Subgroups — C₂×C₆₁×#3486 #29566215

Order 15616.

11254 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3486	C₂	1	32	yes	maximal, direct product factor
4	C₂×D₄×Q₈×C₆₁	C₂	1	8	yes	maximal, complemented
5	C₂×C₆₁×#548	C₂	1	8	yes	maximal, complemented
6	C₂×C₆₁×#536	C₂	1	6	yes	maximal, complemented
7	C₂×C₆₁×#525	C₂	1	6	yes	maximal, complemented
8	C₂×C₆₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴×C₆₁	—	2	16	no	—
10	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	16	yes	complemented
11	C₂²×#41×C₆₁	C₂²	1	6	yes	—
12	C₂²×#41×C₆₁	C₂²	1	4	yes	complemented
13	C₂³×Q₈×C₆₁	—	2	16	no	—
14	C₂³×Q₈×C₆₁	C₂²	1	3	yes	—
15	C₂³×Q₈×C₆₁	C₂²	1	8	yes	—
16	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₆₁	C₂²	1	16	yes	complemented
18	C₂³×Q₈×C₆₁	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₆₁	C₂²	1	3	yes	—
20	D₄×Q₈×C₆₁	C₂²	1	128	yes	complemented
21	C₆₁×#548	C₂²	1	128	yes	complemented
22	C₆₁×#536	C₂²	1	96	yes	complemented
23	C₂×C₄⋊Q₈×C₆₁	C₂²	1	12	yes	complemented
24	C₂×#124×C₆₁	C₂²	1	12	yes	complemented
25	C₆₁×#525	C₂²	1	96	yes	complemented
26	C₂×#123×C₆₁	C₂²	1	12	yes	—
27	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	12	yes	—
28	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	24	yes	complemented
29	C₂×#121×C₆₁	C₂²	1	12	yes	—
30	C₂×#120×C₆₁	C₂²	1	4	yes	—
31	C₆₁×#519	C₂²	1	16	yes	complemented
32	C₂×C₄×Q₈×C₆₁	C₂²	1	8	yes	complemented
33	C₂×C₄×D₄×C₆₁	C₂²	1	24	yes	—
34	C₂×#117×C₆₁	C₂²	1	6	yes	—
35	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
36	2₋¹⁺⁴×C₆₁	—	2	128	no	—
37	C₂×#41×C₆₁	—	2	32	no	—
38	C₂×#41×C₆₁	—	2	48	no	—
39	C₂×#41×C₆₁	C₂³	1	48	yes	—
40	C₂×#41×C₆₁	C₂³	1	32	yes	complemented
41	C₂²×Q₈×C₆₁	—	2	64	no	—
42	C₂²×Q₈×C₆₁	—	2	32	no	—
43	C₂²×Q₈×C₆₁	—	2	128	no	—
44	C₂²×Q₈×C₆₁	—	2	24	no	—
45	C₂²×Q₈×C₆₁	D₄	1	16	yes	complemented
46	C₂²×Q₈×C₆₁	C₂³	1	6	yes	—
47	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
48	C₂²×Q₈×C₆₁	C₂³	1	24	yes	—
49	C₂²×Q₈×C₆₁	C₂³	1	64	yes	—
50	C₂²×Q₈×C₆₁	C₂³	1	4	yes	—
51	C₂²×Q₈×C₆₁	C₂³	1	8	yes	—
52	C₂²×Q₈×C₆₁	C₂³	1	32	yes	complemented
53	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₁	C₂³	1	6	yes	—
55	C₂²×D₄×C₆₁	C₂³	1	8	yes	—
56	C₂³×C₄×C₆₁	—	2	24	no	—
57	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
58	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
59	C₂³×C₄×C₆₁	C₂³	1	24	yes	—
60	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
61	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
