Subgroups — #29598590

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519711	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal
4	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519711	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₆₁	—	2	1	no	—
8	C₄⋊#11758185	—	2	2	no	—
9	#16839525	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
11	C₆₁×#500	—	2	2	no	—
12	C₆₁×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₆₁	—	2	1	no	—
14	C₂×#107×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	C₂×C₄×Dic₆₁	—	4	1	no	—
18	C₂×#11758185	—	2	2	no	—
19	C₂×#11758185	—	2	4	no	—
20	#14519711	—	2	2	no	—
21	#14519711	Q₈	1	4	yes	complemented
22	#14519684	—	4	1	no	—
23	C₂²×Q₈×C₆₁	—	4	1	no	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊Q₈×C₆₁	—	2	4	no	—
27	C₄⋊Q₈×C₆₁	—	4	2	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
30	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
31	#107×C₆₁	—	2	4	no	—
32	#107×C₆₁	—	2	2	no	—
33	#107×C₆₁	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₂×Q₈×C₆₁	—	4	4	no	—
37	C₂×Q₈×C₆₁	—	8	4	no	—
38	C₂²×C₄×C₆₁	—	4	2	no	—
39	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₆₁	#41	1	2	yes	—
49	C₄⋊C₄×C₆₁	—	2	4	no	—
50	C₄⋊C₄×C₆₁	—	4	4	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₆₁	SD₁₆	1	2	yes	—
54	C₄⋊Dic₆₁	—	2	4	no	—
55	C₄⋊Dic₆₁	—	4	2	no	—
56	C₄×Dic₆₁	—	8	2	no	—
57	#11758185	—	4	2	no	—
58	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
59	Q₈×C₆₁	—	8	8	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	4	4	no	—
62	C₂×C₄×C₆₁	—	8	4	no	—
63	C₂×C₄×C₆₁	#102	1	2	yes	—
64	C₂×C₄×C₆₁	#102	1	2	yes	—
65	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
66	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#137	1	2	yes	—
71	C₂×Dic₆₁	—	4	4	no	—
72	C₂×Dic₆₁	—	8	2	no	—
73	C₂×Dic₆₁	—	8	2	no	—
74	C₈×C₆₁	—	2	2	no	—
75	C₈×C₆₁	C₄×Q₈	1	2	yes	—
76	C₈×C₆₁	#124	1	1	yes	characteristic
77	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
78	#13276	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
80	C₂²×C₆₁	#474	1	2	yes	—
81	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
82	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
83	C₂²×C₆₁	#509	1	2	yes	—
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	8	4	no	—
86	C₄×C₆₁	#358	1	4	yes	—
87	C₄×C₆₁	#391	1	1	yes	characteristic
88	C₄×C₆₁	#419	1	1	yes	characteristic
89	C₄×C₆₁	#475	1	2	yes	—
90	Dic₆₁	—	8	4	no	—
91	C₂×#500	D₆₁	1	1	yes	characteristic
92	C₂×C₄⋊C₁₆	—	61	1	no	—
93	C₄×#107	—	61	1	no	—
94	#1380	—	61	4	no	—
95	C₂×C₆₁	#1988	1	1	yes	characteristic
96	C₂×C₆₁	C₂×#358	1	1	yes	characteristic
97	C₂×C₆₁	#2140	1	1	yes	characteristic
98	C₂×C₆₁	#2226	1	2	yes	—
99	C₂×C₆₁	#2227	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂²×C₁₆	—	61	2	no	—
102	#500	—	2	2	no	—
103	#500	Dic₆₁	1	4	yes	complemented
104	C₂×#107	—	2	1	no	—
105	C₂×#107	—	61	2	no	—
106	C₂×#107	C₂×D₆₁	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	122	1	no	—
109	C₄⋊C₁₆	—	122	2	no	—
110	#340	—	122	1	no	—
111	C₆₁	#13276	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	4	1	no	—
113	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	4	no	—
116	C₄⋊Q₈	—	4	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	122	2	no	—
120	C₂×C₄²	—	244	1	no	—
121	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂×C₁₆	—	122	4	no	—
123	C₂×C₁₆	—	122	2	no	—
124	#107	—	2	4	no	—
125	#107	—	2	2	no	—
126	#107	—	61	4	no	—
127	#107	—	122	2	no	—
128	#107	C₂×Dic₆₁	1	4	yes	—
129	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
130	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
131	#95	—	244	1	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	8	4	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	244	2	no	—
136	C₂²×C₄	#11758199	1	1	yes	characteristic
137	C₂²×C₄	#11758199	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
141	C₂×C₈	Q₈×D₆₁	1	2	yes	—
142	C₂×C₈	#11758226	1	2	yes	—
143	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	122	4	no	—
149	C₄⋊C₄	—	244	2	no	—
150	C₄²	—	488	2	no	—
151	C₄²	#11758200	1	2	yes	—
152	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
153	C₁₆	—	244	2	no	—
154	C₂³	C₂×#11758199	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	488	2	no	—
160	C₂×C₄	—	488	2	no	—
161	C₂×C₄	—	244	4	no	—
162	C₂×C₄	#14519698	1	2	yes	—
163	C₂×C₄	#14519698	1	2	yes	—
164	C₂×C₄	#14519720	1	2	yes	—
165	C₂×C₄	#14519723	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
167	C₂×C₄	#14519753	1	2	yes	—
168	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
169	C₂×C₄	#14519808	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	Q₈×Dic₆₁	1	2	yes	—
172	C₈	#14519766	1	1	yes	characteristic
173	C₈	#14519775	1	1	yes	characteristic
174	C₂²	C₂×#14519698	1	1	yes	characteristic
175	C₂²	#16840039	1	2	yes	—
176	C₂²	C₂×#14519720	1	1	yes	characteristic
177	C₂²	C₂×#14519723	1	1	yes	characteristic
178	C₂²	#16840198	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	8	4	no	—
181	C₄	—	488	4	no	—
182	C₄	#16839566	1	4	yes	—
183	C₄	#16839669	1	1	yes	characteristic
184	C₄	#16839771	1	1	yes	characteristic
185	C₄	#16840044	1	2	yes	—
186	C₂	#21446772	1	1	yes	characteristic
187	C₂	C₂×#16839566	1	1	yes	characteristic
188	C₂	#21447535	1	1	yes	characteristic
189	C₂	#21447963	1	2	yes	—
190	C₂	#21447968	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	#13276
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₆₁×#500	C₄	semidirect
21. #14519711	Q₈	semidirect
103. #500	Dic₆₁	semidirect
111. C₆₁	#13276	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758185	2
C₄×#14519711	2
#21443910	2
C₂×C₆₁×#500	2
#13276	61