Subgroups — #29688899

Order 15616.

662 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	#21456343	C₂	1	1	yes	maximal, characteristic
2	#21454753	C₂	1	2	yes	maximal
3	#21446800	C₂	1	2	yes	maximal
4	#21446567	C₂	1	4	yes	maximal
5	#21445994	C₂	1	1	yes	maximal, characteristic
6	#21445703	C₂	1	4	yes	maximal
7	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
8	#16840319	C₂²	1	2	yes	—
9	#16840171	C₄	1	8	yes	complemented
10	#16839892	—	2	4	no	—
11	C₈×Dic₁₂₂	—	2	4	no	—
12	C₂×#14519713	C₂²	1	4	yes	—
13	#16839800	C₂²	1	4	yes	—
14	#16839788	C₂²	1	4	yes	—
15	#16839786	C₂²	1	2	yes	—
16	#16839717	C₂²	1	4	yes	—
17	#16839522	C₂²	1	8	yes	—
18	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#97×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#376	C₂²	1	1	yes	characteristic
23	C₂×#11758194	C₂³	1	4	yes	—
24	#14519777	C₂×C₄	1	4	yes	—
25	#14519769	C₂×C₄	1	4	yes	—
26	#14519756	—	2	4	no	—
27	C₄×Dic₁₂₂	—	2	4	no	—
28	#14519736	C₂³	1	4	yes	—
29	#14519713	—	2	8	no	—
30	#14519711	C₂×C₄	1	8	yes	—
31	#14519709	C₂×C₄	1	8	yes	—
32	C₄⋊#92418	—	2	4	no	—
33	#14519706	—	2	8	no	—
34	#14519687	—	2	4	no	—
35	C₈×Dic₆₁	—	2	4	no	—
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#97×C₆₁	C₂×C₄	1	4	yes	—
42	#97×C₆₁	Q₈	1	8	yes	complemented
43	C₄×C₈×C₆₁	—	2	4	no	—
44	C₄×C₈×C₆₁	D₄	1	4	yes	—
45	#95×C₆₁	—	2	2	no	—
46	C₂²×C₄×C₆₁	—	2	2	no	—
47	C₂²×C₄×C₆₁	#41	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₂₂	—	2	4	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
56	C₂×C₈×C₆₁	#41	1	4	yes	—
57	C₄²×C₆₁	—	2	2	no	—
58	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
59	C₄²×C₆₁	C₂×D₄	1	2	yes	—
60	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
61	C₂²⋊Dic₆₁	—	2	4	no	—
62	C₄⋊Dic₆₁	—	2	4	no	—
63	C₄⋊Dic₆₁	C₂²×C₄	1	8	yes	—
64	#11758196	—	2	8	no	—
65	C₄×Dic₆₁	—	2	4	no	—
66	#11758194	—	2	8	no	—
67	C₂×#92418	—	2	4	no	—
68	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	4	no	—
72	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
77	C₂×Dic₆₁	—	2	8	no	—
78	Dic₁₂₂	—	4	8	no	—
79	C₈×C₆₁	—	2	2	no	—
80	C₈×C₆₁	C₄×Q₈	1	8	yes	—
81	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
82	#92418	—	4	4	no	—
83	#21737	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
86	C₂²×C₆₁	#533	1	2	yes	—
87	C₄×C₆₁	—	2	4	no	—
88	C₄×C₆₁	—	4	4	no	—
89	C₄×C₆₁	C₂×C₄×Q₈	1	2	yes	—
90	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₄×C₆₁	#532	1	1	yes	characteristic
92	Dic₆₁	—	4	8	no	—
93	#3162	—	61	1	no	—
94	#2960	—	61	2	no	—
95	#1995	—	61	2	no	—
96	#1945	—	61	4	no	—
97	#1849	—	61	1	no	—
98	#1773	—	61	4	no	—
99	#1726	D₆₁	1	1	yes	characteristic
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₆₁	#2955	1	2	yes	—
103	#533	—	61	2	no	—
104	#501	—	61	8	no	—
105	#446	—	122	4	no	—
106	C₈×Q₈	—	122	4	no	—
107	C₂×#108	—	61	4	no	—
108	#427	—	61	4	no	—
109	#424	—	61	4	no	—
110	#423	—	61	2	no	—
111	#405	—	61	4	no	—
112	C₂×#97	C₂×D₆₁	1	2	yes	—
113	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
116	#376	C₂×D₆₁	1	1	yes	characteristic
117	#339	—	61	8	no	—
118	C₆₁	#21737	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	61	4	no	—
120	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
123	C₄⋊Q₈	—	61	4	no	—
124	#125	—	61	4	no	—
125	C₂²⋊Q₈	—	122	4	no	—
126	C₄×Q₈	—	122	4	no	—
127	#117	—	61	4	no	—
128	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
129	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
130	#108	—	122	8	no	—
131	#107	—	61	8	no	—
132	#106	—	61	8	no	—
133	C₄⋊C₈	—	122	8	no	—
134	C₄⋊C₈	—	122	4	no	—
135	#97	—	122	4	no	—
136	#97	Dic₁₂₂	1	8	yes	complemented
137	#97	C₄×D₆₁	1	4	yes	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	—	122	4	no	—
140	C₄×C₈	D₂₄₄	1	4	yes	—
141	#95	—	2	2	no	—
142	C₂×Q₈	—	122	4	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	#11758222	1	2	yes	—
145	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
146	M₁₆	—	122	8	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	122	4	no	—
149	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
150	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
151	C₂×C₈	C₂×C₄×D₆₁	1	4	yes	—
152	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
153	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
154	C₂×C₈	#11758222	1	4	yes	—
155	C₄⋊C₄	—	61	8	no	—
156	C₄⋊C₄	—	122	8	no	—
157	C₄⋊C₄	—	122	4	no	—
158	C₂²⋊C₄	—	122	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	122	4	no	—
161	C₄²	C₂×Dic₁₂₂	1	4	yes	—
162	C₄²	C₂×C₄×D₆₁	1	2	yes	—
163	C₄²	C₂×D₂₄₄	1	2	yes	—
164	C₂³	C₂×#11758222	1	1	yes	characteristic
165	Q₈	—	244	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	122	8	no	—
170	C₂×C₄	C₄×D₂₄₄	1	4	yes	—
171	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
173	C₂×C₄	C₂²×Dic₁₂₂	1	2	yes	—
174	C₂×C₄	C₂×#11758222	1	2	yes	—
175	C₈	—	2	2	no	—
176	C₈	—	244	4	no	—
177	C₈	C₄×Dic₁₂₂	1	8	yes	—
178	C₈	#14519777	1	4	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
181	C₂²	#16840319	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	4	4	no	—
184	C₄	—	244	8	no	—
185	C₄	C₂×C₄×Dic₁₂₂	1	2	yes	—
186	C₄	C₂×#14519777	1	1	yes	characteristic
187	C₄	#16840312	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₄×#14519777	1	1	yes	characteristic
190	C₂	#21454723	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1726	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₂×#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1726	D₆₁
hypocenter	C₆₁	#21737
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
9. #16840171	C₄	semidirect
42. #97×C₆₁	Q₈	semidirect
118. C₆₁	#21737	semidirect
136. #97	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456343	2
#21454753	2
#21446800	2
#21446567	2
#21445994	2
#21445703	2
C₆₁×#1726	2
#21737	61