Subgroups — #21454917

Order 7808.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840316	C₂	1	1	yes	maximal, characteristic
2	#16839890	C₂	1	2	yes	maximal, complemented
3	#16839889	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839886	C₂	1	2	yes	maximal, complemented
7	#16839824	C₂	1	2	yes	maximal
8	#16839822	C₂	1	2	yes	maximal
9	#16839814	C₂	1	1	yes	maximal, characteristic
10	#16839813	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	#14519776	C₄	1	1	yes	characteristic
13	#14519775	C₄	1	1	yes	characteristic
14	#14519767	C₄	1	2	yes	—
15	#14519757	C₄	1	1	yes	characteristic
16	#14519755	C₄	1	2	yes	—
17	#14519753	C₄	1	1	yes	characteristic
18	C₄×Dic₁₂₂	C₂²	1	2	yes	—
19	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
20	#14519734	C₂²	1	2	yes	—
21	C₄²×D₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊#92418	C₂²	1	1	yes	characteristic
23	C₄⋊#92418	C₂²	1	2	yes	—
24	#14519706	C₂²	1	1	yes	characteristic
25	#14519706	C₂²	1	1	yes	characteristic
26	#14519706	C₂²	1	2	yes	—
27	#14519706	C₂²	1	2	yes	—
28	#14519689	C₂²	1	1	yes	characteristic
29	#14519689	C₂²	1	1	yes	characteristic
30	#14519689	C₂²	1	2	yes	—
31	#14519687	C₂²	1	2	yes	—
32	C₄×#92418	C₂²	1	1	yes	characteristic
33	C₄×#92418	C₂²	1	2	yes	—
34	C₈×Dic₆₁	C₂²	1	2	yes	—
35	C₈×Dic₆₁	C₂²	1	2	yes	—
36	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
39	#97×C₆₁	C₂²	1	2	yes	—
40	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
46	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂³	1	2	yes	—
50	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	C₄²×C₆₁	C₂³	1	2	yes	—
54	#11758198	C₂×C₄	1	2	yes	—
55	#11758198	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
58	#11758196	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
66	C₄×Dic₆₁	C₂³	1	2	yes	—
67	C₂×#92418	C₂³	1	2	yes	—
68	C₂×#92418	C₂³	1	1	yes	characteristic
69	C₂×#92418	C₂³	1	1	yes	characteristic
70	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
80	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₆₁	—	2	2	no	—
83	C₄×D₆₁	—	2	1	no	—
84	C₄×D₆₁	—	2	1	no	—
85	C₄×D₆₁	—	2	2	no	—
86	Dic₁₂₂	—	2	2	no	—
87	Dic₁₂₂	—	2	4	no	—
88	C₈×C₆₁	—	2	1	no	—
89	C₈×C₆₁	—	2	2	no	—
90	C₈×C₆₁	—	2	1	no	—
91	#92418	—	2	1	no	—
92	#92418	—	2	2	no	—
93	#92418	—	2	1	no	—
94	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₆₁	—	2	1	no	—
96	C₂×D₆₁	—	2	1	no	—
97	C₂×D₆₁	—	2	1	no	—
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	2	no	—
101	C₄×C₆₁	#131	1	2	yes	—
102	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₆₁	—	2	1	no	—
105	Dic₆₁	—	2	2	no	—
106	Dic₆₁	—	2	2	no	—
107	Dic₆₁	—	2	1	no	—
108	Dic₆₁	#131	1	2	yes	—
109	#2986	—	61	1	no	maximal, Sylow
110	C₂×C₆₁	#519	1	1	yes	characteristic
111	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
112	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
113	D₆₁	—	4	1	no	—
114	#533	—	61	1	no	—
115	#446	—	61	1	no	—
116	#446	—	61	2	no	—
117	#446	D₆₁	1	1	yes	characteristic, complemented
