Subgroups — #29596179

Order 15616.

438 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	C₂³×#11758185	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839587	C₂	1	2	yes	maximal, complemented
3	#21445848	C₂	1	1	yes	maximal, characteristic
4	#21443910	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	—	2	4	no	—
7	C₂²×#11758185	C₄	1	4	yes	complemented
8	C₂²×#11758185	C₂²	1	1	yes	characteristic
9	C₂²×#11758185	C₂²	1	1	yes	characteristic
10	C₂×#14519711	C₂²	1	2	yes	—
11	#16839622	—	2	1	no	—
12	#16839587	—	2	4	no	—
13	#16839525	—	2	1	no	—
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#380	—	2	1	no	—
17	C₆₁×#340	—	2	1	no	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×C₂²⋊Dic₆₁	—	4	1	no	—
20	C₂×#11758185	—	2	2	no	—
21	C₂×#11758185	—	2	8	no	—
22	C₂×#11758185	C₂×C₄	1	2	yes	—
23	C₂×#11758185	C₂×C₄	1	2	yes	—
24	C₂×#11758185	D₄	1	4	yes	—
25	#14519711	—	2	2	no	—
26	#14519711	—	2	2	no	—
27	#14519684	—	4	1	no	—
28	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
37	#107×C₆₁	—	2	2	no	—
38	#107×C₆₁	—	2	2	no	—
39	#95×C₆₁	—	4	1	no	—
40	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	—	2	2	no	—
42	C₂²×C₄×C₆₁	—	2	2	no	—
43	C₂²×C₄×C₆₁	—	4	2	no	—
44	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	2	4	no	—
53	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₁	#41	1	2	yes	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₂²⋊C₄×C₆₁	—	8	2	no	—
61	C₂²⋊Dic₆₁	—	8	2	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	#11758185	—	2	2	no	—
65	#11758185	C₂²⋊C₄	1	4	yes	complemented
66	C₂³×C₆₁	—	2	2	no	—
67	C₂³×C₆₁	—	2	2	no	—
68	C₂³×C₆₁	#107	1	2	yes	—
69	C₂³×C₆₁	D₁₆	1	2	yes	—
70	C₂³×C₆₁	Q₃₂	1	2	yes	—
71	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	4	no	—
76	C₂×C₄×C₆₁	—	8	2	no	—
77	C₂×C₄×C₆₁	—	8	2	no	—
78	C₂×C₄×C₆₁	—	8	2	no	—
79	C₂×C₄×C₆₁	#107	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#121	1	2	yes	—
82	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#135	1	2	yes	—
84	C₂×Dic₆₁	—	8	2	no	—
85	C₂×Dic₆₁	—	8	2	no	—
86	C₂×Dic₆₁	—	8	2	no	—
87	C₈×C₆₁	—	2	2	no	—
88	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₆₁	#117	1	1	yes	characteristic
90	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
91	#12834	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	—	2	2	no	—
93	C₂²×C₆₁	—	2	2	no	—
94	C₂²×C₆₁	—	2	4	no	—
95	C₂²×C₆₁	—	2	4	no	—
96	C₂²×C₆₁	#366	1	4	yes	complemented
97	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
98	C₂²×C₆₁	#466	1	2	yes	—
99	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
100	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
101	C₂²×C₆₁	#508	1	2	yes	—
102	C₄×C₆₁	—	2	2	no	—
103	C₄×C₆₁	—	16	2	no	—
104	C₄×C₆₁	#380	1	1	yes	characteristic
105	C₄×C₆₁	#427	1	1	yes	characteristic
106	C₄×C₆₁	#467	1	2	yes	—
107	Dic₆₁	—	16	2	no	—
108	C₂³×C₁₆	—	61	1	no	—
109	C₂×#366	—	61	2	no	—
110	#1808	—	61	1	no	—
111	#1808	D₆₁	1	1	yes	characteristic
112	#1380	—	61	2	no	—
113	C₂×C₆₁	—	2	4	no	—
114	C₂×C₆₁	#1808	1	1	yes	characteristic
115	C₂×C₆₁	C₂×#366	1	1	yes	characteristic
116	C₂×C₆₁	#2158	1	1	yes	characteristic
117	C₂×C₆₁	#2215	1	2	yes	—
118	C₂×C₆₁	#2216	1	2	yes	—
