Subgroups — #29598585

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 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×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal
4	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519777	—	2	1	no	—
6	C₂²×#11758185	C₂²	1	2	yes	—
7	#16840167	—	2	2	no	—
8	#16840167	C₄	1	4	yes	complemented
9	C₂×#14519711	—	2	1	no	—
10	C₂×#14519711	C₂²	1	2	yes	—
11	C₄⋊#11758185	—	2	2	no	—
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
15	C₆₁×#340	—	2	1	no	—
16	C₂²×Dic₁₂₂	—	4	1	no	—
17	#14519777	—	2	4	no	—
18	#14519777	—	4	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	4	1	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂×#11758185	—	2	4	no	—
23	#14519711	—	2	4	no	—
24	#14519711	—	2	2	no	—
25	#14519711	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄²×C₆₁	—	4	1	no	—
30	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
31	#107×C₆₁	—	2	2	no	—
32	#107×C₆₁	Q₈	1	4	yes	complemented
33	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	#95×C₆₁	—	4	1	no	—
36	C₂²×C₄×C₆₁	—	4	2	no	—
37	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	4	2	no	—
41	C₂×Dic₁₂₂	—	4	4	no	—
42	C₂×Dic₁₂₂	—	8	2	no	—
43	C₂×Dic₁₂₂	—	8	2	no	—
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₆₁	#41	1	2	yes	—
50	C₄⋊C₄×C₆₁	—	2	4	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄²×C₆₁	—	8	2	no	—
53	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₆₁	SD₁₆	1	2	yes	—
55	C₄⋊Dic₆₁	—	2	4	no	—
56	C₄⋊Dic₆₁	—	4	4	no	—
57	C₄⋊Dic₆₁	—	4	2	no	—
58	#11758185	—	4	2	no	—
59	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	4	4	no	—
62	C₂×C₄×C₆₁	—	8	2	no	—
63	C₂×C₄×C₆₁	—	8	2	no	—
64	C₂×C₄×C₆₁	#102	1	2	yes	—
65	C₂×C₄×C₆₁	#102	1	2	yes	—
66	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
67	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
68	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#137	1	2	yes	—
72	C₂×Dic₆₁	—	4	4	no	—
73	C₂×Dic₆₁	—	8	2	no	—
74	C₂×Dic₆₁	—	8	2	no	—
75	Dic₁₂₂	—	8	8	no	—
76	C₈×C₆₁	—	2	2	no	—
77	C₈×C₆₁	C₄×Q₈	1	2	yes	—
78	C₈×C₆₁	#124	1	1	yes	characteristic
79	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
80	#13275	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
82	C₂²×C₆₁	#474	1	2	yes	—
83	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
84	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
85	C₂²×C₆₁	#510	1	2	yes	—
86	C₄×C₆₁	—	2	2	no	—
87	C₄×C₆₁	—	8	4	no	—
88	C₄×C₆₁	#357	1	4	yes	—
89	C₄×C₆₁	#391	1	1	yes	characteristic
90	C₄×C₆₁	#419	1	1	yes	characteristic
91	C₄×C₆₁	#475	1	2	yes	—
92	Dic₆₁	—	8	4	no	—
93	C₂×#500	—	61	1	no	—
94	C₂×C₄⋊C₁₆	—	61	1	no	—
95	C₄×#107	D₆₁	1	1	yes	characteristic
96	#1380	—	61	4	no	—
97	C₂×C₆₁	#1988	1	1	yes	characteristic
98	C₂×C₆₁	C₂×#357	1	1	yes	characteristic
99	C₂×C₆₁	#2141	1	1	yes	characteristic
100	C₂×C₆₁	#2224	1	2	yes	—
101	C₂×C₆₁	#2225	1	2	yes	—
102	C₂×C₄⋊Q₈	—	122	1	no	—
103	C₂²×C₁₆	—	61	2	no	—
104	#500	—	61	4	no	—
105	#500	—	122	2	no	—
106	C₂×#107	—	61	2	no	—
107	C₂×#107	—	122	1	no	—
108	C₂×#107	C₂×D₆₁	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₁₆	—	122	2	no	—
112	#340	—	2	1	no	—
113	C₆₁	#13275	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	244	1	no	—
115	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	C₄⋊Q₈	—	122	4	no	—
118	C₄⋊Q₈	—	244	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	122	2	no	—
121	C₂×C₄⋊C₄	—	244	1	no	—
122	C₂×C₄²	—	4	1	no	—
123	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
124	C₂×C₁₆	—	122	2	no	—
125	C₂×C₁₆	—	122	4	no	—
126	#107	—	2	2	no	—
127	#107	—	61	4	no	—
128	#107	—	122	4	no	—
129	#107	—	122	2	no	—
130	#107	Dic₁₂₂	1	4	yes	complemented
131	C₄×C₈	C₄×D₆₁	1	2	yes	—
132	C₄×C₈	D₂₄₄	1	2	yes	—
133	#95	—	4	1	no	—
134	C₂×Q₈	—	488	2	no	—
135	C₂×Q₈	—	488	2	no	—
136	C₂×Q₈	—	244	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	244	2	no	—
139	C₂²×C₄	#11758199	1	1	yes	characteristic
140	C₂²×C₄	#11758199	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	#11758198	1	2	yes	—
144	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
146	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
147	C₂×C₈	#11758222	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	122	4	no	—
151	C₄⋊C₄	—	244	4	no	—
152	C₄⋊C₄	—	244	2	no	—
153	C₄²	—	8	2	no	—
154	C₄²	#11758198	1	2	yes	—
155	C₄²	#11758201	1	2	yes	—
156	C₁₆	—	244	2	no	—
157	C₂³	C₂×#11758199	1	1	yes	characteristic
158	Q₈	—	488	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	488	2	no	—
164	C₂×C₄	—	488	2	no	—
165	C₂×C₄	—	244	4	no	—
166	C₂×C₄	#14519697	1	2	yes	—
167	C₂×C₄	#14519697	1	2	yes	—
168	C₂×C₄	#14519718	1	2	yes	—
169	C₂×C₄	#14519720	1	2	yes	—
170	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
171	C₂×C₄	#14519756	1	2	yes	—
172	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
173	C₂×C₄	#14519807	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	C₄×Dic₁₂₂	1	2	yes	—
176	C₈	#14519765	1	1	yes	characteristic
177	C₈	#14519777	1	1	yes	characteristic
178	C₂²	C₂×#14519697	1	1	yes	characteristic
179	C₂²	#16840042	1	2	yes	—
180	C₂²	C₂×#14519718	1	1	yes	characteristic
181	C₂²	C₂×#14519720	1	1	yes	characteristic
182	C₂²	#16840202	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	488	4	no	—
186	C₄	#16839562	1	4	yes	—
187	C₄	#16839668	1	1	yes	characteristic
188	C₄	#16839770	1	1	yes	characteristic
189	C₄	#16840047	1	2	yes	—
190	C₂	#21446771	1	1	yes	characteristic
191	C₂	C₂×#16839562	1	1	yes	characteristic
192	C₂	#21447538	1	1	yes	characteristic
193	C₂	#21447956	1	2	yes	—
194	C₂	#21447961	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×#107×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×#107	D₆₁
hypocenter	C₆₁	#13275
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
8. #16840167	C₄	semidirect
32. #107×C₆₁	Q₈	semidirect
113. C₆₁	#13275	semidirect
130. #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×C₄⋊#11758185	2
#21443910	2
C₄×#107×C₆₁	2
#13275	61