Group #29516739 — subgroups

Subgroups — C₆₁×#13306 #29516739

Order 15616.

566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#467	C₂	1	2	yes	maximal
3	C₂×C₆₁×#358	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
9	C₂×Q₃₂×C₆₁	—	2	4	no	—
10	C₂×Q₃₂×C₆₁	—	2	2	no	—
11	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#467	—	2	4	no	—
14	C₆₁×#467	—	2	4	no	—
15	C₂×#107×C₆₁	C₂²	1	2	yes	—
16	C₂×#103×C₆₁	—	2	2	no	—
17	C₆₁×#358	—	2	4	no	—
18	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
19	C₂²×Q₈×C₆₁	—	2	2	no	—
20	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
21	C₂×Q₁₆×C₆₁	—	2	2	no	—
22	C₂×Q₁₆×C₆₁	—	2	2	no	—
23	C₂×Q₁₆×C₆₁	—	2	2	no	—
24	C₂×Q₁₆×C₆₁	—	2	2	no	—
25	C₂×Q₁₆×C₆₁	—	2	2	no	—
26	C₂×Q₁₆×C₆₁	D₄	1	4	yes	—
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	—	2	1	no	—
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂²⋊Q₈×C₆₁	—	2	8	no	—
34	C₂²⋊Q₈×C₆₁	—	4	4	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₂×C₂²⋊C₄×C₆₁	—	4	2	no	—
38	Q₃₂×C₆₁	—	2	8	no	—
39	C₂×C₁₆×C₆₁	—	2	2	no	—
40	C₂×C₁₆×C₆₁	—	2	1	no	—
41	C₂×C₁₆×C₆₁	—	2	2	no	—
42	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
43	#107×C₆₁	—	2	2	no	—
44	#107×C₆₁	D₄	1	4	yes	—
45	#103×C₆₁	—	2	8	no	—
46	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
47	C₂×Q₈×C₆₁	—	2	4	no	—
48	C₂×Q₈×C₆₁	—	4	2	no	—
49	C₂×Q₈×C₆₁	—	4	2	no	—
50	C₂×Q₈×C₆₁	—	4	2	no	—
51	C₂×Q₈×C₆₁	—	4	2	no	—
52	C₂×Q₈×C₆₁	—	4	2	no	—
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	—	2	1	no	—
55	C₂²×C₄×C₆₁	—	2	1	no	—
56	C₂²×C₄×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	—	4	2	no	—
59	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₆₁	—	2	4	no	—
62	Q₁₆×C₆₁	—	2	4	no	—
63	Q₁₆×C₆₁	—	4	2	no	—
64	Q₁₆×C₆₁	—	4	2	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	2	1	no	—
67	C₂×C₈×C₆₁	—	4	1	no	—
68	C₂×C₈×C₆₁	—	4	2	no	—
69	C₂×C₈×C₆₁	—	4	1	no	—
70	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
74	C₄⋊C₄×C₆₁	—	2	4	no	—
75	C₄⋊C₄×C₆₁	—	4	8	no	—
76	C₄⋊C₄×C₆₁	—	4	2	no	—
77	C₂²⋊C₄×C₆₁	—	4	8	no	—
78	C₁₆×C₆₁	—	2	2	no	—
79	C₁₆×C₆₁	—	4	1	no	—
80	C₂³×C₆₁	—	2	2	no	—
81	C₂³×C₆₁	—	2	1	no	—
82	C₂³×C₆₁	—	2	1	no	—
83	C₂³×C₆₁	—	2	2	no	—
84	C₂³×C₆₁	Q₃₂	1	2	yes	complemented
85	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₆₁	—	4	4	no	—
87	Q₈×C₆₁	—	4	4	no	—
88	Q₈×C₆₁	—	8	2	no	—
89	Q₈×C₆₁	—	8	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	4	1	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	—	4	1	no	—
95	C₂×C₄×C₆₁	—	4	4	no	—
96	C₂×C₄×C₆₁	—	4	4	no	—
97	C₂×C₄×C₆₁	—	8	2	no	—
98	C₂×C₄×C₆₁	—	8	2	no	—
99	C₂×C₄×C₆₁	—	8	2	no	—
100	C₂×C₄×C₆₁	—	8	2	no	—
101	C₂×C₄×C₆₁	#120	1	2	yes	—
102	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
104	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
105	C₈×C₆₁	—	4	1	no	—
106	C₈×C₆₁	#120	1	1	yes	characteristic
107	C₈×C₆₁	#121	1	2	yes	—
108	C₈×C₆₁	#127	1	1	yes	characteristic
109	#13306	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₆₁	—	2	4	no	—
111	C₂²×C₆₁	—	2	2	no	—
112	C₂²×C₆₁	—	4	2	no	—
113	C₂²×C₆₁	—	4	1	no	—
114	C₂²×C₆₁	—	4	1	no	—
115	C₂²×C₆₁	#447	1	2	yes	—
