Subgroups — #21462620

Order 7808.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840814	C₂	1	4	yes	maximal, complemented
2	#16840812	C₂	1	2	yes	maximal, complemented
3	#16840811	C₂	1	2	yes	maximal, complemented
4	#16840765	C₂	1	1	yes	maximal, characteristic
5	#16840765	C₂	1	2	yes	maximal
6	C₄×#11758217	C₂	1	1	yes	maximal, characteristic
7	C₄×#11758217	C₂	1	2	yes	maximal
8	C₂×#14519742	C₂	1	1	yes	maximal, characteristic
9	C₂×#11758217	C₂²	1	1	yes	characteristic
10	C₂×#11758217	C₂²	1	2	yes	—
11	C₂×#11758217	C₂²	1	1	yes	characteristic
12	#14519844	C₂²	1	4	yes	—
13	#14519844	C₂²	1	2	yes	—
14	C₄⋊#92420	C₂²	1	2	yes	—
15	C₄⋊#92420	C₂²	1	4	yes	—
16	#14519832	C₂²	1	4	yes	—
17	#14519832	C₂²	1	2	yes	—
18	C₄×#92420	C₂²	1	2	yes	—
19	C₄×#92420	C₂²	1	4	yes	—
20	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
21	#14519742	C₄	1	2	yes	—
22	#14519742	C₄	1	1	yes	characteristic
23	#14519742	C₄	1	1	yes	characteristic
24	#14519742	C₂²	1	1	yes	characteristic
25	#14519742	C₂²	1	1	yes	characteristic
26	#14519742	C₂²	1	2	yes	—
27	C₂×#11758196	C₄	1	2	yes	—
28	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
34	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
36	C₂×Dic₁₂₂	—	2	4	no	—
37	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
40	#11758217	—	2	2	no	—
41	#11758217	—	2	4	no	—
42	#11758217	Q₈	1	4	yes	complemented
43	C₂×#92420	C₂³	1	2	yes	—
44	C₂×#92420	C₂³	1	4	yes	—
45	C₂×#92420	C₂³	1	2	yes	—
46	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
49	#11758196	C₂×C₄	1	2	yes	—
50	#11758196	C₂×C₄	1	2	yes	—
51	#11758196	C₂×C₄	1	2	yes	—
52	#11758196	C₂×C₄	1	2	yes	—
53	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
58	C₄×Dic₆₁	C₂³	1	2	yes	—
59	C₄×Dic₆₁	C₂³	1	2	yes	—
60	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
61	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	4	2	no	—
64	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₆₁	—	2	4	no	—
69	C₂×Dic₆₁	—	4	1	no	—
70	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
76	C₂×Dic₆₁	#41	1	2	yes	—
77	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
78	Dic₁₂₂	—	2	2	no	—
79	Dic₁₂₂	—	2	4	no	—
80	Dic₁₂₂	M₁₆	1	4	yes	complemented
81	#92420	—	2	2	no	—
82	#92420	—	2	4	no	—
83	#92420	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	—	2	1	no	—
86	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
87	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	2	2	no	—
91	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
92	Dic₆₁	—	2	2	no	—
93	Dic₆₁	—	2	1	no	—
94	Dic₆₁	—	2	1	no	—
95	Dic₆₁	C₄×Q₈	1	2	yes	—
96	Dic₆₁	C₂×M₁₆	1	4	yes	—
97	Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
98	Dic₆₁	C₂×#41	1	1	yes	characteristic
99	Q₈×M₁₆	—	61	1	no	maximal, Sylow
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₆₁	#568	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	61	1	no	—
105	#446	—	61	2	no	—
106	#446	—	61	4	no	—
