Subgroups — #21454986

Order 7808.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840296	C₂	1	1	yes	maximal, characteristic
2	#16839891	C₂	1	2	yes	maximal, complemented
3	#16839838	C₂	1	2	yes	maximal, complemented
4	#16839832	C₂	1	2	yes	maximal, complemented
5	#16839831	C₂	1	2	yes	maximal, complemented
6	#16839822	C₂	1	2	yes	maximal, complemented
7	#16839819	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839787	C₂	1	2	yes	maximal, complemented
9	C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758189	C₂²	1	2	yes	—
11	C₂×C₈×D₆₁	C₂²	1	2	yes	—
12	#14519771	C₄	1	2	yes	—
13	#14519769	C₄	1	1	yes	characteristic
14	#14519765	C₄	1	1	yes	characteristic
15	#14519760	C₄	1	2	yes	—
16	#14519756	C₄	1	1	yes	characteristic
17	#14519751	C₄	1	1	yes	characteristic
18	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
19	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₆₁	C₂²	1	2	yes	—
21	#14519734	C₂²	1	2	yes	—
22	C₄⋊#92418	C₂²	1	1	yes	characteristic
23	C₄⋊#92418	C₂²	1	1	yes	characteristic
24	#14519706	C₂²	1	2	yes	—
25	#14519706	C₂²	1	2	yes	—
26	C₂²⋊#92418	C₂²	1	2	yes	—
27	#14519689	C₂²	1	2	yes	—
28	#14519689	C₂²	1	2	yes	—
29	#14519688	C₂²	1	2	yes	—
30	#14519687	C₂²	1	2	yes	—
31	C₈×Dic₆₁	C₂²	1	2	yes	—
32	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
36	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₆₁	C₂×C₄	1	2	yes	—
41	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₆₁	C₂³	1	2	yes	—
43	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
44	M₁₆×C₆₁	—	4	1	no	—
45	M₁₆×C₆₁	—	4	1	no	—
46	C₂×C₈×C₆₁	C₂³	1	2	yes	—
47	C₂×C₈×C₆₁	C₂³	1	2	yes	—
48	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	C₄²×C₆₁	C₂³	1	1	yes	characteristic
52	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
53	#11758198	C₂×C₄	1	2	yes	—
54	#11758198	C₂×C₄	1	2	yes	—
55	#11758198	C₂×C₄	1	2	yes	—
56	#11758198	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
59	#11758196	C₂×C₄	1	2	yes	—
60	#11758196	C₂×C₄	1	2	yes	—
61	#11758196	C₂×C₄	1	2	yes	—
62	#11758196	C₂×C₄	1	2	yes	—
63	C₄×Dic₆₁	C₂³	1	2	yes	—
64	C₂×#92418	C₂³	1	2	yes	—
65	C₂×#92418	C₂³	1	2	yes	—
66	#11758189	—	4	2	no	—
67	C₈×D₆₁	—	4	2	no	—
68	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₆₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	4	1	no	—
73	C₂×C₄×C₆₁	—	4	1	no	—
74	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₆₁	—	4	2	no	—
81	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
84	D₂₄₄	—	4	1	no	—
85	C₄×D₆₁	—	2	2	no	—
86	C₄×D₆₁	—	2	2	no	—
87	Dic₁₂₂	—	4	1	no	—
88	C₈×C₆₁	—	2	2	no	—
89	C₈×C₆₁	—	2	2	no	—
90	#92418	—	2	2	no	—
91	#92418	—	2	2	no	—
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	—	2	1	no	—
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₆₁	—	2	2	no	—
97	C₂×D₆₁	—	2	2	no	—
98	C₂×D₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	2	1	no	—
104	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₆₁	—	2	2	no	—
107	Dic₆₁	—	2	2	no	—
108	Dic₆₁	—	2	2	no	—
109	#2995	—	61	1	no	maximal, Sylow
110	C₂×C₆₁	—	4	1	no	—
111	C₂×C₆₁	#520	1	1	yes	characteristic
112	C₂×C₆₁	#568	1	1	yes	characteristic