116	C₂²×C₆₁	#466	1	1	yes	characteristic
117	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
118	C₂²×C₆₁	#508	1	1	yes	characteristic
119	C₂²×C₆₁	#510	1	2	yes	—
120	C₄×C₆₁	—	4	1	no	—
121	C₄×C₆₁	—	8	4	no	—
122	C₄×C₆₁	—	8	4	no	—
123	C₄×C₆₁	#392	1	1	yes	characteristic
124	C₄×C₆₁	#449	1	2	yes	—
125	C₄×C₆₁	#469	1	1	yes	characteristic
126	C₂²×Q₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
127	C₂×#467	C₂×C₆₁	1	2	yes	—
128	C₂×#358	C₂×C₆₁	1	2	yes	complemented
129	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic
130	#1380	C₂×C₆₁	1	1	yes	characteristic, complemented
131	C₂×C₆₁	—	4	2	no	—
132	C₂×C₆₁	#1999	1	1	yes	characteristic
133	C₂×C₆₁	#2188	1	2	yes	—
134	C₂×C₆₁	#2189	1	2	yes	—
135	C₂×C₆₁	#2216	1	1	yes	characteristic
136	C₂×C₆₁	#2219	1	1	yes	characteristic
137	C₂²×Q₁₆	C₂²×C₆₁	1	2	yes	—
138	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
139	C₂×C₂²⋊Q₈	—	2	2	no	—
140	C₂×Q₃₂	—	2	4	no	—
141	C₂×Q₃₂	—	2	2	no	—
142	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
144	#467	—	2	4	no	—
145	#467	—	2	4	no	—
146	C₂×#107	C₂²×C₆₁	1	2	yes	—
147	C₂×#103	—	2	2	no	—
148	#358	—	2	4	no	—
149	C₂²⋊C₁₆	—	2	2	no	—
150	C₆₁	#13306	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×Q₈	—	2	2	no	—
152	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	—	2	2	no	—
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×Q₁₆	—	2	2	no	—
158	C₂×Q₁₆	D₄×C₆₁	1	4	yes	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
165	C₂²⋊Q₈	—	2	8	no	—
166	C₂²⋊Q₈	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	2	no	—
169	C₂×C₂²⋊C₄	—	4	2	no	—
170	Q₃₂	—	2	8	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	D₄×C₆₁	1	2	yes	—
175	#107	—	2	2	no	—
176	#107	D₄×C₆₁	1	4	yes	—
177	#103	—	2	8	no	—
178	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
205	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	8	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	4	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	2	no	—
216	C₂³	Q₃₂×C₆₁	1	2	yes	complemented
217	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	4	4	no	—
220	Q₈	—	8	2	no	—
221	Q₈	—	8	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	#120×C₆₁	1	2	yes	—
234	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
235	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
236	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
237	C₈	—	4	1	no	—
238	C₈	#120×C₆₁	1	1	yes	characteristic
239	C₈	#121×C₆₁	1	2	yes	—
240	C₈	#127×C₆₁	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	C₆₁×#447	1	2	yes	—
247	C₂²	C₆₁×#466	1	1	yes	characteristic
248	C₂²	C₂×Q₃₂×C₆₁	1	1	yes	characteristic
249	C₂²	C₆₁×#508	1	1	yes	characteristic
250	C₂²	C₆₁×#510	1	2	yes	—
251	C₄	—	4	1	no	—
252	C₄	—	8	4	no	—
253	C₄	—	8	4	no	—
254	C₄	C₆₁×#392	1	1	yes	characteristic
255	C₄	C₆₁×#449	1	2	yes	—
256	C₄	C₆₁×#469	1	1	yes	characteristic
257	C₂	—	4	2	no	—
258	C₂	C₆₁×#1999	1	1	yes	characteristic
259	C₂	C₆₁×#2188	1	2	yes	—
260	C₂	C₆₁×#2189	1	2	yes	—
261	C₂	C₆₁×#2216	1	1	yes	characteristic
262	C₂	C₆₁×#2219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×C₆₁×#467

