Subgroups — C₂×C₆₁×#3402 #29564832

Order 15616.

1938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#556	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3402	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₆₁×#497	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×#495	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#464	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#462	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#463	C₂	1	2	yes	maximal
8	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₆₁×#440	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#556	—	2	8	no	—
18	C₆₁×#556	C₂²	1	16	yes	complemented
19	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
20	C₂×#126×C₆₁	—	2	2	no	—
21	C₂×#126×C₆₁	—	2	2	no	—
22	C₂×#124×C₆₁	C₂²	1	2	yes	—
23	C₂×#124×C₆₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
25	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
29	C₂×#117×C₆₁	—	2	2	no	—
30	C₆₁×#497	C₂²	1	16	yes	complemented
31	C₆₁×#495	C₂²	1	8	yes	—
32	C₆₁×#464	C₂²	1	16	yes	complemented
33	C₆₁×#463	C₂²	1	16	yes	—
34	C₆₁×#462	C₂²	1	16	yes	complemented
35	C₆₁×#446	C₂²	1	8	yes	complemented
36	C₆₁×#440	C₂²	1	16	yes	complemented
37	C₄×Q₁₆×C₆₁	C₂²	1	8	yes	complemented
38	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#103×C₆₁	C₂²	1	2	yes	—
42	C₂×#103×C₆₁	C₂²	1	2	yes	—
43	C₂×#102×C₆₁	C₂²	1	2	yes	—
44	C₂×#97×C₆₁	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
46	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
47	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
48	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₁	—	2	2	no	—
51	C₂×Q₁₆×C₆₁	—	2	4	no	—
52	C₂×Q₁₆×C₆₁	—	4	2	no	—
53	C₂×Q₁₆×C₆₁	—	4	4	no	—
54	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
55	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
56	C₂×Q₁₆×C₆₁	C₂³	1	8	yes	—
57	C₂×SD₁₆×C₆₁	—	2	8	no	—
58	C₂×SD₁₆×C₆₁	—	4	4	no	—
59	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
60	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
61	C₂×D₈×C₆₁	—	4	2	no	—
62	C₂×D₈×C₆₁	C₂³	1	4	yes	—
63	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₁	—	2	4	no	—
67	C₄⋊Q₈×C₆₁	C₂³	1	8	yes	—
68	#126×C₆₁	—	2	16	no	—
69	#126×C₆₁	—	2	16	no	—
70	#124×C₆₁	—	2	4	no	—
71	#124×C₆₁	—	2	4	no	—
72	#124×C₆₁	C₂³	1	8	yes	—
73	#124×C₆₁	C₂³	1	8	yes	—
74	C₂²⋊Q₈×C₆₁	—	2	16	no	—
75	C₄×Q₈×C₆₁	—	2	4	no	—
76	C₄×Q₈×C₆₁	—	2	4	no	—
77	C₄×Q₈×C₆₁	D₄	1	4	yes	—
78	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
79	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
80	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
81	C₄×D₄×C₆₁	—	2	4	no	—
82	C₄×D₄×C₆₁	C₂³	1	8	yes	—
83	#117×C₆₁	—	2	16	no	—
84	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
91	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
96	C₂×C₄²×C₆₁	—	2	2	no	—
97	C₂×C₄²×C₆₁	—	2	2	no	—
98	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
