Subgroups — #29598603

Order 15616.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447575	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
3	#21443927	C₂	1	4	yes	maximal
4	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₂²	1	2	yes	—
6	C₂×#14519713	C₂²	1	2	yes	—
7	C₄×#11758194	—	2	1	no	—
8	C₄⋊#11758185	—	2	1	no	—
9	C₄⋊#11758185	—	2	1	no	—
10	#16839527	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
12	C₆₁×#500	—	2	1	no	—
13	C₆₁×#500	—	2	1	no	—
14	C₆₁×#500	C₄	1	2	yes	—
15	C₆₁×#500	C₄	1	2	yes	—
16	C₂×#107×C₆₁	—	2	1	no	—
17	C₂×#107×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#11758194	—	2	2	no	—
20	#14519717	—	2	4	no	—
21	C₂×#11758185	—	2	2	no	—
22	#14519713	—	2	2	no	—
23	#14519713	Q₈	1	4	yes	complemented
24	#14519688	—	4	1	no	—
25	C₄×#92418	—	4	1	no	—
26	C₂²×Q₈×C₆₁	—	4	1	no	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₁	—	2	2	no	—
30	C₄⋊Q₈×C₆₁	—	2	2	no	—
31	C₄⋊Q₈×C₆₁	—	4	1	no	—
32	C₄⋊Q₈×C₆₁	—	4	1	no	—
33	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	#107×C₆₁	—	2	2	no	—
37	#107×C₆₁	—	2	2	no	—
38	#107×C₆₁	—	2	2	no	—
39	#107×C₆₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₆₁	—	4	2	no	—
45	C₂×Q₈×C₆₁	—	4	2	no	—
46	C₂×Q₈×C₆₁	—	8	4	no	—
47	C₂²×C₄×C₆₁	—	4	2	no	—
48	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₆₁	#41	1	2	yes	—
58	C₄⋊C₄×C₆₁	—	2	4	no	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₄⋊C₄×C₆₁	—	4	2	no	—
61	C₄⋊C₄×C₆₁	—	4	2	no	—
62	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₁	D₈	1	1	yes	characteristic
65	C₄²×C₆₁	D₈	1	1	yes	characteristic
66	#11758194	—	2	4	no	—
67	#11758194	—	4	2	no	—
68	C₂×#92418	—	4	2	no	—
69	#11758185	—	4	2	no	—
70	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₆₁	—	8	4	no	—
72	Q₈×C₆₁	—	8	4	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	4	4	no	—
76	C₂×C₄×C₆₁	—	8	4	no	—
77	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#137	1	2	yes	—
85	C₈×C₆₁	—	2	1	no	—
86	C₈×C₆₁	—	2	1	no	—
87	C₈×C₆₁	C₄×Q₈	1	2	yes	—
88	C₈×C₆₁	#124	1	1	yes	characteristic
89	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
90	#92418	—	4	4	no	—
91	#13279	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	—	2	1	no	—
94	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
95	C₂²×C₆₁	#475	1	2	yes	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	—	8	4	no	—
99	C₄×C₆₁	#361	1	2	yes	—
100	C₄×C₆₁	#362	1	2	yes	—
101	C₄×C₆₁	#391	1	1	yes	characteristic
102	C₄×C₆₁	#419	1	1	yes	characteristic
103	C₄×C₆₁	#474	1	2	yes	—
104	C₂×#500	D₆₁	1	1	yes	characteristic
105	#2150	—	61	1	no	—
106	C₄×#108	—	61	1	no	—
107	#1387	—	61	4	no	—
108	C₂×C₆₁	—	2	2	no	—
109	C₂×C₆₁	#1988	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#361	1	1	yes	characteristic
111	C₂×C₆₁	C₂×#362	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×M₃₂	—	61	2	no	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	Dic₆₁	1	2	yes	—
117	#500	Dic₆₁	1	2	yes	—
118	C₂×#108	—	61	2	no	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂×D₆₁	1	2	yes	—
121	C₄×M₁₆	—	122	1	no	—
122	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
123	C₄⋊C₁₆	—	122	1	no	—
124	C₄⋊C₁₆	—	122	1	no	—
125	#341	—	122	1	no	—
126	C₆₁	#13279	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	122	2	no	—
129	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
138	M₃₂	—	122	4	no	—
139	C₂×C₁₆	—	122	2	no	—
140	#108	—	61	4	no	—
141	#108	—	122	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×Dic₆₁	1	4	yes	—
146	#97	—	244	1	no	—
147	C₄×C₈	—	244	1	no	—
148	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
149	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#11758200	1	1	yes	characteristic
157	C₂²×C₄	#11758200	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
159	M₁₆	—	122	4	no	—
160	M₁₆	—	244	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	244	2	no	—
163	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₆₁	1	2	yes	—
166	C₂×C₈	#11758226	1	2	yes	—
167	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	#11758199	1	1	yes	characteristic
174	C₄²	#11758199	1	1	yes	characteristic
175	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
177	C₁₆	—	244	2	no	—
178	C₂³	C₂×#11758200	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#14519698	1	1	yes	characteristic
186	C₂×C₄	#14519698	1	1	yes	characteristic
187	C₂×C₄	#14519698	1	1	yes	characteristic
188	C₂×C₄	#14519698	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
190	C₂×C₄	#14519753	1	2	yes	—
191	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
192	C₂×C₄	#14519808	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	244	4	no	—
196	C₈	Q₈×Dic₆₁	1	2	yes	—
197	C₈	#14519766	1	1	yes	characteristic
198	C₈	#14519775	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#14519698	1	1	yes	characteristic
202	C₂²	#16840044	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#16839575	1	2	yes	—
207	C₄	#16839578	1	2	yes	—
208	C₄	#16839669	1	1	yes	characteristic
209	C₄	#16839771	1	1	yes	characteristic
210	C₄	#16840039	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#21446772	1	1	yes	characteristic
213	C₂	C₂×#16839575	1	1	yes	characteristic
214	C₂	C₂×#16839578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14519698
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×C₆₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	#13279
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
23. #14519713	Q₈	semidirect
126. C₆₁	#13279	semidirect

Maximal subgroups

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

subgroup	index
#21447575	2
C₄×#14519713	2
#21443927	2
C₂×C₆₁×#500	2
#13279	61