62	C₄⋊Q₈×C₆₁	C₂³	1	96	yes	complemented
63	#124×C₆₁	C₂³	1	96	yes	complemented
64	#123×C₆₁	C₂³	1	96	yes	—
65	C₂²⋊Q₈×C₆₁	C₂³	1	96	yes	—
66	C₂²⋊Q₈×C₆₁	C₂³	1	192	yes	complemented
67	#121×C₆₁	C₂³	1	96	yes	—
68	#120×C₆₁	C₂³	1	32	yes	—
69	C₄×Q₈×C₆₁	C₂³	1	64	yes	complemented
70	C₄×D₄×C₆₁	C₂³	1	192	yes	—
71	#117×C₆₁	C₂³	1	48	yes	—
72	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
73	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
74	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
77	C₂×C₄²×C₆₁	C₂³	1	12	yes	—
78	C₂⁴×C₆₁	—	2	4	no	—
79	C₂⁴×C₆₁	—	2	8	no	—
80	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
81	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
82	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
83	C₂⁴×C₆₁	C₂⁴	1	2	yes	—
84	#41×C₆₁	—	2	128	no	—
85	#41×C₆₁	—	2	192	no	—
86	C₂×Q₈×C₆₁	—	2	256	no	—
87	C₂×Q₈×C₆₁	—	2	128	no	—
88	C₂×Q₈×C₆₁	—	2	48	no	—
89	C₂×Q₈×C₆₁	—	2	96	no	—
90	C₂×Q₈×C₆₁	—	2	24	no	—
91	C₂×Q₈×C₆₁	C₂×D₄	1	32	yes	—
92	C₂×Q₈×C₆₁	C₂×D₄	1	64	yes	complemented
93	C₂×Q₈×C₆₁	C₂⁴	1	24	yes	—
94	C₂×Q₈×C₆₁	C₂⁴	1	48	yes	—
95	C₂×Q₈×C₆₁	C₂⁴	1	16	yes	—
96	C₂×D₄×C₆₁	—	2	16	no	—
97	C₂×D₄×C₆₁	—	2	32	no	—
98	C₂×D₄×C₆₁	—	2	24	no	—
99	C₂×D₄×C₆₁	C₂⁴	1	32	yes	—
100	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
101	C₂×D₄×C₆₁	C₂⁴	1	24	yes	—
102	C₂²×C₄×C₆₁	—	2	12	no	—
103	C₂²×C₄×C₆₁	—	2	48	no	—
104	C₂²×C₄×C₆₁	—	2	96	no	—
105	C₂²×C₄×C₆₁	—	2	48	no	—
106	C₂²×C₄×C₆₁	C₂×D₄	1	24	yes	—
107	C₂²×C₄×C₆₁	C₂⁴	1	48	yes	—
108	C₂²×C₄×C₆₁	C₂⁴	1	48	yes	—
109	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
110	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
112	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
114	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
115	C₄⋊C₄×C₆₁	C₂⁴	1	48	yes	—
116	C₄⋊C₄×C₆₁	C₂⁴	1	96	yes	—
117	C₂²⋊C₄×C₆₁	C₂⁴	1	16	yes	—
118	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
119	C₂²⋊C₄×C₆₁	C₂⁴	1	96	yes	—
120	C₄²×C₆₁	C₂⁴	1	48	yes	—
121	C₂³×C₆₁	—	2	32	no	—
122	C₂³×C₆₁	—	2	8	no	—
123	C₂³×C₆₁	—	2	4	no	—
124	C₂³×C₆₁	—	2	4	no	—
125	C₂³×C₆₁	—	2	4	no	—
126	C₂³×C₆₁	—	2	16	no	—
127	C₂³×C₆₁	—	2	8	no	—
128	C₂³×C₆₁	—	2	16	no	—
129	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
130	C₂³×C₆₁	2₋¹⁺⁴	1	8	yes	complemented
131	C₂³×C₆₁	C₂⁵	1	8	yes	—
132	C₂³×C₆₁	C₂⁵	1	2	yes	—
133	C₂³×C₆₁	C₂⁵	1	4	yes	—
134	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
135	Q₈×C₆₁	—	2	96	no	—
136	Q₈×C₆₁	—	2	48	no	—
137	Q₈×C₆₁	C₂²×D₄	1	64	yes	—
