Subgroups — #29599630

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839589	C₂	1	2	yes	maximal
3	#21443910	C₂	1	2	yes	maximal
4	#21443897	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1942	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	—	2	1	no	—
7	C₂²×#11758185	C₂²	1	2	yes	—
8	#16840167	—	2	4	no	—
9	C₂×#14519711	—	2	1	no	—
10	C₂×#14519711	C₂²	1	2	yes	—
11	#16839589	—	2	4	no	—
12	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#389	—	2	1	no	—
16	C₆₁×#340	—	2	1	no	—
17	C₂²×Dic₁₂₂	—	4	1	no	—
18	#14519777	—	2	4	no	—
19	#14519777	—	4	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	4	1	no	—
22	C₂×#11758185	—	2	2	no	—
23	C₂×#11758185	—	2	2	no	—
24	C₂×#11758185	Q₈	1	4	yes	complemented
25	#14519711	—	2	4	no	—
26	#14519711	—	2	2	no	—
27	#14519711	D₄	1	4	yes	—
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
33	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
34	#107×C₆₁	—	2	1	no	—
35	#107×C₆₁	—	2	2	no	—
36	#107×C₆₁	Q₈	1	2	yes	complemented
37	C₄×C₈×C₆₁	—	2	2	no	—
38	#95×C₆₁	—	4	1	no	—
39	C₂²×C₄×C₆₁	—	4	1	no	—
40	C₂²×C₄×C₆₁	—	4	1	no	—
41	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₆₁	—	4	2	no	—
45	C₂×Dic₁₂₂	—	4	4	no	—
46	C₂×Dic₁₂₂	—	8	2	no	—
47	C₂×Dic₁₂₂	—	8	2	no	—
48	C₂×C₈×C₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	—	2	1	no	—
50	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₆₁	—	2	2	no	—
56	C₄⋊C₄×C₆₁	—	4	1	no	—
57	C₄⋊C₄×C₆₁	—	4	1	no	—
58	C₄⋊C₄×C₆₁	—	4	1	no	—
59	C₄⋊C₄×C₆₁	—	8	2	no	—
60	C₄²×C₆₁	—	2	1	no	—
61	C₄²×C₆₁	—	2	1	no	—
62	C₄⋊Dic₆₁	—	2	4	no	—
63	C₄⋊Dic₆₁	—	4	4	no	—
64	C₄⋊Dic₆₁	—	4	2	no	—
65	#11758185	—	2	4	no	—
66	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	8	1	no	—
73	C₂×C₄×C₆₁	—	8	1	no	—
74	C₂×C₄×C₆₁	—	8	1	no	—
75	C₂×C₄×C₆₁	—	8	1	no	—
76	C₂×C₄×C₆₁	—	8	1	no	—
77	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
78	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
79	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
85	C₂×Dic₆₁	—	4	4	no	—
86	C₂×Dic₆₁	—	8	2	no	—
87	C₂×Dic₆₁	—	8	2	no	—
88	Dic₁₂₂	—	8	8	no	—
89	C₈×C₆₁	—	2	2	no	—
90	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
92	#13452	—	61	1	no	maximal, Sylow
93	C₂²×C₆₁	#447	1	1	yes	characteristic
94	C₂²×C₆₁	#474	1	1	yes	characteristic
95	C₂²×C₆₁	#500	1	1	yes	characteristic
96	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
98	C₂²×C₆₁	#509	1	1	yes	characteristic
99	C₂²×C₆₁	#510	1	1	yes	characteristic
100	C₄×C₆₁	—	2	2	no	—
101	C₄×C₆₁	—	2	2	no	—
102	C₄×C₆₁	—	8	2	no	—
103	C₄×C₆₁	—	16	1	no	—
104	C₄×C₆₁	#395	1	1	yes	characteristic
105	C₄×C₆₁	#451	1	1	yes	characteristic
106	C₄×C₆₁	#477	1	1	yes	characteristic
107	C₄×C₆₁	#501	1	1	yes	characteristic
108	Dic₆₁	—	8	4	no	—
109	C₂×#500	—	61	1	no	—
110	C₂×#367	—	61	2	no	—
111	#1942	D₆₁	1	1	yes	characteristic
112	#1380	—	61	2	no	—
113	#1374	—	61	1	no	—
114	C₂×C₆₁	#2057	1	1	yes	characteristic
115	C₂×C₆₁	#2185	1	1	yes	characteristic
116	C₂×C₆₁	#2187	1	1	yes	characteristic
117	C₂×C₆₁	#2225	1	1	yes	characteristic
118	C₂×C₆₁	#2226	1	1	yes	characteristic
119	C₂×C₆₁	#2254	1	1	yes	characteristic
120	C₂×C₆₁	#2255	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	—	122	1	no	—
122	C₂²×C₁₆	—	61	2	no	—
123	#500	—	122	4	no	—