99	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
100	#107×C₆₁	C₂³	1	4	yes	—
101	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
102	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
103	#103×C₆₁	C₂³	1	8	yes	—
104	#103×C₆₁	C₂³	1	8	yes	—
105	#102×C₆₁	C₂³	1	8	yes	—
106	#97×C₆₁	C₂³	1	8	yes	—
107	C₄×C₈×C₆₁	C₂³	1	4	yes	—
108	C₂⁴×C₆₁	—	2	2	no	—
109	C₂×Q₈×C₆₁	—	2	2	no	—
110	C₂×Q₈×C₆₁	—	2	2	no	—
111	C₂×Q₈×C₆₁	—	2	4	no	—
112	C₂×Q₈×C₆₁	—	2	4	no	—
113	C₂×Q₈×C₆₁	—	2	4	no	—
114	C₂×Q₈×C₆₁	—	2	4	no	—
115	C₂×Q₈×C₆₁	—	4	2	no	—
116	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
117	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
118	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
119	C₂×D₄×C₆₁	—	2	2	no	—
120	C₂×D₄×C₆₁	—	2	4	no	—
121	C₂×D₄×C₆₁	—	2	4	no	—
122	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₆₁	—	2	2	no	—
124	C₂²×C₄×C₆₁	—	2	2	no	—
125	C₂²×C₄×C₆₁	—	2	2	no	—
126	C₂²×C₄×C₆₁	—	2	8	no	—
127	C₂²×C₄×C₆₁	—	4	8	no	—
128	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₆₁	—	2	8	no	—
134	Q₁₆×C₆₁	—	4	4	no	—
135	Q₁₆×C₆₁	—	4	8	no	—
136	SD₁₆×C₆₁	—	2	16	no	—
137	SD₁₆×C₆₁	—	4	8	no	—
138	D₈×C₆₁	—	4	4	no	—
139	C₂×C₈×C₆₁	—	2	2	no	—
140	C₂×C₈×C₆₁	—	2	4	no	—
141	C₂×C₈×C₆₁	#41	1	4	yes	—
142	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
143	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
144	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
145	C₄⋊C₄×C₆₁	—	2	8	no	—
146	C₄⋊C₄×C₆₁	—	2	8	no	—
147	C₄⋊C₄×C₆₁	—	2	8	no	—
148	C₄⋊C₄×C₆₁	—	2	8	no	—
149	C₄⋊C₄×C₆₁	—	2	2	no	—
150	C₄⋊C₄×C₆₁	C₂×D₄	1	8	yes	—
151	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
152	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
153	C₂²⋊C₄×C₆₁	—	2	8	no	—
154	C₂²⋊C₄×C₆₁	—	2	8	no	—
155	C₂²⋊C₄×C₆₁	—	2	8	no	—
156	C₂²⋊C₄×C₆₁	—	2	8	no	—
157	C₂²⋊C₄×C₆₁	—	2	8	no	—
158	C₄²×C₆₁	—	2	8	no	—
159	C₄²×C₆₁	—	2	8	no	—
160	C₄²×C₆₁	C₂×D₄	1	4	yes	—
161	C₄²×C₆₁	C₂×D₄	1	2	yes	—
162	C₄²×C₆₁	C₂⁴	1	2	yes	—
163	C₄²×C₆₁	C₂⁴	1	4	yes	—
164	C₂³×C₆₁	—	2	4	no	—
165	C₂³×C₆₁	—	4	4	no	—
166	C₂³×C₆₁	—	4	4	no	—
167	C₂³×C₆₁	—	4	4	no	—
168	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₆₁	—	2	4	no	—
170	Q₈×C₆₁	—	2	4	no	—
171	Q₈×C₆₁	—	2	4	no	—
172	Q₈×C₆₁	—	2	4	no	—
173	Q₈×C₆₁	—	4	4	no	—
174	Q₈×C₆₁	—	4	4	no	—
175	Q₈×C₆₁	—	4	4	no	—
176	D₄×C₆₁	—	2	4	no	—
177	D₄×C₆₁	—	2	4	no	—
178	D₄×C₆₁	—	4	4	no	—
179	C₂×C₄×C₆₁	—	2	4	no	—
180	C₂×C₄×C₆₁	—	2	2	no	—
181	C₂×C₄×C₆₁	—	2	2	no	—
182	C₂×C₄×C₆₁	—	2	4	no	—
183	C₂×C₄×C₆₁	—	2	4	no	—
184	C₂×C₄×C₆₁	—	2	4	no	—
185	C₂×C₄×C₆₁	—	2	4	no	—
186	C₂×C₄×C₆₁	—	4	16	no	—
187	C₂×C₄×C₆₁	—	4	4	no	—
188	C₂×C₄×C₆₁	—	4	4	no	—
189	C₂×C₄×C₆₁	—	4	4	no	—
190	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
191	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
193	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
196	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
197	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
198	C₈×C₆₁	—	2	2	no	—