C₂

yes

maximal

C₂×C₆₁×#358

C₂

yes

maximal, complemented

C₂×C₆₁×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₆₁×#1380

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₆₁

C₂²

yes

—

C₂³×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₂²⋊Q₈×C₆₁

—

C₂×Q₃₂×C₆₁

—

C₂×Q₃₂×C₆₁

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#467

—

C₆₁×#467

—

C₂×#107×C₆₁

C₂²

yes

—

C₂×#103×C₆₁

—

C₆₁×#358

—

C₆₁×C₂²⋊C₁₆

—

C₂²×Q₈×C₆₁

—

C₂³×C₄×C₆₁

D₄

yes

characteristic

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊Q₈×C₆₁

—

C₂²⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₂²⋊C₄×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#107×C₆₁

—

#107×C₆₁

D₄

yes

—

#103×C₆₁

—

C₂⁴×C₆₁

D₈

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

Q₃₂

yes

complemented

C₂³×C₆₁

C₂×D₈

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

100

C₂×C₄×C₆₁

—

101

C₂×C₄×C₆₁

#120

yes

—

102

C₂×C₄×C₆₁

#121

yes

characteristic

103

C₂×C₄×C₆₁

#135

yes

characteristic

104

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

105

C₈×C₆₁

—

106

C₈×C₆₁

#120

yes

characteristic

107

C₈×C₆₁

#121

yes

—

108

C₈×C₆₁

#127

yes

characteristic

109

#13306

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

110

C₂²×C₆₁

—

111

C₂²×C₆₁

—

112

C₂²×C₆₁

—

113

C₂²×C₆₁

—

114

C₂²×C₆₁

—

115

C₂²×C₆₁

#447

yes

—

116

C₂²×C₆₁

#466

yes

characteristic

117

C₂²×C₆₁

C₂×Q₃₂

yes

characteristic

118

C₂²×C₆₁

#508

yes

characteristic

119

C₂²×C₆₁

#510

yes

—

120

C₄×C₆₁

—

121

C₄×C₆₁

—

122

C₄×C₆₁

—

123

C₄×C₆₁

#392

yes

characteristic

124

C₄×C₆₁

#449

yes

—

125

C₄×C₆₁

#469

yes

characteristic

126

C₂²×Q₃₂

C₂×C₆₁

yes

characteristic, complemented

127

C₂×#467

C₂×C₆₁

yes

—

128

C₂×#358

C₂×C₆₁

yes

complemented

129

C₂×C₂²⋊C₁₆

C₂×C₆₁

yes

characteristic

130

#1380

C₂×C₆₁

yes

characteristic, complemented

131

C₂×C₆₁

—

132

C₂×C₆₁

#1999

yes

characteristic

133

C₂×C₆₁

#2188

yes

—

134

C₂×C₆₁

#2189

yes

—

135

C₂×C₆₁

#2216

yes

characteristic

136

C₂×C₆₁

#2219

yes

characteristic

137

C₂²×Q₁₆

C₂²×C₆₁

yes

—

138

C₂³×C₈

C₂²×C₆₁

yes

characteristic

139

C₂×C₂²⋊Q₈

—

140

C₂×Q₃₂

—

141

C₂×Q₃₂

—

142

C₂²×C₁₆

C₂²×C₆₁

yes

characteristic

143

C₂²×C₁₆

C₂²×C₆₁

yes

characteristic

144

#467

—

145

#467

—

146

C₂×#107

C₂²×C₆₁

yes

—

147

C₂×#103

—

148

#358

—

149

C₂²⋊C₁₆

—

150

C₆₁

#13306

yes

characteristic, direct product factor, Sylow

151

C₂²×Q₈

—

152

C₂³×C₄

D₄×C₆₁

yes

characteristic

153

C₂×Q₁₆

—

154

C₂×Q₁₆

—

155

C₂×Q₁₆

—

156

C₂×Q₁₆

—

157

C₂×Q₁₆

—

158

C₂×Q₁₆

D₄×C₆₁

yes

—

159

C₂²×C₈

—

160

C₂²×C₈

—

161

C₂²×C₈

—

162

C₂²×C₈

—

163

C₂²×C₈

D₄×C₆₁

yes

characteristic

164

C₂²×C₈

C₂³×C₆₁

yes

characteristic

165

C₂²⋊Q₈

—

166

C₂²⋊Q₈

—

167

C₂×C₄⋊C₄

—

168

C₂×C₄⋊C₄

—

169

C₂×C₂²⋊C₄

—

170

Q₃₂

—

171

C₂×C₁₆

—

172

C₂×C₁₆

—

173

C₂×C₁₆

—

174

C₂×C₁₆

D₄×C₆₁

yes

—

