Subgroups — #21462755

Order 7808.

356 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	#16840814	C₂	1	2	yes	maximal, complemented
2	#16840793	C₂	1	2	yes	maximal, complemented
3	#16840793	C₂	1	2	yes	maximal, complemented
4	D₄×#92420	C₂	1	2	yes	maximal, complemented
5	#16840786	C₂	1	2	yes	maximal, complemented
6	#16840783	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840768	C₂	1	2	yes	maximal, complemented
8	C₂×#14519844	C₂	1	1	yes	maximal, characteristic, complemented
9	#16840293	C₂	1	1	yes	maximal, characteristic
10	C₂×#11758217	C₂²	1	2	yes	—
11	C₂²×#92420	C₂²	1	1	yes	characteristic
12	C₂²×#92420	C₂²	1	1	yes	characteristic
13	C₂²×#92420	C₂²	1	2	yes	—
14	#14519844	C₂²	1	2	yes	—
15	#14519844	C₂²	1	2	yes	—
16	#14519844	C₂²	1	2	yes	—
17	C₄⋊#92420	C₂²	1	1	yes	characteristic
18	C₄⋊#92420	C₂²	1	2	yes	—
19	C₄⋊#92420	C₂²	1	1	yes	characteristic
20	C₂²⋊#92420	C₂²	1	2	yes	—
21	C₂²⋊#92420	C₂²	1	2	yes	—
22	C₂²⋊#92420	C₂²	1	2	yes	—
23	#14519832	C₂²	1	2	yes	—
24	#14519832	C₂²	1	2	yes	—
25	C₄×#92420	C₂²	1	2	yes	—
26	#14519770	C₄	1	2	yes	—
27	#14519768	C₄	1	1	yes	characteristic
28	#14519764	C₄	1	1	yes	characteristic
29	#14519758	C₄	1	2	yes	—
30	#14519756	C₄	1	1	yes	characteristic
31	#14519742	C₂²	1	1	yes	characteristic
32	D₄×Dic₆₁	C₂²	1	2	yes	—
33	D₄×Dic₆₁	C₂²	1	1	yes	characteristic
34	#14519733	C₂²	1	2	yes	—
35	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
36	#121×C₆₁	C₄	1	1	yes	characteristic
37	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₆₁	C₂³	1	2	yes	—
43	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
44	#11758217	—	4	2	no	—
45	C₂×#92420	—	4	1	no	—
46	C₂×#92420	—	4	2	no	—
47	C₂×#92420	—	4	1	no	—
48	C₂×#92420	C₂³	1	2	yes	—
49	C₂×#92420	C₂³	1	2	yes	—
50	C₂×#92420	C₂³	1	2	yes	—
51	C₂×#92420	C₂³	1	2	yes	—
52	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
54	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₆₁	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	2	yes	—
66	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
67	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
68	C₂³×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	D₄×C₆₁	—	2	4	no	—
71	D₄×C₆₁	—	4	1	no	—
72	C₂×C₄×C₆₁	—	4	1	no	—
73	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₆₁	—	2	2	no	—
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	—	2	2	no	—
79	C₂×Dic₆₁	—	2	1	no	—
80	C₂×Dic₆₁	—	2	2	no	—
81	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
85	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
86	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
87	Dic₁₂₂	—	4	1	no	—
88	#92420	—	2	2	no	—
89	#92420	—	2	2	no	—
90	#92420	—	2	2	no	—
91	#92420	—	2	2	no	—
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	—	2	1	no	—
94	C₂²×C₆₁	—	2	2	no	—
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	—	2	2	no	—
97	C₂²×C₆₁	#131	1	2	yes	—
98	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	Dic₆₁	—	2	1	no	—
103	Dic₆₁	—	2	2	no	—
104	Dic₆₁	—	2	1	no	—
105	Dic₆₁	—	2	1	no	—
106	Dic₆₁	—	2	2	no	—
107	Dic₆₁	H(5,𝔽₂)	1	1	yes	characteristic
108	Dic₆₁	2₋¹⁺⁴	1	1	yes	characteristic
