Subgroups — #29578586

Order 15616.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446538	C₂	1	2	yes	maximal
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839484	C₂	1	1	yes	maximal, characteristic
4	#21443660	C₂	1	2	yes	maximal
5	C₄×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	—	2	2	no	—
7	C₂×C₄⋊#92418	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839517	—	2	2	no	—
11	#16839515	—	2	2	no	—
12	#16839484	—	2	2	no	—
13	C₂×#106×C₆₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#340	C₄	1	1	yes	characteristic
17	C₆₁×#340	C₄	1	1	yes	characteristic
18	C₂²×#92418	—	2	1	no	—
19	C₂²×#92418	—	2	2	no	—
20	C₂²×#92418	D₄	1	1	yes	characteristic
21	C₂²×#92418	D₄	1	1	yes	characteristic
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	4	4	no	—
25	C₄×#92418	—	2	1	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	Q₈	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	—	2	2	no	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#106×C₆₁	—	2	4	no	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₄×C₈×C₆₁	Q₈	1	2	yes	—
38	#95×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
48	C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	—
52	C₄²×C₆₁	—	4	2	no	—
53	C₄²×C₆₁	—	4	2	no	—
54	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₁	#41	1	1	yes	characteristic
58	C₂×#92418	—	2	2	no	—
59	C₂×#92418	—	2	2	no	—
60	C₂×#92418	—	2	2	no	—
61	C₂×#92418	—	2	1	no	—
62	C₂×#92418	—	2	1	no	—
63	C₂×#92418	—	2	2	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂×#92418	—	4	2	no	—
67	C₂×#92418	—	4	2	no	—
68	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	4	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	—	4	2	no	—
75	C₂×C₄×C₆₁	—	4	2	no	—
76	C₂×C₄×C₆₁	—	4	2	no	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₆₁	#131	1	2	yes	—
85	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄×C₆₁	#137	1	2	yes	—
87	C₈×C₆₁	—	2	2	no	—
88	C₈×C₆₁	—	2	2	no	—
89	#92418	—	2	4	no	—
90	#92418	—	4	1	no	—
91	#92418	—	4	1	no	—
92	#92418	—	8	2	no	—
93	#9600	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#389	1	1	yes	characteristic
95	C₂²×C₆₁	#435	1	2	yes	—
96	C₂²×C₆₁	#440	1	2	yes	—
97	C₂²×C₆₁	#443	1	1	yes	characteristic
98	C₂²×C₆₁	#443	1	1	yes	characteristic
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	4	4	no	—
102	C₄×C₆₁	—	8	2	no	—
103	C₄×C₆₁	#423	1	1	yes	characteristic
104	C₄×C₆₁	#424	1	1	yes	characteristic
105	C₄×C₆₁	#446	1	2	yes	—
106	C₄×C₆₁	#469	1	1	yes	characteristic
107	C₄×C₆₁	#470	1	1	yes	characteristic
108	C₄×C₆₁	#501	1	2	yes	—
109	#1939	—	61	2	no	—
110	C₄×#106	D₆₁	1	1	yes	characteristic
111	C₂×#335	—	61	1	no	—
112	C₂×#322	—	61	1	no	—
113	#1279	—	61	2	no	—
114	C₂×C₆₁	#1585	1	2	yes	—
115	C₂×C₆₁	#1588	1	2	yes	—
116	C₂×C₆₁	#1940	1	1	yes	characteristic
117	C₂×C₆₁	#1944	1	1	yes	characteristic
118	C₂×C₆₁	#1945	1	1	yes	characteristic
119	C₂×#106	Dic₆₁	1	2	yes	—
120	C₂×C₄⋊C₈	—	61	2	no	—
121	C₂×C₄⋊C₈	—	122	2	no	—
122	C₂×C₄×C₈	—	61	1	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
126	#340	Dic₆₁	1	1	yes	characteristic
127	#340	Dic₆₁	1	1	yes	characteristic
128	#336	—	122	2	no	—
129	#335	—	122	2	no	—
130	#322	—	122	2	no	—
131	C₆₁	#9600	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	61	1	no	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	122	1	no	—
135	C₂²×C₈	—	122	2	no	—
136	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	122	2	no	—
144	C₄⋊C₈	—	122	2	no	—
145	C₄⋊C₈	—	244	4	no	—
146	C₄×C₈	—	61	2	no	—
147	C₄×C₈	—	122	1	no	—
148	C₄×C₈	—	122	1	no	—
149	C₄×C₈	—	122	1	no	—
150	C₄×C₈	Dic₁₂₂	1	2	yes	—
151	C₄×C₈	D₂₄₄	1	2	yes	—
152	#95	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
156	C₂²×C₄	#11758224	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	122	2	no	—
161	C₂×C₈	—	122	2	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	1	no	—
164	C₂×C₈	—	122	1	no	—
165	C₂×C₈	—	122	2	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	244	2	no	—
168	C₂×C₈	—	244	2	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
171	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	#11758194	1	4	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
179	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
180	C₄²	Q₈×D₆₁	1	1	yes	characteristic
181	C₄²	#11758226	1	1	yes	characteristic
182	C₂³	D₄×Dic₆₁	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
192	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
193	C₂×C₄	#14519752	1	1	yes	characteristic
194	C₂×C₄	#14519757	1	1	yes	characteristic
195	C₂×C₄	#14519768	1	1	yes	characteristic
196	C₂×C₄	#14519776	1	1	yes	characteristic
197	C₂×C₄	C₂×#11758194	1	2	yes	—
198	C₂×C₄	#14519784	1	2	yes	—
199	C₂×C₄	#14519804	1	2	yes	—
200	C₂×C₄	#14519806	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	122	4	no	—
204	C₈	—	488	2	no	—
205	C₈	—	244	1	no	—
206	C₈	—	244	1	no	—
207	C₂²	#16839661	1	1	yes	characteristic
208	C₂²	#16839834	1	2	yes	—
209	C₂²	#16839863	1	2	yes	—
210	C₂²	#16839879	1	1	yes	characteristic
211	C₂²	#16839879	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	#16839786	1	1	yes	characteristic
217	C₄	#16839789	1	1	yes	characteristic
218	C₄	#16839893	1	2	yes	—
219	C₄	#16840016	1	1	yes	characteristic
220	C₄	#16840021	1	1	yes	characteristic
221	C₄	#16840169	1	2	yes	—
222	C₂	#21444677	1	2	yes	—
223	C₂	#21444698	1	2	yes	—
224	C₂	#21446543	1	1	yes	characteristic
225	C₂	#21446563	1	1	yes	characteristic
226	C₂	#21446568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#106×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#106	D₆₁
hypocenter	C₆₁	#9600
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
131. C₆₁	#9600	semidirect

Maximal subgroups

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

subgroup	index
#21446538	2
C₂×#16839515	2
C₂×#16839484	2
#21443660	2
C₄×#106×C₆₁	2
#9600	61