199	C₈×C₆₁	—	2	4	no	—
200	C₈×C₆₁	C₂×#41	1	4	yes	—
201	C₂×#3402	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂²×C₆₁	—	4	4	no	—
203	C₂²×C₆₁	—	4	4	no	—
204	C₂²×C₆₁	—	4	4	no	—
205	C₂²×C₆₁	—	8	8	no	—
206	C₂²×C₆₁	#550	1	2	yes	—
207	C₂²×C₆₁	#575	1	2	yes	—
208	C₂²×C₆₁	#578	1	2	yes	—
209	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
210	C₄×C₆₁	—	2	4	no	—
211	C₄×C₆₁	—	2	2	no	—
212	C₄×C₆₁	—	2	2	no	—
213	C₄×C₆₁	—	2	4	no	—
214	C₄×C₆₁	—	4	4	no	—
215	C₄×C₆₁	—	4	4	no	—
216	C₄×C₆₁	—	4	4	no	—
217	C₄×C₆₁	#550	1	2	yes	—
218	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
219	C₄×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×#556	C₂×C₆₁	1	2	yes	complemented
221	#3402	C₂×C₆₁	1	16	yes	direct product factor
222	C₂×#497	C₂×C₆₁	1	2	yes	complemented
223	C₂×#495	C₂×C₆₁	1	1	yes	characteristic
224	C₂×#464	C₂×C₆₁	1	2	yes	complemented
225	C₂×#462	C₂×C₆₁	1	2	yes	complemented
226	C₂×#463	C₂×C₆₁	1	2	yes	—
227	C₂×#446	C₂×C₆₁	1	1	yes	characteristic, complemented
228	C₂×#440	C₂×C₆₁	1	2	yes	complemented
229	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
230	C₂×C₆₁	—	8	4	no	—
231	C₂×C₆₁	#3402	1	4	yes	direct product factor
232	C₂×C₆₁	C₂×#550	1	1	yes	characteristic
233	C₂×C₆₁	C₂×#575	1	1	yes	characteristic
234	C₂×C₆₁	C₂×#578	1	1	yes	characteristic
235	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
236	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
237	C₂²×Q₁₆	C₂²×C₆₁	1	2	yes	—
238	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
239	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
240	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
241	#556	—	2	8	no	—
242	#556	C₂²×C₆₁	1	16	yes	complemented
243	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
244	C₂×#126	—	2	2	no	—
245	C₂×#126	—	2	2	no	—
246	C₂×#124	C₂²×C₆₁	1	2	yes	—
247	C₂×#124	C₂²×C₆₁	1	2	yes	—
248	C₂×C₂²⋊Q₈	—	2	2	no	—
249	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
250	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
251	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
252	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
253	C₂×#117	—	2	2	no	—
254	#497	C₂²×C₆₁	1	16	yes	complemented
255	#495	C₂²×C₆₁	1	8	yes	—
256	#464	C₂²×C₆₁	1	16	yes	complemented
257	#463	C₂²×C₆₁	1	16	yes	—
258	#462	C₂²×C₆₁	1	16	yes	complemented
259	#446	C₂²×C₆₁	1	8	yes	complemented
260	#440	C₂²×C₆₁	1	16	yes	complemented
261	C₄×Q₁₆	C₂²×C₆₁	1	8	yes	complemented
262	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
263	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
264	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
265	C₂×#103	C₂²×C₆₁	1	2	yes	—
266	C₂×#103	C₂²×C₆₁	1	2	yes	—
267	C₂×#102	C₂²×C₆₁	1	2	yes	—
268	C₂×#97	C₂²×C₆₁	1	2	yes	—
269	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
270	C₆₁	C₂×#3402	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
272	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