138	D₄×C₆₁	—	2	32	no	—
139	D₄×C₆₁	—	2	64	no	—
140	D₄×C₆₁	—	2	48	no	—
141	C₂×C₄×C₆₁	—	2	8	no	—
142	C₂×C₄×C₆₁	—	2	24	no	—
143	C₂×C₄×C₆₁	—	2	12	no	—
144	C₂×C₄×C₆₁	—	2	16	no	—
145	C₂×C₄×C₆₁	—	2	96	no	—
146	C₂×C₄×C₆₁	—	2	96	no	—
147	C₂×C₄×C₆₁	C₂²×D₄	1	24	yes	—
148	C₂×C₄×C₆₁	C₂²×D₄	1	48	yes	—
149	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
150	C₂×C₄×C₆₁	C₂⁵	1	12	yes	—
151	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
152	C₂×C₄×C₆₁	C₂⁵	1	12	yes	—
153	C₂×#3486	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₆₁	—	2	32	no	—
155	C₂²×C₆₁	—	2	4	no	—
156	C₂²×C₆₁	—	2	4	no	—
157	C₂²×C₆₁	—	2	4	no	—
158	C₂²×C₆₁	—	2	8	no	—
159	C₂²×C₆₁	—	2	16	no	—
160	C₂²×C₆₁	—	2	8	no	—
161	C₂²×C₆₁	—	2	16	no	—
162	C₂²×C₆₁	—	4	16	no	—
163	C₂²×C₆₁	C₂³×D₄	1	4	yes	—
164	C₂²×C₆₁	C₂³×D₄	1	2	yes	—
165	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	4	yes	—
166	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	8	yes	complemented
167	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
168	C₄×C₆₁	—	2	8	no	—
169	C₄×C₆₁	—	2	12	no	—
170	C₄×C₆₁	—	2	16	no	—
171	C₄×C₆₁	C₂³×D₄	1	24	yes	—
172	C₂²×2₋¹⁺⁴	C₂×C₆₁	1	1	yes	characteristic, complemented
173	C₂⁴×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
174	#3486	C₂×C₆₁	1	32	yes	direct product factor
175	C₂×D₄×Q₈	C₂×C₆₁	1	8	yes	complemented
176	C₂×#548	C₂×C₆₁	1	8	yes	complemented
177	C₂×#536	C₂×C₆₁	1	6	yes	complemented
178	C₂×#525	C₂×C₆₁	1	6	yes	complemented
179	C₂×#519	C₂×C₆₁	1	1	yes	characteristic, complemented
180	C₂×C₆₁	—	2	4	no	—
181	C₂×C₆₁	—	2	8	no	—
182	C₂×C₆₁	—	4	8	no	—
183	C₂×C₆₁	#3486	1	4	yes	direct product factor
184	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
185	C₂×C₆₁	C₂²×2₋¹⁺⁴	1	2	yes	—
186	C₂×2₋¹⁺⁴	—	2	16	no	—
187	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	16	yes	complemented
188	C₂²×#41	C₂²×C₆₁	1	6	yes	—
189	C₂²×#41	C₂²×C₆₁	1	4	yes	complemented
190	C₂³×Q₈	—	2	16	no	—
191	C₂³×Q₈	C₂²×C₆₁	1	3	yes	—
192	C₂³×Q₈	C₂²×C₆₁	1	8	yes	—
193	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
194	C₂³×Q₈	C₂²×C₆₁	1	16	yes	complemented
195	C₂³×Q₈	C₂²×C₆₁	1	4	yes	complemented
196	C₂⁴×C₄	C₂²×C₆₁	1	3	yes	—
197	D₄×Q₈	C₂²×C₆₁	1	128	yes	complemented
198	#548	C₂²×C₆₁	1	128	yes	complemented
199	#536	C₂²×C₆₁	1	96	yes	complemented
200	C₂×C₄⋊Q₈	C₂²×C₆₁	1	12	yes	complemented
201	C₂×#124	C₂²×C₆₁	1	12	yes	complemented
202	#525	C₂²×C₆₁	1	96	yes	complemented
203	C₂×#123	C₂²×C₆₁	1	12	yes	—