175

#107

—

176

#107

D₄×C₆₁

yes

—

177

#103

—

178

C₂⁴

D₈×C₆₁

yes

characteristic

179

C₂×Q₈

—

180

C₂×Q₈

—

181

C₂×Q₈

—

182

C₂×Q₈

—

183

C₂×Q₈

—

184

C₂×Q₈

—

185

C₂²×C₄

—

186

C₂²×C₄

—

187

C₂²×C₄

—

188

C₂²×C₄

—

189

C₂²×C₄

—

190

C₂²×C₄

—

191

C₂²×C₄

D₈×C₆₁

yes

characteristic

192

C₂²×C₄

C₂×D₄×C₆₁

yes

characteristic

193

Q₁₆

—

194

Q₁₆

—

195

Q₁₆

—

196

Q₁₆

—

197

C₂×C₈

—

198

C₂×C₈

—

199

C₂×C₈

—

200

C₂×C₈

—

201

C₂×C₈

—

202

C₂×C₈

C₂×D₄×C₆₁

yes

—

203

C₂×C₈

C₂×D₄×C₆₁

yes

characteristic

204

C₂×C₈

C₂×D₄×C₆₁

yes

—

205

C₂×C₈

#41×C₆₁

yes

characteristic

206

C₄⋊C₄

—

207

C₄⋊C₄

—

208

C₄⋊C₄

—

209

C₂²⋊C₄

—

210

C₁₆

—

211

C₁₆

—

212

C₂³

—

213

C₂³

—

214

C₂³

—

215

C₂³

—

216

C₂³

Q₃₂×C₆₁

yes

complemented

217

C₂³

C₂×D₈×C₆₁

yes

characteristic

218

Q₈

—

219

Q₈

—

220

Q₈

—

221

Q₈

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

#120×C₆₁

yes

—

234

C₂×C₄

#121×C₆₁

yes

characteristic

235

C₂×C₄

#135×C₆₁

yes

characteristic

236

C₂×C₄

AGL(1,ℤ/8)×C₆₁

yes

—

237

C₈

—

238

C₈

#120×C₆₁

yes

characteristic

239

C₈

#121×C₆₁

yes

—

240

C₈

#127×C₆₁

yes

characteristic

241

C₂²

—

242

C₂²

—

243

C₂²

—

244

C₂²

—

245

C₂²

—

246

C₂²

C₆₁×#447

yes

—

247

C₂²

C₆₁×#466

yes

characteristic

248

C₂²

C₂×Q₃₂×C₆₁

yes

characteristic

249

C₂²

C₆₁×#508

yes

characteristic

250

C₂²

C₆₁×#510

yes

—

251

C₄

—

252

C₄

—

253

C₄

—

254

C₄

C₆₁×#392

yes

characteristic

255

C₄

C₆₁×#449

yes

—

256

C₄

C₆₁×#469

yes

characteristic

257

C₂

—

258

C₂

C₆₁×#1999

yes

characteristic

259

C₂

C₆₁×#2188

yes

—

260

C₂

C₆₁×#2189

yes

—

261

C₂

C₆₁×#2216

yes

characteristic

262

C₂

C₆₁×#2219

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₁

C₂×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₆₁

prime radical

C₂⁴×C₆₁

D₈

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₆₁	C₂	semidirect
3. C₂×C₆₁×#358	C₂	semidirect
5. C₆₁×#1380	C₂	semidirect
84. C₂³×C₆₁	Q₃₂	semidirect
109. #13306	C₆₁	direct (N × H)
126. C₂²×Q₃₂	C₂×C₆₁	semidirect
128. C₂×#358	C₂×C₆₁	semidirect
130. #1380	C₂×C₆₁	semidirect
150. C₆₁	#13306	direct (N × H)
216. C₂³	Q₃₂×C₆₁	semidirect

H ≅ G/N

type

1. C₂²×Q₃₂×C₆₁

C₂

semidirect

3. C₂×C₆₁×#358

C₂

semidirect

5. C₆₁×#1380

C₂

semidirect

84. C₂³×C₆₁

Q₃₂

semidirect

109. #13306

C₆₁

direct (N × H)

126. C₂²×Q₃₂

C₂×C₆₁

semidirect

128. C₂×#358

C₂×C₆₁

semidirect

130. #1380

C₂×C₆₁

semidirect

150. C₆₁

#13306

direct (N × H)

216. C₂³

Q₃₂×C₆₁

semidirect

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

subgroup	index
C₂²×Q₃₂×C₆₁	2
C₂×C₆₁×#467	2
C₂×C₆₁×#358	2
C₂×C₆₁×C₂²⋊C₁₆	2
C₆₁×#1380	2
#13306	61

subgroup

index

C₂²×Q₃₂×C₆₁

C₂×C₆₁×#467

C₂×C₆₁×#358

C₂×C₆₁×C₂²⋊C₁₆

C₆₁×#1380

#13306

Subgroups — C₆₁×#13306 #29516739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#13306 #29516739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13306 #29516739