Subgroups — #29835513

Order 15616.

482 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	#21456333	C₂	1	1	yes	maximal, characteristic
2	#21448106	C₂	1	4	yes	maximal, complemented
3	#21448102	C₂	1	4	yes	maximal, complemented
4	#21447601	C₂	1	4	yes	maximal
5	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic
7	#16840319	—	2	1	no	—
8	C₂²×#11758185	C₂²	1	2	yes	—
9	#16840167	C₂²	1	2	yes	—
10	#16840164	—	2	1	no	—
11	#16840164	C₂²	1	2	yes	—
12	#16840160	—	2	1	no	—
13	#16839800	—	2	1	no	—
14	#16839800	C₂²	1	2	yes	—
15	#16839589	C₂²	1	8	yes	—
16	#16839587	C₂²	1	8	yes	—
17	#16839538	C₂²	1	2	yes	—
18	#16839538	C₂²	1	2	yes	—
19	C₆₁×#533	—	2	1	no	—
20	C₆₁×#500	C₂²	1	2	yes	—
21	C₆₁×#499	—	2	1	no	—
22	C₆₁×#499	C₂²	1	2	yes	—
23	C₆₁×#498	—	2	1	no	—
24	C₆₁×#427	—	2	1	no	—
25	C₆₁×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	#14519777	—	2	2	no	—
28	#14519769	—	2	2	no	—
29	#14519756	—	4	2	no	—
30	C₄×Dic₁₂₂	—	4	2	no	—
31	#14519736	—	2	2	no	—
32	C₂×#11758185	Q₈	1	8	yes	complemented
33	C₂×#11758185	C₂³	1	4	yes	—
34	#14519711	—	2	2	no	—
35	#14519711	C₂³	1	4	yes	—
36	#14519709	—	2	2	no	—
37	#14519709	—	2	2	no	—
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₁	—	2	2	no	—
41	#125×C₆₁	—	2	2	no	—
42	C₂²⋊Q₈×C₆₁	—	4	2	no	—
43	C₄×Q₈×C₆₁	—	4	2	no	—
44	#117×C₆₁	—	2	2	no	—
45	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
46	#107×C₆₁	—	2	2	no	—
47	#107×C₆₁	C₂³	1	4	yes	—
48	#106×C₆₁	—	2	2	no	—
49	#106×C₆₁	—	2	2	no	—
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₆₁	—	4	2	no	—
55	C₂²×C₄×C₆₁	D₈	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₁₂₂	—	4	2	no	—
58	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₆₁	—	2	4	no	—
65	C₄⋊C₄×C₆₁	—	4	4	no	—
66	C₄⋊C₄×C₆₁	—	4	2	no	—
67	C₂²⋊C₄×C₆₁	—	4	2	no	—
68	C₄²×C₆₁	—	4	2	no	—
69	C₄²×C₆₁	D₈	1	2	yes	—
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²⋊Dic₆₁	—	4	2	no	—
73	C₄⋊Dic₆₁	—	2	4	no	—
74	C₄⋊Dic₆₁	—	4	2	no	—
75	#11758196	—	4	4	no	—
76	C₄×Dic₆₁	—	4	2	no	—
77	#11758185	C₂×Q₈	1	8	yes	—
78	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₆₁	—	8	4	no	—
80	C₂×C₄×C₆₁	—	4	2	no	—
81	C₂×C₄×C₆₁	—	4	4	no	—
82	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₆₁	—	4	4	no	—
89	Dic₁₂₂	—	8	4	no	—
90	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
91	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
92	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
93	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
94	#33728	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	#500	1	2	yes	—
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	—	8	4	no	—
101	C₄×C₆₁	#500	1	2	yes	—
102	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₆₁	—	8	4	no	—
105	#3160	—	61	1	no	—
106	#3160	D₆₁	1	1	yes	characteristic
107	#2255	—	61	4	no	—
108	#2255	—	61	4	no	—
109	#2158	—	61	4	no	—
110	C₂×#346	—	61	1	no	—
111	C₂×C₆₁	—	2	2	no	—
112	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
113	C₂×C₆₁	#3414	1	2	yes	—
114	#533	—	2	1	no	—
115	#533	—	122	1	no	—
116	C₂²×C₁₆	—	61	2	no	—
117	#500	—	61	2	no	—
118	#500	C₂×D₆₁	1	2	yes	—
119	#499	—	2	1	no	—
120	#499	—	61	2	no	—
121	#499	—	122	1	no	—
122	#499	C₂×D₆₁	1	2	yes	—
123	#498	—	2	1	no	—
124	#498	—	122	1	no	—
125	#427	—	2	1	no	—
126	#427	—	61	2	no	—
127	#427	—	122	1	no	—
128	#427	C₂×D₆₁	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
130	#367	—	61	8	no	—
131	#366	—	61	8	no	—
132	#346	—	61	2	no	—
133	#346	—	61	2	no	—
134	C₆₁	#33728	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	122	2	no	—
139	#125	—	2	2	no	—
140	#125	—	122	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	244	2	no	—
143	C₄×Q₈	—	4	2	no	—
144	C₄×Q₈	—	244	2	no	—
145	#117	—	2	2	no	—
146	#117	—	122	2	no	—
147	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
148	C₂×C₁₆	—	61	8	no	—
149	C₂×C₁₆	—	61	4	no	—
150	#107	—	2	2	no	—
151	#107	—	61	4	no	—
152	#107	—	122	2	no	—
153	#107	C₂²×D₆₁	1	4	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	—	122	2	no	—
157	#106	—	122	2	no	—
158	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	244	2	no	—
164	C₂²×C₄	#11758199	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
166	C₂×C₈	#11758202	1	4	yes	—
167	C₂×C₈	Q₈×D₆₁	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	122	4	no	—
176	C₄⋊C₄	—	244	4	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	244	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	244	2	no	—
182	C₄²	#11758199	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
185	C₁₆	—	61	8	no	—
186	C₂³	C₂×#11758199	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	488	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	244	4	no	—
192	C₂×C₄	#14519775	1	2	yes	—
193	C₂×C₄	C₂×#11758199	1	2	yes	—
194	C₂×C₄	C₂×#11758199	1	2	yes	—
195	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
196	C₂×C₄	C₂×#11758202	1	2	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
198	C₈	#14519775	1	2	yes	—
199	C₈	C₂×#11758202	1	2	yes	—
200	C₈	C₂×#11758202	1	2	yes	—
201	C₈	C₂×Q₈×D₆₁	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	#16840165	1	2	yes	—
204	C₂²	C₂²×#11758199	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	8	4	no	—
208	C₄	—	488	4	no	—
209	C₄	#16840165	1	2	yes	—
210	C₄	C₂×#14519775	1	1	yes	characteristic
211	C₄	C₂²×#11758202	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₂×#16840165	1	1	yes	characteristic
214	C₂	#21458953	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3160	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#3160	D₆₁
hypocenter	C₆₁	#33728
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #21448106	C₂	semidirect
3. #21448102	C₂	semidirect
32. C₂×#11758185	Q₈	semidirect
134. C₆₁	#33728	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
#21448106	2
#21448102	2
#21447601	2
C₂×#16839538	2
C₆₁×#3160	2
#33728	61