109	#2992	—	61	1	no	maximal, Sylow
110	C₂×C₆₁	—	2	2	no	—
111	C₂×C₆₁	—	4	2	no	—
112	C₂×C₆₁	#520	1	1	yes	characteristic
113	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
114	C₂×C₆₁	#568	1	1	yes	characteristic
115	#529	—	61	1	no	—
116	#446	—	61	2	no	—
117	#435	—	61	2	no	—
118	#435	—	61	2	no	—
119	C₈×D₄	—	61	2	no	—
120	#433	—	61	2	no	—
121	#432	—	61	1	no	—
122	#424	—	61	2	no	—
123	C₂×C₄⋊C₈	—	61	1	no	—
124	C₆₁	#2992	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₁₆	—	61	2	no	—
126	C₂²×C₈	—	61	1	no	—
127	C₂²×C₈	—	61	1	no	—
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	#6923	1	1	yes	characteristic
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	—	61	1	no	—
140	C₄⋊C₈	—	61	1	no	—
141	C₄⋊C₈	—	61	2	no	—
142	C₄⋊C₈	—	61	1	no	—
143	C₄⋊C₈	—	61	2	no	—
144	C₄⋊C₈	—	61	2	no	—
145	C₄⋊C₈	—	61	2	no	—
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	C₂×#6923	1	2	yes	—
154	C₂×D₄	C₂×#6923	1	1	yes	characteristic
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	61	2	no	—
157	C₂²×C₄	—	61	1	no	—
158	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
159	M₁₆	—	244	2	no	—
160	C₂×C₈	—	61	2	no	—
161	C₂×C₈	—	61	2	no	—
162	C₂×C₈	—	61	2	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	—	244	1	no	—
165	C₂×C₈	—	244	2	no	—
166	C₂×C₈	—	244	1	no	—
167	C₄⋊C₄	—	61	2	no	—
168	C₄⋊C₄	—	61	2	no	—
169	C₄⋊C₄	—	61	2	no	—
170	C₄⋊C₄	—	61	1	no	—
171	C₄⋊C₄	—	61	2	no	—
172	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
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	C₂×#6923	1	2	yes	—
178	C₄²	—	61	2	no	—
179	C₄²	—	61	1	no	—
180	C₄²	—	61	1	no	—
181	C₄²	—	61	1	no	—
182	C₄²	—	61	1	no	—
183	C₂³	C₂²×#6923	1	2	yes	—
184	C₂³	C₂²×#6923	1	1	yes	characteristic
185	Q₈	—	244	1	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	61	1	no	—
190	C₂×C₄	—	61	1	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	61	1	no	—
193	C₂×C₄	—	61	2	no	—
194	C₂×C₄	—	61	2	no	—
195	C₂×C₄	—	122	1	no	—
196	C₂×C₄	—	122	2	no	—
197	C₂×C₄	—	122	1	no	—
198	C₂×C₄	—	122	2	no	—
199	C₂×C₄	—	122	2	no	—
200	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
201	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
202	C₂×C₄	C₂²×#6923	1	2	yes	—
203	C₈	—	122	2	no	—
204	C₈	—	122	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	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#14519864	1	2	yes	—
213	C₂²	C₂³×#6923	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	61	1	no	—
218	C₄	—	61	1	no	—
219	C₄	—	122	1	no	—
220	C₄	—	122	2	no	—
221	C₄	—	122	1	no	—
222	C₄	—	122	1	no	—
223	C₄	—	122	2	no	—
224	C₂	—	2	2	no	—
225	C₂	—	4	2	no	—
226	C₂	#16840831	1	1	yes	characteristic
227	C₂	C₂×#14519864	1	1	yes	characteristic
228	C₂	#16840840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	#121×C₆₁	C₄
hypercenter	#121	#6923
hypocenter	C₆₁	#2992
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #16840814	C₂	semidirect
2. #16840793	C₂	semidirect
3. #16840793	C₂	semidirect
4. D₄×#92420	C₂	semidirect
5. #16840786	C₂	semidirect
6. #16840783	C₂	semidirect
7. #16840768	C₂	semidirect
8. C₂×#14519844	C₂	semidirect
124. C₆₁	#2992	semidirect