124	C₂×#107	—	61	2	no	—
125	C₂×#107	—	122	1	no	—
126	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
127	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
129	#389	—	2	1	no	—
130	#367	—	122	4	no	—
131	#340	—	2	1	no	—
132	C₆₁	#13452	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	244	1	no	—
134	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	122	4	no	—
137	C₄⋊Q₈	—	244	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	122	2	no	—
142	C₂×C₄⋊C₄	—	244	1	no	—
143	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
144	C₂×C₁₆	—	61	4	no	—
145	C₂×C₁₆	—	122	2	no	—
146	C₂×C₁₆	—	122	2	no	—
147	#107	—	2	1	no	—
148	#107	—	2	2	no	—
149	#107	—	61	4	no	—
150	#107	—	122	4	no	—
151	#107	—	122	2	no	—
152	#107	Dic₁₂₂	1	2	yes	complemented
153	C₄×C₈	—	2	2	no	—
154	#95	—	4	1	no	—
155	C₂×Q₈	—	488	2	no	—
156	C₂×Q₈	—	488	2	no	—
157	C₂×Q₈	—	244	4	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	244	2	no	—
161	C₂²×C₄	#11758199	1	1	yes	characteristic
162	C₂²×C₄	#11758199	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	#11758202	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
168	C₂×C₈	#11758222	1	1	yes	characteristic
169	C₂×C₈	D₄×D₆₁	1	2	yes	—
170	C₂×C₈	Q₈×D₆₁	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	8	2	no	—
176	C₄⋊C₄	—	122	4	no	—
177	C₄⋊C₄	—	244	4	no	—
178	C₄⋊C₄	—	244	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	122	4	no	—
182	C₂³	C₂×#11758199	1	1	yes	characteristic
183	Q₈	—	488	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	488	2	no	—
195	C₂×C₄	—	488	2	no	—
196	C₂×C₄	—	244	4	no	—
197	C₂×C₄	#14519720	1	2	yes	—
198	C₂×C₄	#14519721	1	2	yes	—
199	C₂×C₄	#14519745	1	1	yes	characteristic
200	C₂×C₄	#14519756	1	1	yes	characteristic
201	C₂×C₄	#14519775	1	1	yes	characteristic
202	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
203	C₂×C₄	#14519802	1	1	yes	characteristic
204	C₂×C₄	#14519807	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#14519755	1	2	yes	—
207	C₈	#14519776	1	2	yes	—
208	C₂²	#16839895	1	1	yes	characteristic
209	C₂²	#16840042	1	1	yes	characteristic
210	C₂²	#16840165	1	1	yes	characteristic
211	C₂²	C₂×#14519720	1	1	yes	characteristic
212	C₂²	C₂×#14519721	1	1	yes	characteristic
213	C₂²	#16840197	1	1	yes	characteristic
214	C₂²	#16840202	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	2	no	—
218	C₄	—	16	1	no	—
219	C₄	—	488	4	no	—
220	C₄	#16839682	1	1	yes	characteristic
221	C₄	#16839917	1	1	yes	characteristic
222	C₄	#16840057	1	1	yes	characteristic
223	C₄	#16840172	1	1	yes	characteristic
224	C₂	#21447147	1	1	yes	characteristic
225	C₂	#21447735	1	1	yes	characteristic
226	C₂	#21447747	1	1	yes	characteristic
227	C₂	#21447961	1	1	yes	characteristic
228	C₂	#21447966	1	1	yes	characteristic
229	C₂	#21448099	1	1	yes	characteristic
230	C₂	#21448106	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1942	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#1942	D₆₁
hypocenter	C₆₁	#13452
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
24. C₂×#11758185	Q₈	semidirect
36. #107×C₆₁	Q₈	semidirect
132. C₆₁	#13452	semidirect
152. #107	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840167	2
C₂×#16839589	2
#21443910	2
#21443897	2
C₆₁×#1942	2
#13452	61