Subgroups — #30607908

Order 15616.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462160	C₂	1	2	yes	maximal
2	C₂×#16840684	C₂	1	1	yes	maximal, characteristic
3	#21462025	C₂	1	1	yes	maximal, characteristic
4	#21460870	C₂	1	2	yes	maximal
5	#21445701	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	2	yes	—
7	C₂×#14519851	C₂²	1	1	yes	characteristic
8	C₂×#14519851	C₂²	1	1	yes	characteristic
9	#16840684	—	2	1	no	—
10	#16840684	—	2	1	no	—
11	#16840684	—	2	4	no	—
12	#16840651	—	2	1	no	—
13	#16840648	—	2	1	no	—
14	#16839800	C₄	1	1	yes	characteristic
15	#16839720	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
17	#16839523	C₄	1	2	yes	—
18	C₂×#108×C₆₁	C₄	1	1	yes	characteristic
19	#14519851	—	2	1	no	—
20	#14519851	—	2	1	no	—
21	#14519851	—	2	2	no	—
22	#14519851	—	2	2	no	—
23	#14519851	D₄	1	2	yes	—
24	#14519851	Q₈	1	2	yes	—
25	C₂×#11758187	—	2	1	no	—
26	C₂×#11758187	—	2	1	no	—
27	C₂×#11758187	—	2	2	no	—
28	C₂²×#92418	C₂³	1	1	yes	characteristic
29	#14519736	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	#14519711	—	2	1	no	—
32	#14519709	—	2	1	no	—
33	#14519687	—	2	2	no	—
34	C₈×Dic₆₁	—	2	2	no	—
35	C₈×Dic₆₁	D₄	1	1	yes	characteristic
36	C₈×Dic₆₁	D₄	1	1	yes	characteristic
37	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
38	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
39	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	#108×C₆₁	C₈	1	2	yes	—
42	#108×C₆₁	C₈	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₆₁	—	2	1	no	—
45	M₁₆×C₆₁	—	2	2	no	—
46	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
54	C₂²⋊Dic₆₁	—	2	2	no	—
55	C₄⋊Dic₆₁	—	2	2	no	—
56	C₄×Dic₆₁	—	2	1	no	—
57	C₄×Dic₆₁	—	2	2	no	—
58	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#92418	C₂×D₄	1	1	yes	characteristic
61	C₂×#92418	C₂×D₄	1	1	yes	characteristic
62	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
63	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
64	C₂×#92418	#41	1	1	yes	characteristic
65	C₂×#92418	#41	1	1	yes	characteristic
66	#11758187	—	2	2	no	—
67	#11758187	—	2	2	no	—
68	#11758187	—	4	2	no	—
69	C₂³×C₆₁	#131	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
76	C₂×Dic₆₁	—	2	1	no	—
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	—	4	2	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₈×C₆₁	—	2	4	no	—
81	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
85	#92418	#121	1	1	yes	characteristic
86	#92418	C₂²⋊Q₈	1	1	yes	characteristic
87	#92418	#125	1	1	yes	characteristic
88	#92418	C₄⋊Q₈	1	1	yes	characteristic
89	#13164	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	4	1	no	—
91	C₂²×C₆₁	#424	1	1	yes	characteristic
92	C₂²×C₆₁	#436	1	1	yes	characteristic
93	C₂²×C₆₁	C₈×Q₈	1	1	yes	characteristic
94	C₄×C₆₁	#389	1	1	yes	characteristic
95	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
97	C₄×C₆₁	#446	1	1	yes	characteristic
98	Dic₆₁	—	2	2	no	—
99	Dic₆₁	—	4	1	no	—
100	Dic₆₁	—	8	2	no	—
101	#2153	—	61	2	no	—
102	C₂×#364	—	61	1	no	—
103	C₄×M₃₂	—	61	1	no	—
104	#1772	—	61	1	no	—
105	#1378	—	61	2	no	—
106	C₂×C₆₁	—	2	1	no	—
107	C₂×C₆₁	—	2	1	no	—
108	C₂×C₆₁	—	2	1	no	—
109	C₂×C₆₁	#1939	1	1	yes	characteristic
110	C₂×M₃₂	—	61	1	no	—
111	C₂×M₃₂	—	61	1	no	—
112	C₂×M₃₂	—	61	2	no	—
113	C₂×#108	#6923	1	1	yes	characteristic