113	C₂×C₆₁	#568	1	1	yes	characteristic
114	D₆₁	—	4	2	no	—
115	#529	—	61	1	no	—
116	#446	—	61	2	no	—
117	#436	—	61	2	no	—
118	#435	—	61	2	no	—
119	#435	—	61	2	no	—
120	#433	—	61	2	no	—
121	#432	—	61	1	no	—
122	#424	—	61	2	no	—
123	#423	D₆₁	1	1	yes	characteristic, complemented
124	C₆₁	#2995	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₁₆	—	61	2	no	—
126	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
127	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
128	C₂²×C₈	—	61	2	no	—
129	#126	—	61	2	no	—
130	#125	—	61	1	no	—
131	#124	—	61	1	no	—
132	#123	—	61	2	no	—
133	C₂²⋊Q₈	—	61	1	no	—
134	#121	—	61	1	no	—
135	C₄×Q₈	—	61	1	no	—
136	C₄×D₄	—	61	2	no	—
137	C₄×D₄	—	61	1	no	—
138	#117	—	61	2	no	—
139	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
140	C₄⋊C₈	—	61	2	no	—
141	C₄⋊C₈	—	61	2	no	—
142	C₄⋊C₈	—	61	1	no	—
143	C₄⋊C₈	—	61	1	no	—
144	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
145	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
146	C₂²⋊C₈	—	61	2	no	—
147	C₂²⋊C₈	—	61	2	no	—
148	C₂²⋊C₈	—	61	2	no	—
149	#97	—	61	2	no	—
150	#97	—	61	2	no	—
151	C₄×C₈	—	61	2	no	—
152	C₂×Q₈	—	61	1	no	—
153	C₂×D₄	—	61	2	no	—
154	C₂×D₄	—	61	1	no	—
155	C₂²×C₄	—	61	2	no	—
156	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	244	2	no	—
162	C₂×C₈	—	61	2	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	—	244	2	no	—
165	C₂×C₈	C₂²×D₆₁	1	2	yes	—
166	C₂×C₈	C₂²×D₆₁	1	2	yes	—
167	C₄⋊C₄	—	61	2	no	—
168	C₄⋊C₄	—	61	2	no	—
169	C₄⋊C₄	—	61	2	no	—
170	C₄⋊C₄	—	61	2	no	—
171	C₄⋊C₄	—	61	1	no	—
172	C₄⋊C₄	—	61	1	no	—
173	C₂²⋊C₄	—	61	2	no	—
174	C₂²⋊C₄	—	61	2	no	—
175	C₂²⋊C₄	—	61	2	no	—
176	C₂²⋊C₄	—	61	2	no	—
177	C₂²⋊C₄	—	61	2	no	—
178	C₄²	—	61	2	no	—
179	C₄²	C₄×D₆₁	1	1	yes	characteristic
180	C₄²	C₄×D₆₁	1	1	yes	characteristic
181	C₄²	C₂²×D₆₁	1	1	yes	characteristic
182	C₄²	C₂²×D₆₁	1	1	yes	characteristic
183	C₂³	—	61	2	no	—
184	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
185	Q₈	—	244	1	no	—
186	D₄	—	244	1	no	—
187	D₄	—	244	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	61	2	no	—
193	C₂×C₄	—	61	2	no	—
194	C₂×C₄	—	61	2	no	—
195	C₂×C₄	—	122	2	no	—
196	C₂×C₄	—	122	2	no	—
197	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	122	2	no	—
206	C₈	—	122	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	122	2	no	—
211	C₂²	—	122	2	no	—
212	C₂²	—	122	2	no	—
213	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	122	2	no	—
220	C₄	—	122	2	no	—
221	C₄	—	122	2	no	—
222	C₄	#14519822	1	1	yes	characteristic
223	C₄	#14519825	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	244	2	no	—
226	C₂	#16840243	1	1	yes	characteristic
227	C₂	#16840528	1	1	yes	characteristic
228	C₂	#16840528	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#423	C₂
hypercenter	#423	D₆₁
hypocenter	C₆₁	#2995
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #16839891	C₂	semidirect
3. #16839838	C₂	semidirect
4. #16839832	C₂	semidirect
5. #16839831	C₂	semidirect
6. #16839822	C₂	semidirect
7. #16839819	C₂	semidirect
8. #16839787	C₂	semidirect
9. C₆₁×#423	C₂	semidirect
123. #423	D₆₁	semidirect
124. C₆₁	#2995	semidirect