273	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
274	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
275	C₂³×C₄	—	2	2	no	—
276	C₂×Q₁₆	—	2	4	no	—
277	C₂×Q₁₆	—	4	2	no	—
278	C₂×Q₁₆	—	4	4	no	—
279	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
280	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
281	C₂×Q₁₆	C₂³×C₆₁	1	8	yes	—
282	C₂×SD₁₆	—	2	8	no	—
283	C₂×SD₁₆	—	4	4	no	—
284	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
285	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
286	C₂×D₈	—	4	2	no	—
287	C₂×D₈	C₂³×C₆₁	1	4	yes	—
288	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
291	C₄⋊Q₈	—	2	4	no	—
292	C₄⋊Q₈	C₂³×C₆₁	1	8	yes	—
293	#126	—	2	16	no	—
294	#126	—	2	16	no	—
295	#124	—	2	4	no	—
296	#124	—	2	4	no	—
297	#124	C₂³×C₆₁	1	8	yes	—
298	#124	C₂³×C₆₁	1	8	yes	—
299	C₂²⋊Q₈	—	2	16	no	—
300	C₄×Q₈	—	2	4	no	—
301	C₄×Q₈	—	2	4	no	—
302	C₄×Q₈	D₄×C₆₁	1	4	yes	—
303	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
304	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
305	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
306	C₄×D₄	—	2	4	no	—
307	C₄×D₄	C₂³×C₆₁	1	8	yes	—
308	#117	—	2	16	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
314	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	—	2	2	no	—
323	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
324	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
325	#107	C₂³×C₆₁	1	4	yes	—
326	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
327	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
328	#103	C₂³×C₆₁	1	8	yes	—
329	#103	C₂³×C₆₁	1	8	yes	—
330	#102	C₂³×C₆₁	1	8	yes	—
331	#97	C₂³×C₆₁	1	8	yes	—
332	C₄×C₈	C₂³×C₆₁	1	4	yes	—
333	C₂⁴	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	4	no	—
338	C₂×Q₈	—	2	4	no	—
339	C₂×Q₈	—	2	4	no	—
340	C₂×Q₈	—	4	2	no	—
341	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
342	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
343	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	4	8	no	—
353	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
358	Q₁₆	—	2	8	no	—
359	Q₁₆	—	4	4	no	—
360	Q₁₆	—	4	8	no	—
361	SD₁₆	—	2	16	no	—
362	SD₁₆	—	4	8	no	—
363	D₈	—	4	4	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	4	no	—
366	C₂×C₈	#41×C₆₁	1	4	yes	—
367	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
368	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
369	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
370	C₄⋊C₄	—	2	8	no	—
371	C₄⋊C₄	—	2	8	no	—
372	C₄⋊C₄	—	2	8	no	—
373	C₄⋊C₄	—	2	8	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	C₂×D₄×C₆₁	1	8	yes	—
376	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
377	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
378	C₂²⋊C₄	—	2	8	no	—
379	C₂²⋊C₄	—	2	8	no	—
380	C₂²⋊C₄	—	2	8	no	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	2	8	no	—
383	C₄²	—	2	8	no	—
384	C₄²	—	2	8	no	—
385	C₄²	C₂×D₄×C₆₁	1	4	yes	—
386	C₄²	C₂×D₄×C₆₁	1	2	yes	—