107	C₈×Q₈	—	61	2	no	—
108	#423	—	61	1	no	—
109	#423	—	61	2	no	—
110	C₄×M₁₆	—	61	1	no	—
111	C₄×M₁₆	—	61	2	no	—
112	C₆₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	61	1	no	—
114	C₂×M₁₆	—	61	1	no	—
115	C₂×M₁₆	—	61	2	no	—
116	C₂×M₁₆	—	61	1	no	—
117	C₄×Q₈	—	61	1	no	—
118	C₄×Q₈	—	61	1	no	—
119	C₄×Q₈	—	61	2	no	—
120	C₄×Q₈	—	61	1	no	—
121	C₄×Q₈	—	61	2	no	—
122	C₄×Q₈	—	61	1	no	—
123	C₂×C₄⋊C₄	—	61	2	no	—
124	C₂×C₄⋊C₄	#6923	1	1	yes	characteristic
125	C₂×C₄²	—	61	2	no	—
126	C₂×C₄²	—	61	1	no	—
127	C₄⋊C₈	—	61	2	no	—
128	C₄⋊C₈	—	61	4	no	—
129	C₄⋊C₈	—	61	4	no	—
130	C₄⋊C₈	—	61	2	no	—
131	#97	—	61	4	no	—
132	#97	—	61	2	no	—
133	C₄×C₈	—	61	2	no	—
134	C₄×C₈	—	61	4	no	—
135	C₂×Q₈	—	61	1	no	—
136	C₂×Q₈	—	61	2	no	—
137	C₂×Q₈	—	61	1	no	—
138	C₂×Q₈	—	122	4	no	—
139	C₂²×C₄	—	61	1	no	—
140	C₂²×C₄	—	61	2	no	—
141	C₂²×C₄	—	61	1	no	—
142	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
143	C₂²×C₄	C₂×#6923	1	2	yes	—
144	M₁₆	—	61	4	no	—
145	M₁₆	—	122	4	no	—
146	M₁₆	—	122	2	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	—	61	4	no	—
149	C₂×C₈	—	61	2	no	—
150	C₄⋊C₄	—	61	2	no	—
151	C₄⋊C₄	—	61	2	no	—
152	C₄⋊C₄	—	61	2	no	—
153	C₄⋊C₄	—	61	2	no	—
154	C₄⋊C₄	C₂×#6923	1	2	yes	—
155	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
156	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
157	C₄²	—	61	2	no	—
158	C₄²	—	61	1	no	—
159	C₄²	—	61	1	no	—
160	C₄²	—	61	2	no	—
161	C₄²	—	61	2	no	—
162	C₄²	—	61	1	no	—
163	C₄²	—	61	1	no	—
164	C₄²	—	61	2	no	—
165	C₂³	C₂²×#6923	1	1	yes	characteristic
166	Q₈	—	61	4	no	—
167	Q₈	—	122	4	no	—
168	Q₈	—	122	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	61	2	no	—
172	C₂×C₄	—	61	2	no	—
173	C₂×C₄	—	61	1	no	—
174	C₂×C₄	—	61	2	no	—
175	C₂×C₄	—	61	1	no	—
176	C₂×C₄	—	61	1	no	—
177	C₂×C₄	—	61	1	no	—
178	C₂×C₄	—	61	2	no	—
179	C₂×C₄	—	122	4	no	—
180	C₂×C₄	—	244	1	no	—
181	C₂×C₄	C₂²×#6923	1	2	yes	—
182	C₂×C₄	C₂²×#6923	1	2	yes	—
183	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
184	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
185	C₈	—	61	4	no	—
186	C₈	—	122	4	no	—
187	C₈	—	122	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	Q₈×#6923	1	2	yes	—
191	C₂²	C₂³×#6923	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	61	2	no	—
196	C₄	—	61	1	no	—
197	C₄	—	61	1	no	—
198	C₄	—	61	4	no	—
199	C₄	—	122	2	no	—
200	C₄	—	122	1	no	—
201	C₄	—	122	1	no	—
202	C₄	D₆₁⋊M₁₆	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×Q₈×#6923	1	1	yes	characteristic
205	C₂	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
206	C₂	#16840840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×C₄⋊C₄×C₆₁	C₄
hypercenter	C₂×C₄⋊C₄	#6923
hypocenter	C₆₁	Q₈×M₁₆
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #16840814	C₂	semidirect
2. #16840812	C₂	semidirect
3. #16840811	C₂	semidirect
42. #11758217	Q₈	semidirect
80. Dic₁₂₂	M₁₆	semidirect
83. #92420	C₂×Q₈	semidirect
112. C₆₁	Q₈×M₁₆	semidirect