118	C₈×Q₈	—	61	1	no	—
119	C₈×Q₈	—	61	2	no	—
120	C₈×Q₈	—	61	1	no	—
121	#433	—	61	2	no	—
122	#433	—	61	2	no	—
123	#431	—	61	1	no	—
124	#431	—	61	1	no	—
125	C₆₁	#2986	1	1	yes	characteristic, complemented, Sylow
126	C₄⋊Q₈	—	61	1	no	—
127	C₄⋊Q₈	—	61	1	no	—
128	#125	—	61	2	no	—
129	C₂²⋊Q₈	—	61	1	no	—
130	C₂²⋊Q₈	—	61	2	no	—
131	C₂²⋊Q₈	—	61	1	no	—
132	C₄×Q₈	—	61	2	no	—
133	C₄×Q₈	—	61	1	no	—
134	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
135	#117	—	61	2	no	—
136	C₂×C₄²	—	61	1	no	—
137	C₄⋊C₈	—	61	1	no	—
138	C₄⋊C₈	—	61	1	no	—
139	C₄⋊C₈	—	61	2	no	—
140	C₄⋊C₈	—	61	2	no	—
141	C₄⋊C₈	—	61	1	no	—
142	C₄⋊C₈	—	61	2	no	—
143	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
145	C₂²⋊C₈	—	61	2	no	—
146	C₂²⋊C₈	—	61	1	no	—
147	C₂²⋊C₈	—	61	1	no	—
148	#97	—	61	2	no	—
149	#97	C₂×D₆₁	1	2	yes	—
150	C₄×C₈	—	61	2	no	—
151	C₄×C₈	—	61	2	no	—
152	C₄×C₈	—	61	1	no	—
153	C₄×C₈	—	61	2	no	—
154	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
155	C₂×Q₈	—	61	2	no	—
156	C₂×Q₈	—	61	1	no	—
157	C₂×Q₈	C₄×D₆₁	1	1	yes	characteristic
158	C₂²×C₄	—	61	1	no	—
159	C₂²×C₄	—	61	1	no	—
160	C₂²×C₄	—	61	1	no	—
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	61	2	no	—
163	C₂×C₈	—	61	1	no	—
164	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₆₁	1	2	yes	—
166	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
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	2	no	—
172	C₄⋊C₄	—	61	2	no	—
173	C₄⋊C₄	—	61	1	no	—
174	C₄⋊C₄	C₄×D₆₁	1	2	yes	—
175	C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
176	C₂²⋊C₄	—	61	2	no	—
177	C₂²⋊C₄	—	61	2	no	—
178	C₄²	—	61	1	no	—
179	C₄²	—	61	1	no	—
180	C₄²	—	61	1	no	—
181	C₄²	—	61	2	no	—
182	C₄²	C₂²×D₆₁	1	1	yes	characteristic
183	C₄²	C₂²×D₆₁	1	2	yes	—
184	C₂³	—	61	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	122	4	no	—
187	Q₈	—	122	2	no	—
188	C₂×C₄	—	61	1	no	—
189	C₂×C₄	—	61	1	no	—
190	C₂×C₄	—	61	1	no	—
191	C₂×C₄	—	61	2	no	—
192	C₂×C₄	—	61	2	no	—
193	C₂×C₄	—	122	2	no	—
194	C₂×C₄	—	122	1	no	—
195	C₂×C₄	—	122	1	no	—
196	C₂×C₄	—	122	2	no	—
197	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	122	1	no	—
206	C₈	—	122	2	no	—
207	C₈	—	122	1	no	—
208	C₂²	—	122	1	no	—
209	C₂²	—	122	1	no	—
210	C₂²	—	122	1	no	—
211	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	61	2	no	—
216	C₄	—	122	1	no	—
217	C₄	—	122	2	no	—
218	C₄	—	122	1	no	—
219	C₄	—	122	2	no	—
220	C₄	#14519785	1	2	yes	—
221	C₄	#14519824	1	1	yes	characteristic
222	C₄	#14519825	1	1	yes	characteristic
223	C₂	—	244	1	no	—
224	C₂	#16840238	1	1	yes	characteristic
225	C₂	#131×D₆₁	1	1	yes	characteristic
226	C₂	C₂×#14519785	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#446	C₂
hypercenter	#446	D₆₁
hypocenter	C₆₁	#2986
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #16839890	C₂	semidirect
3. #16839889	C₂	semidirect
4. Q₈×#92418	C₂	semidirect
5. C₈×Dic₁₂₂	C₂	semidirect
6. #16839886	C₂	semidirect
11. C₆₁×#446	C₂	semidirect
117. #446	D₆₁	semidirect
125. C₆₁	#2986	semidirect