204	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	12	yes	—
205	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	24	yes	complemented
206	C₂×#121	C₂²×C₆₁	1	12	yes	—
207	C₂×#120	C₂²×C₆₁	1	4	yes	—
208	#519	C₂²×C₆₁	1	16	yes	complemented
209	C₂×C₄×Q₈	C₂²×C₆₁	1	8	yes	complemented
210	C₂×C₄×D₄	C₂²×C₆₁	1	24	yes	—
211	C₂×#117	C₂²×C₆₁	1	6	yes	—
212	C₆₁	C₂×#3486	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
214	2₋¹⁺⁴	—	2	128	no	—
215	C₂×#41	—	2	32	no	—
216	C₂×#41	—	2	48	no	—
217	C₂×#41	C₂³×C₆₁	1	48	yes	—
218	C₂×#41	C₂³×C₆₁	1	32	yes	complemented
219	C₂²×Q₈	—	2	64	no	—
220	C₂²×Q₈	—	2	32	no	—
221	C₂²×Q₈	—	2	128	no	—
222	C₂²×Q₈	—	2	24	no	—
223	C₂²×Q₈	D₄×C₆₁	1	16	yes	complemented
224	C₂²×Q₈	C₂³×C₆₁	1	6	yes	—
225	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
226	C₂²×Q₈	C₂³×C₆₁	1	24	yes	—
227	C₂²×Q₈	C₂³×C₆₁	1	64	yes	—
228	C₂²×Q₈	C₂³×C₆₁	1	4	yes	—
229	C₂²×Q₈	C₂³×C₆₁	1	8	yes	—
230	C₂²×Q₈	C₂³×C₆₁	1	32	yes	complemented
231	C₂²×D₄	C₂³×C₆₁	1	8	yes	—
232	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
233	C₂²×D₄	C₂³×C₆₁	1	6	yes	—
234	C₂³×C₄	—	2	24	no	—
235	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
236	C₂³×C₄	C₂³×C₆₁	1	12	yes	—
237	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
238	C₂³×C₄	C₂³×C₆₁	1	24	yes	—
239	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
240	C₄⋊Q₈	C₂³×C₆₁	1	96	yes	complemented
241	#124	C₂³×C₆₁	1	96	yes	complemented
242	#123	C₂³×C₆₁	1	96	yes	—
243	C₂²⋊Q₈	C₂³×C₆₁	1	96	yes	—
244	C₂²⋊Q₈	C₂³×C₆₁	1	192	yes	complemented
245	#121	C₂³×C₆₁	1	96	yes	—
246	#120	C₂³×C₆₁	1	32	yes	—
247	C₄×Q₈	C₂³×C₆₁	1	64	yes	complemented
248	C₄×D₄	C₂³×C₆₁	1	192	yes	—
249	#117	C₂³×C₆₁	1	48	yes	—
250	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
251	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
254	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
255	C₂×C₄²	C₂³×C₆₁	1	12	yes	—
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	2	8	no	—
258	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
259	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
260	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
261	C₂⁴	C₂⁴×C₆₁	1	2	yes	—
262	#41	—	2	128	no	—
263	#41	—	2	192	no	—
264	C₂×Q₈	—	2	256	no	—
265	C₂×Q₈	—	2	128	no	—
266	C₂×Q₈	—	2	48	no	—
267	C₂×Q₈	—	2	96	no	—
268	C₂×Q₈	—	2	24	no	—
269	C₂×Q₈	C₂×D₄×C₆₁	1	32	yes	—
270	C₂×Q₈	C₂×D₄×C₆₁	1	64	yes	complemented
271	C₂×Q₈	C₂⁴×C₆₁	1	24	yes	—
272	C₂×Q₈	C₂⁴×C₆₁	1	48	yes	—
273	C₂×Q₈	C₂⁴×C₆₁	1	16	yes	—