387	C₄²	C₂⁴×C₆₁	1	2	yes	—
388	C₄²	C₂⁴×C₆₁	1	4	yes	—
389	C₂³	—	2	4	no	—
390	C₂³	—	4	4	no	—
391	C₂³	—	4	4	no	—
392	C₂³	—	4	4	no	—
393	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
394	Q₈	—	2	4	no	—
395	Q₈	—	2	4	no	—
396	Q₈	—	2	4	no	—
397	Q₈	—	2	4	no	—
398	Q₈	—	4	4	no	—
399	Q₈	—	4	4	no	—
400	Q₈	—	4	4	no	—
401	D₄	—	2	4	no	—
402	D₄	—	2	4	no	—
403	D₄	—	4	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	4	16	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
416	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
417	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
419	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
421	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
422	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
423	C₈	—	2	2	no	—
424	C₈	—	2	4	no	—
425	C₈	C₂×#41×C₆₁	1	4	yes	—
426	C₂²	—	4	4	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	8	8	no	—
430	C₂²	C₆₁×#550	1	2	yes	—
431	C₂²	C₆₁×#575	1	2	yes	—
432	C₂²	C₆₁×#578	1	2	yes	—
433	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
434	C₄	—	2	4	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	4	no	—
438	C₄	—	4	4	no	—
439	C₄	—	4	4	no	—
440	C₄	—	4	4	no	—
441	C₄	C₆₁×#550	1	2	yes	—
442	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
443	C₄	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
444	C₂	—	8	4	no	—
445	C₂	C₆₁×#3402	1	4	yes	direct product factor
446	C₂	C₂×C₆₁×#550	1	1	yes	characteristic
447	C₂	C₂×C₆₁×#575	1	1	yes	characteristic
448	C₂	C₂×C₆₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂²×D₈×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#556	C₂	semidirect
2. C₆₁×#3402	C₂	direct (N × H)
3. C₂×C₆₁×#497	C₂	semidirect
5. C₂×C₆₁×#464	C₂	semidirect
6. C₂×C₆₁×#462	C₂	semidirect
8. C₂×C₆₁×#446	C₂	semidirect
9. C₂×C₆₁×#440	C₂	semidirect
10. C₂×C₄×Q₁₆×C₆₁	C₂	semidirect
18. C₆₁×#556	C₂²	semidirect
30. C₆₁×#497	C₂²	semidirect
32. C₆₁×#464	C₂²	semidirect
34. C₆₁×#462	C₂²	semidirect
35. C₆₁×#446	C₂²	semidirect
36. C₆₁×#440	C₂²	semidirect
37. C₄×Q₁₆×C₆₁	C₂²	semidirect
201. C₂×#3402	C₆₁	direct (N × H)
220. C₂×#556	C₂×C₆₁	semidirect
221. #3402	C₂×C₆₁	direct (N × H)
222. C₂×#497	C₂×C₆₁	semidirect
224. C₂×#464	C₂×C₆₁	semidirect
225. C₂×#462	C₂×C₆₁	semidirect
227. C₂×#446	C₂×C₆₁	semidirect
228. C₂×#440	C₂×C₆₁	semidirect
229. C₂×C₄×Q₁₆	C₂×C₆₁	semidirect
231. C₂×C₆₁	#3402	direct (N × H)
242. #556	C₂²×C₆₁	semidirect
254. #497	C₂²×C₆₁	semidirect
256. #464	C₂²×C₆₁	semidirect
258. #462	C₂²×C₆₁	semidirect
259. #446	C₂²×C₆₁	semidirect
260. #440	C₂²×C₆₁	semidirect
261. C₄×Q₁₆	C₂²×C₆₁	semidirect
270. C₆₁	C₂×#3402	direct (N × H)
445. C₂	C₆₁×#3402	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×#556	2
C₆₁×#3402	2
C₂×C₆₁×#497	2
C₂×C₆₁×#495	2
C₂×C₆₁×#464	2
C₂×C₆₁×#462	2
C₂×C₆₁×#463	2
C₂×C₆₁×#446	2
C₂×C₆₁×#440	2
C₂×C₄×Q₁₆×C₆₁	2
C₂×#3402	61

Subgroups — C2×C61×#3402 #29564832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3402 #29564832