119	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
120	C₂²×C₁₆	—	61	1	no	—
121	C₂²×C₁₆	—	61	4	no	—
122	C₂²×C₁₆	—	61	1	no	—
123	C₂²×C₁₆	—	122	4	no	—
124	C₂×#107	—	61	2	no	—
125	C₂×#107	C₂×D₆₁	1	2	yes	—
126	#380	—	2	1	no	—
127	#380	—	122	1	no	—
128	#366	—	122	4	no	—
129	#340	—	2	1	no	—
130	#340	—	122	1	no	—
131	C₆₁	#12834	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	Dic₁₂₂	1	2	yes	—
136	C₂²×C₈	C₄×D₆₁	1	2	yes	—
137	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	122	2	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	—	244	1	no	—
143	C₂×C₁₆	—	61	2	no	—
144	C₂×C₁₆	—	61	4	no	—
145	C₂×C₁₆	—	61	2	no	—
146	C₂×C₁₆	—	122	8	no	—
147	C₂×C₁₆	—	122	2	no	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	—	122	2	no	—
151	#107	—	122	2	no	—
152	#95	—	4	1	no	—
153	#95	—	244	1	no	—
154	C₂⁴	#11758199	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	244	2	no	—
159	C₂²×C₄	#11758196	1	2	yes	—
160	C₂²×C₄	#11758199	1	1	yes	characteristic
161	C₂²×C₄	#11758202	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	#11758196	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
170	C₂×C₈	D₄×D₆₁	1	2	yes	—
171	C₂×C₈	#11758224	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	244	2	no	—
175	C₄⋊C₄	—	244	2	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₂²⋊C₄	—	488	2	no	—
178	C₁₆	—	61	4	no	—
179	C₁₆	—	122	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#14519710	1	2	yes	—
183	C₂³	#14519718	1	2	yes	—
184	C₂³	#14519723	1	2	yes	—
185	C₂³	C₂×#11758199	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	488	2	no	—
194	C₂×C₄	—	488	2	no	—
195	C₂×C₄	—	488	2	no	—
196	C₂×C₄	#14519710	1	2	yes	—
197	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
198	C₂×C₄	#14519749	1	2	yes	—
199	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
200	C₂×C₄	#14519797	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
203	C₈	#14519733	1	1	yes	characteristic
204	C₈	#14519754	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	#16839587	1	4	yes	complemented
210	C₂²	C₂×#14519710	1	1	yes	characteristic
211	C₂²	#16839996	1	2	yes	—
212	C₂²	C₂×#14519718	1	1	yes	characteristic
213	C₂²	C₂×#14519723	1	1	yes	characteristic
214	C₂²	#16840192	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	16	2	no	—
217	C₄	—	976	2	no	—
218	C₄	#16839620	1	1	yes	characteristic
219	C₄	#16839798	1	1	yes	characteristic
220	C₄	#16840001	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	#21445846	1	1	yes	characteristic
223	C₂	C₂×#16839587	1	1	yes	characteristic
224	C₂	#21447601	1	1	yes	characteristic
225	C₂	#21447903	1	2	yes	—
226	C₂	#21447908	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1808	C₂
socle	C₂³×C₆₁	C₂×D₈
prime radical	C₂⁴×C₆₁	D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1808	D₆₁
hypocenter	C₆₁	#12834
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#16839587	C₂	semidirect
4. #21443910	C₂	semidirect
7. C₂²×#11758185	C₄	semidirect
65. #11758185	C₂²⋊C₄	semidirect
96. C₂²×C₆₁	#366	semidirect
131. C₆₁	#12834	semidirect
209. C₂²	#16839587	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11758185	2
C₂×#16839587	2
#21445848	2
#21443910	2
C₆₁×#1808	2
#12834	61