274	C₂×D₄	—	2	16	no	—
275	C₂×D₄	—	2	32	no	—
276	C₂×D₄	—	2	24	no	—
277	C₂×D₄	C₂⁴×C₆₁	1	32	yes	—
278	C₂×D₄	C₂⁴×C₆₁	1	16	yes	—
279	C₂×D₄	C₂⁴×C₆₁	1	24	yes	—
280	C₂²×C₄	—	2	12	no	—
281	C₂²×C₄	—	2	48	no	—
282	C₂²×C₄	—	2	96	no	—
283	C₂²×C₄	—	2	48	no	—
284	C₂²×C₄	C₂×D₄×C₆₁	1	24	yes	—
285	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
286	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
287	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
288	C₂²×C₄	C₂⁴×C₆₁	1	48	yes	—
289	C₂²×C₄	C₂⁴×C₆₁	1	48	yes	—
290	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
291	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
292	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
293	C₄⋊C₄	C₂⁴×C₆₁	1	48	yes	—
294	C₄⋊C₄	C₂⁴×C₆₁	1	96	yes	—
295	C₂²⋊C₄	C₂⁴×C₆₁	1	16	yes	—
296	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
297	C₂²⋊C₄	C₂⁴×C₆₁	1	96	yes	—
298	C₄²	C₂⁴×C₆₁	1	48	yes	—
299	C₂³	—	2	32	no	—
300	C₂³	—	2	8	no	—
301	C₂³	—	2	4	no	—
302	C₂³	—	2	4	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	2	16	no	—
305	C₂³	—	2	8	no	—
306	C₂³	—	2	16	no	—
307	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
308	C₂³	2₋¹⁺⁴×C₆₁	1	8	yes	complemented
309	C₂³	C₂⁵×C₆₁	1	8	yes	—
310	C₂³	C₂⁵×C₆₁	1	2	yes	—
311	C₂³	C₂⁵×C₆₁	1	4	yes	—
312	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
313	Q₈	—	2	96	no	—
314	Q₈	—	2	48	no	—
315	Q₈	C₂²×D₄×C₆₁	1	64	yes	—
316	D₄	—	2	32	no	—
317	D₄	—	2	64	no	—
318	D₄	—	2	48	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	24	no	—
321	C₂×C₄	—	2	12	no	—
322	C₂×C₄	—	2	16	no	—
323	C₂×C₄	—	2	96	no	—
324	C₂×C₄	—	2	96	no	—
325	C₂×C₄	C₂²×D₄×C₆₁	1	24	yes	—
326	C₂×C₄	C₂²×D₄×C₆₁	1	48	yes	—
327	C₂×C₄	C₂⁵×C₆₁	1	16	yes	—
328	C₂×C₄	C₂⁵×C₆₁	1	12	yes	—
329	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
330	C₂×C₄	C₂⁵×C₆₁	1	12	yes	—
331	C₂²	—	2	32	no	—
332	C₂²	—	2	4	no	—
333	C₂²	—	2	4	no	—
334	C₂²	—	2	4	no	—
335	C₂²	—	2	8	no	—
336	C₂²	—	2	16	no	—
337	C₂²	—	2	8	no	—
338	C₂²	—	2	16	no	—
339	C₂²	—	4	16	no	—
340	C₂²	C₂³×D₄×C₆₁	1	4	yes	—
341	C₂²	C₂³×D₄×C₆₁	1	2	yes	—
342	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	4	yes	—
343	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	8	yes	complemented
344	C₂²	C₂⁶×C₆₁	1	1	yes	characteristic
345	C₄	—	2	8	no	—
346	C₄	—	2	12	no	—
347	C₄	—	2	16	no	—
348	C₄	C₂³×D₄×C₆₁	1	24	yes	—
349	C₂	—	2	4	no	—
350	C₂	—	2	8	no	—
351	C₂	—	4	8	no	—
352	C₂	C₆₁×#3486	1	4	yes	direct product factor
353	C₂	C₂⁴×D₄×C₆₁	1	1	yes	characteristic
