
408	C₂⁴	C₂×D₄×C₆₁	1	4	yes	—
409	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
410	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	8	no	—
414	C₂×Q₈	—	4	8	no	—
415	C₂×Q₈	—	4	4	no	—
416	C₂×Q₈	—	4	8	no	—
417	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	32	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	—	4	8	no	—
429	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	—	4	8	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
454	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
460	SD₁₆	—	4	16	no	—
461	M₁₆	—	4	8	no	—
462	C₂×C₈	—	2	8	no	—
463	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
471	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	16	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	4	no	—
492	C₂³	—	4	8	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	8	16	no	—
495	C₂³	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
496	C₂³	#137×C₆₁	1	4	yes	—
497	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
498	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
499	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
500	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
501	Q₈	—	2	8	no	—
502	Q₈	—	4	8	no	—
503	Q₈	—	4	4	no	—
504	D₄	—	2	8	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	16	no	—
508	D₄	—	4	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
532	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
534	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
535	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
536	C₈	—	2	8	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	8	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	4	16	no	—
547	C₂²	—	4	4	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	8	16	no	—
552	C₂²	—	8	8	no	—
553	C₂²	C₆₁×#534	1	2	yes	—
554	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	4	yes	—
555	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
556	C₂²	C₂×#137×C₆₁	1	4	yes	—
557	C₂²	C₂×#137×C₆₁	1	2	yes	—
558	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	4	4	no	—
563	C₄	—	4	4	no	—
564	C₄	—	4	4	no	—
565	C₄	C₆₁×#534	1	2	yes	—
566	C₄	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
567	C₂	—	2	4	no	—
568	C₂	—	2	4	no	—
569	C₂	—	4	2	no	—
570	C₂	—	8	4	no	—
571	C₂	C₆₁×#3193	1	4	yes	direct product factor
572	C₂	C₂×C₆₁×#534	1	1	yes	characteristic
573	C₂	C₂²×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
574	C₂	C₂²×#137×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂²×#121×C₆₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊Q₈×C₆₁	C₂	semidirect
3. C₆₁×#3193	C₂	direct (N × H)
4. C₂×C₆₁×#484	C₂	semidirect
5. C₂×C₆₁×#480	C₂	semidirect
6. C₂×C₆₁×#468	C₂	semidirect
7. C₂×C₆₁×#448	C₂	semidirect
8. C₂×C₆₁×#450	C₂	semidirect
9. C₂×C₆₁×#407	C₂	semidirect
10. C₂³×Q₈×C₆₁	C₂²	semidirect
14. C₂²×SD₁₆×C₆₁	C₂²	semidirect
15. C₂²×M₁₆×C₆₁	C₂²	semidirect
20. C₂×C₂²⋊Q₈×C₆₁	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₆₁	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₆₁	C₂²	semidirect
33. C₆₁×#484	C₂²	semidirect
34. C₆₁×#480	C₂²	semidirect
35. C₆₁×#468	C₂²	semidirect
36. C₆₁×#450	C₂²	semidirect
37. C₆₁×#448	C₂²	semidirect
38. C₂×#107×C₆₁	C₂²	semidirect
39. C₂×#103×C₆₁	C₂²	semidirect
40. C₂×#102×C₆₁	C₂²	semidirect
41. C₆₁×#407	C₂²	semidirect
43. C₂×C₂²⋊C₈×C₆₁	C₂²	semidirect
50. C₂²×Q₈×C₆₁	C₂³	semidirect
73. C₂×SD₁₆×C₆₁	C₂³	semidirect
75. C₂×M₁₆×C₆₁	C₂³	semidirect
82. C₂²⋊Q₈×C₆₁	C₂³	semidirect
99. C₂×C₄⋊C₄×C₆₁	C₂³	semidirect
106. #107×C₆₁	C₂³	semidirect
107. #103×C₆₁	C₂³	semidirect
108. #102×C₆₁	C₂³	semidirect
110. C₂²⋊C₈×C₆₁	C₂³	semidirect
249. C₂×#3193	C₆₁	direct (N × H)
280. C₂²×C₂²⋊Q₈	C₂×C₆₁	semidirect
282. #3193	C₂×C₆₁	direct (N × H)
283. C₂×#484	C₂×C₆₁	semidirect
284. C₂×#480	C₂×C₆₁	semidirect
285. C₂×#468	C₂×C₆₁	semidirect
286. C₂×#448	C₂×C₆₁	semidirect
287. C₂×#450	C₂×C₆₁	semidirect
288. C₂×#407	C₂×C₆₁	semidirect
293. C₂×C₆₁	#3193	direct (N × H)
297. C₂³×Q₈	C₂²×C₆₁	semidirect
301. C₂²×SD₁₆	C₂²×C₆₁	semidirect
302. C₂²×M₁₆	C₂²×C₆₁	semidirect
307. C₂×C₂²⋊Q₈	C₂²×C₆₁	semidirect
308. C₂×C₂²⋊Q₈	C₂²×C₆₁	semidirect
317. C₂²×C₄⋊C₄	C₂²×C₆₁	semidirect
320. #484	C₂²×C₆₁	semidirect
321. #480	C₂²×C₆₁	semidirect
322. #468	C₂²×C₆₁	semidirect
323. #450	C₂²×C₆₁	semidirect
324. #448	C₂²×C₆₁	semidirect
325. C₂×#107	C₂²×C₆₁	semidirect
326. C₂×#103	C₂²×C₆₁	semidirect
327. C₂×#102	C₂²×C₆₁	semidirect
328. #407	C₂²×C₆₁	semidirect
330. C₂×C₂²⋊C₈	C₂²×C₆₁	semidirect
331. C₆₁	C₂×#3193	direct (N × H)
338. C₂²×Q₈	C₂³×C₆₁	semidirect
361. C₂×SD₁₆	C₂³×C₆₁	semidirect
363. C₂×M₁₆	C₂³×C₆₁	semidirect
370. C₂²⋊Q₈	C₂³×C₆₁	semidirect
387. C₂×C₄⋊C₄	C₂³×C₆₁	semidirect
394. #107	C₂³×C₆₁	semidirect
395. #103	C₂³×C₆₁	semidirect
396. #102	C₂³×C₆₁	semidirect
398. C₂²⋊C₈	C₂³×C₆₁	semidirect
571. C₂	C₆₁×#3193	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
C₂²×#121×C₆₁	2
C₆₁×#3193	2
C₂×C₆₁×#484	2
C₂×C₆₁×#480	2
C₂×C₆₁×#468	2
C₂×C₆₁×#448	2
C₂×C₆₁×#450	2
C₂×C₆₁×#407	2
C₂×#3193	61

Subgroups — C2×C61×#3193 #29564169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3193 #29564169