Subgroups — #21445529

Order 7808.

404 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	C₂²×#11758194	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519736	C₂	1	1	yes	maximal, characteristic
3	#16839522	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	—	2	2	no	—
6	C₂×#11758194	—	2	1	no	—
7	C₂×#11758194	C₄	1	4	yes	complemented
8	C₂×#11758194	C₂²	1	2	yes	—
9	C₂²×#92418	—	2	1	no	—
10	#14519736	—	2	1	no	—
11	#14519736	—	2	1	no	—
12	#14519736	C₄	1	2	yes	complemented
13	#14519736	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₆₁	C₄	1	2	yes	complemented
15	C₂×C₂²⋊Dic₆₁	—	2	1	no	—
16	C₂×C₄×Dic₆₁	—	2	1	no	—
17	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	#117×C₆₁	—	2	2	no	—
19	#117×C₆₁	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
22	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
23	C₂⁴×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	—	2	2	no	—
25	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	D₄	1	2	yes	—
33	C₂²×C₄×C₆₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₆₁	—	2	2	no	—
36	C₄⋊C₄×C₆₁	—	2	4	no	—
37	C₂²⋊C₄×C₆₁	—	2	4	no	—
38	C₄²×C₆₁	—	2	2	no	—
39	C₄²×C₆₁	C₂×C₄	1	4	yes	complemented
40	C₂²⋊Dic₆₁	—	2	2	no	—
41	C₂²⋊Dic₆₁	—	2	2	no	—
42	C₄⋊Dic₆₁	—	2	2	no	—
43	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
44	C₄×Dic₆₁	—	2	2	no	—
45	C₄×Dic₆₁	—	2	2	no	—
46	#11758194	—	2	4	no	—
47	#11758194	—	2	2	no	—
48	#11758194	C₂×C₄	1	4	yes	complemented
49	C₂×#92418	—	2	4	no	—
50	C₂×#92418	—	2	2	no	—
51	C₂³×C₆₁	—	4	2	no	—
52	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	—	2	4	no	—
58	C₂×C₄×C₆₁	—	4	4	no	—
59	C₂×C₄×C₆₁	C₄²	1	2	yes	—
60	C₂×C₄×C₆₁	C₄²	1	2	yes	—
61	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₆₁	—	2	4	no	—
78	C₂×Dic₆₁	—	4	4	no	—
79	#92418	—	2	4	no	—
80	C₂²×C₆₁	—	2	2	no	—
81	C₂²×C₆₁	—	2	2	no	—
82	C₂²×C₆₁	—	2	2	no	—
83	C₂²×C₆₁	—	4	2	no	—
84	C₂²×C₆₁	—	4	2	no	—
85	C₂²×C₆₁	#95	1	2	yes	—
86	C₂²×C₆₁	#95	1	2	yes	—
87	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₆₁	—	4	4	no	—
91	C₄×C₆₁	#95	1	2	yes	—
92	C₄×C₆₁	#95	1	2	yes	—
93	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₆₁	—	4	4	no	—
98	#1733	—	61	1	no	maximal, Sylow
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	—	2	2	no	—
102	C₂×C₆₁	—	2	1	no	—
103	C₂×C₆₁	C₂×#95	1	1	yes	characteristic
104	C₂×C₆₁	#421	1	2	yes	—
105	C₂²×M₁₆	—	61	1	no	—
106	C₂×#117	—	61	1	no	—
107	C₂×#117	D₆₁	1	1	yes	characteristic
108	#339	—	61	4	no	—
109	C₆₁	#1733	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
111	C₂×M₁₆	—	61	4	no	—
112	C₂×M₁₆	—	61	2	no	—
113	C₂×M₁₆	—	122	1	no	—
114	C₂×M₁₆	—	122	2	no	—
115	C₂²×C₈	—	122	1	no	—
116	#117	—	2	2	no	—
117	#117	—	61	2	no	—
118	#117	—	61	2	no	—
119	#117	—	122	1	no	—
120	#117	—	122	1	no	—
121	#117	Dic₆₁	1	4	yes	complemented
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	61	2	no	—
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₂²⋊C₄	—	122	1	no	—
126	C₂×C₄²	—	122	1	no	—
127	C₂×C₄²	C₂×D₆₁	1	2	yes	—
128	C₂⁴	C₂²⋊D₆₁	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	122	2	no	—
131	C₂²×C₄	Dic₁₂₂	1	2	yes	—
132	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₆₁	1	2	yes	—
135	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
136	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
138	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
140	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
141	M₁₆	—	61	4	no	—
142	M₁₆	—	122	4	no	—
143	M₁₆	—	122	2	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	—	122	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	61	4	no	—
148	C₄⋊C₄	—	122	2	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	122	2	no	—
151	C₂²⋊C₄	—	122	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	122	2	no	—
154	C₄²	—	122	2	no	—
155	C₄²	C₂×Dic₆₁	1	4	yes	complemented
156	C₂³	—	4	2	no	—
157	C₂³	#11758196	1	2	yes	—
158	C₂³	#11758198	1	1	yes	characteristic
159	C₂³	#11758198	1	1	yes	characteristic
160	C₂³	C₂²⋊Dic₆₁	1	2	yes	—
161	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	122	4	no	—
165	C₂×C₄	—	244	4	no	—
166	C₂×C₄	C₄×Dic₆₁	1	2	yes	—
167	C₂×C₄	C₄×Dic₆₁	1	2	yes	—
168	C₂×C₄	#11758196	1	2	yes	—
169	C₂×C₄	#11758196	1	2	yes	—
170	C₂×C₄	#11758196	1	2	yes	—
171	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
172	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
173	C₂×C₄	#11758198	1	1	yes	characteristic
174	C₂×C₄	#11758198	1	1	yes	characteristic
175	C₂×C₄	#11758198	1	2	yes	—
176	C₂×C₄	#11758198	1	2	yes	—
177	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
178	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₈	—	122	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	#14519684	1	2	yes	—
191	C₂²	#14519684	1	2	yes	—
192	C₂²	C₂×#11758198	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
194	C₂²	C₂×#11758196	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	244	4	no	—
197	C₄	#14519684	1	2	yes	—
198	C₄	#14519684	1	2	yes	—
199	C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
200	C₄	C₂×#11758198	1	1	yes	characteristic
201	C₄	C₂×#11758196	1	1	yes	characteristic
202	C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	2	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	C₂×#14519684	1	1	yes	characteristic
208	C₂	#16839780	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#117×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₆₁
hypocenter	C₆₁	#1733
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #16839522	C₂	semidirect
7. C₂×#11758194	C₄	semidirect
12. #14519736	C₄	semidirect
14. C₂×C₄⋊Dic₆₁	C₄	semidirect
19. #117×C₆₁	C₄	semidirect
39. C₄²×C₆₁	C₂×C₄	semidirect
48. #11758194	C₂×C₄	semidirect
109. C₆₁	#1733	semidirect
121. #117	Dic₆₁	semidirect
155. C₄²	C₂×Dic₆₁	semidirect