354	C₂	C₂²×2₋¹⁺⁴×C₆₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₁
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
1. C₂²×2₋¹⁺⁴×C₆₁	C₂	semidirect
2. C₂⁴×Q₈×C₆₁	C₂	semidirect
3. C₆₁×#3486	C₂	direct (N × H)
4. C₂×D₄×Q₈×C₆₁	C₂	semidirect
5. C₂×C₆₁×#548	C₂	semidirect
6. C₂×C₆₁×#536	C₂	semidirect
7. C₂×C₆₁×#525	C₂	semidirect
8. C₂×C₆₁×#519	C₂	semidirect
10. C₂×2₋¹⁺⁴×C₆₁	C₂²	semidirect
12. C₂²×#41×C₆₁	C₂²	semidirect
17. C₂³×Q₈×C₆₁	C₂²	semidirect
18. C₂³×Q₈×C₆₁	C₂²	semidirect
20. D₄×Q₈×C₆₁	C₂²	semidirect
21. C₆₁×#548	C₂²	semidirect
22. C₆₁×#536	C₂²	semidirect
23. C₂×C₄⋊Q₈×C₆₁	C₂²	semidirect
24. C₂×#124×C₆₁	C₂²	semidirect
25. C₆₁×#525	C₂²	semidirect
28. C₂×C₂²⋊Q₈×C₆₁	C₂²	semidirect
31. C₆₁×#519	C₂²	semidirect
32. C₂×C₄×Q₈×C₆₁	C₂²	semidirect
40. C₂×#41×C₆₁	C₂³	semidirect
45. C₂²×Q₈×C₆₁	D₄	semidirect
52. C₂²×Q₈×C₆₁	C₂³	semidirect
62. C₄⋊Q₈×C₆₁	C₂³	semidirect
63. #124×C₆₁	C₂³	semidirect
66. C₂²⋊Q₈×C₆₁	C₂³	semidirect
69. C₄×Q₈×C₆₁	C₂³	semidirect
92. C₂×Q₈×C₆₁	C₂×D₄	semidirect
130. C₂³×C₆₁	2₋¹⁺⁴	semidirect
153. C₂×#3486	C₆₁	direct (N × H)
166. C₂²×C₆₁	C₂×2₋¹⁺⁴	semidirect
172. C₂²×2₋¹⁺⁴	C₂×C₆₁	semidirect
173. C₂⁴×Q₈	C₂×C₆₁	semidirect
174. #3486	C₂×C₆₁	direct (N × H)
175. C₂×D₄×Q₈	C₂×C₆₁	semidirect
176. C₂×#548	C₂×C₆₁	semidirect
177. C₂×#536	C₂×C₆₁	semidirect
178. C₂×#525	C₂×C₆₁	semidirect
179. C₂×#519	C₂×C₆₁	semidirect
183. C₂×C₆₁	#3486	direct (N × H)
187. C₂×2₋¹⁺⁴	C₂²×C₆₁	semidirect
189. C₂²×#41	C₂²×C₆₁	semidirect
194. C₂³×Q₈	C₂²×C₆₁	semidirect
195. C₂³×Q₈	C₂²×C₆₁	semidirect
197. D₄×Q₈	C₂²×C₆₁	semidirect
198. #548	C₂²×C₆₁	semidirect
199. #536	C₂²×C₆₁	semidirect
200. C₂×C₄⋊Q₈	C₂²×C₆₁	semidirect
201. C₂×#124	C₂²×C₆₁	semidirect
202. #525	C₂²×C₆₁	semidirect
205. C₂×C₂²⋊Q₈	C₂²×C₆₁	semidirect
208. #519	C₂²×C₆₁	semidirect
209. C₂×C₄×Q₈	C₂²×C₆₁	semidirect
212. C₆₁	C₂×#3486	direct (N × H)
218. C₂×#41	C₂³×C₆₁	semidirect
223. C₂²×Q₈	D₄×C₆₁	semidirect
230. C₂²×Q₈	C₂³×C₆₁	semidirect
240. C₄⋊Q₈	C₂³×C₆₁	semidirect
241. #124	C₂³×C₆₁	semidirect
244. C₂²⋊Q₈	C₂³×C₆₁	semidirect
247. C₄×Q₈	C₂³×C₆₁	semidirect
270. C₂×Q₈	C₂×D₄×C₆₁	semidirect
308. C₂³	2₋¹⁺⁴×C₆₁	semidirect
343. C₂²	C₂×2₋¹⁺⁴×C₆₁	semidirect
352. C₂	C₆₁×#3486	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₆₁	2
C₂⁴×Q₈×C₆₁	2
C₆₁×#3486	2
C₂×D₄×Q₈×C₆₁	2
C₂×C₆₁×#548	2
C₂×C₆₁×#536	2
C₂×C₆₁×#525	2
C₂×C₆₁×#519	2
C₂×#3486	61

Subgroups — C2×C61×#3486 #29566215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3486 #29566215