114	#427	—	61	1	no	—
115	#405	—	61	2	no	—
116	C₂×C₄×C₈	—	61	1	no	—
117	#364	—	122	1	no	—
118	#364	—	122	1	no	—
119	#364	—	122	4	no	—
120	#346	—	122	1	no	—
121	C₄×C₁₆	—	122	1	no	—
122	#339	—	61	2	no	—
123	C₆₁	#13164	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	C₂×#6923	1	2	yes	—
125	C₂²×C₈	—	61	1	no	—
126	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
127	#117	—	61	2	no	—
128	C₂×C₄²	—	61	1	no	—
129	M₃₂	—	61	2	no	—
130	M₃₂	—	61	2	no	—
131	M₃₂	—	122	1	no	—
132	M₃₂	—	122	1	no	—
133	M₃₂	—	122	2	no	—
134	M₃₂	—	122	2	no	—
135	C₂×C₁₆	—	122	1	no	—
136	C₂×C₁₆	—	122	1	no	—
137	C₂×C₁₆	—	122	2	no	—
138	#108	#92420	1	2	yes	—
139	#108	#92420	1	2	yes	—
140	#107	—	122	1	no	—
141	#106	—	122	1	no	—
142	#97	—	122	2	no	—
143	C₄×C₈	—	61	1	no	—
144	C₄×C₈	—	61	1	no	—
145	C₄×C₈	—	61	1	no	—
146	C₄×C₈	—	61	1	no	—
147	C₄×C₈	—	122	2	no	—
148	C₂²×C₄	—	122	1	no	—
149	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×#92420	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	61	1	no	—
154	C₂×C₈	—	61	1	no	—
155	C₂×C₈	—	61	1	no	—
156	C₂×C₈	—	61	1	no	—
157	C₂×C₈	—	61	1	no	—
158	C₂×C₈	—	61	1	no	—
159	C₂×C₈	#11758213	1	1	yes	characteristic
160	C₂×C₈	#11758213	1	1	yes	characteristic
161	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
162	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
163	C₂×C₈	C₂×#92420	1	1	yes	characteristic
164	C₂×C₈	C₂×#92420	1	1	yes	characteristic
165	C₄⋊C₄	—	122	2	no	—
166	C₂²⋊C₄	—	122	2	no	—
167	C₄²	—	61	1	no	—
168	C₄²	—	61	1	no	—
169	C₄²	—	122	1	no	—
170	C₄²	—	122	2	no	—
171	C₁₆	—	122	2	no	—
172	C₁₆	—	122	2	no	—
173	C₁₆	—	244	2	no	—
174	C₂³	#14519865	1	1	yes	characteristic
175	C₂×C₄	—	122	1	no	—
176	C₂×C₄	—	122	1	no	—
177	C₂×C₄	—	244	2	no	—
178	C₂×C₄	—	244	2	no	—
179	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
180	C₂×C₄	D₄×#6923	1	1	yes	characteristic
181	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
182	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
183	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
184	C₂×C₄	#14519864	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	61	1	no	—
187	C₈	—	61	1	no	—
188	C₈	—	61	1	no	—
189	C₈	—	61	1	no	—
190	C₈	C₄⋊#92420	1	1	yes	characteristic
191	C₈	C₄⋊#92420	1	1	yes	characteristic
192	C₈	C₄⋊#92420	1	1	yes	characteristic
193	C₈	C₄⋊#92420	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#16840769	1	1	yes	characteristic
196	C₂²	#16840796	1	1	yes	characteristic
197	C₂²	Q₈×#92420	1	1	yes	characteristic
198	C₄	—	122	2	no	—
199	C₄	—	488	2	no	—
200	C₄	—	244	1	no	—
201	C₄	#16840713	1	1	yes	characteristic
202	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
203	C₄	D₄×#92420	1	1	yes	characteristic
204	C₄	#16840812	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#21461916	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16840713
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#108×C₆₁	C₄
socle	C₂×C₆₁	#1939
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#108	#6923
hypocenter	C₆₁	#13164
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
123. C₆₁	#13164	semidirect

Maximal subgroups

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

subgroup	index
#21462160	2
C₂×#16840684	2
#21462025	2
#21460870	2
#21445701	2
#13164	61