Subgroups — C₂²×#19127762 #36268121

Order 21248.

2310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127762	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂²×#14581703	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#117×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈₃⋊C₈	C₂²	1	2	yes	—
8	C₂³×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#19127762	C₂²	1	64	yes	direct product factor
10	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	24	yes	complemented
11	C₂×C₂²⋊(C₈₃⋊C₈)	C₂²	1	24	yes	—
12	C₂×#14581703	C₂²	1	12	yes	complemented
13	C₂×C₄×C₈₃⋊C₈	C₂²	1	12	yes	complemented
14	C₂⁴×C₄×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₈₃	C₄	1	12	yes	—
16	C₂×#117×C₈₃	C₂²	1	12	yes	—
17	C₂²×C₄⋊C₄×C₈₃	C₄	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₈₃	C₄	1	2	yes	—
19	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	—	2	8	no	—
22	C₂²×C₈₃⋊C₈	—	2	8	no	—
23	C₂²×C₈₃⋊C₈	C₂³	1	12	yes	—
24	C₂²×C₈₃⋊C₈	C₂³	1	12	yes	—
25	C₄⋊(C₈₃⋊C₈)	C₂³	1	32	yes	complemented
26	C₂²⋊(C₈₃⋊C₈)	C₂³	1	32	yes	—
27	#14581703	C₂³	1	16	yes	complemented
28	C₄×C₈₃⋊C₈	C₂³	1	16	yes	complemented
29	C₂⁵×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₈₃	—	2	8	no	—
31	C₂³×C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₈₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₈₃	C₂×C₄	1	6	yes	—
35	C₂³×C₄×C₈₃	C₂³	1	6	yes	—
36	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
37	#117×C₈₃	C₂×C₄	1	48	yes	—
38	#117×C₈₃	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	24	yes	—
40	C₂×C₂²⋊C₄×C₈₃	C₂×C₄	1	24	yes	—
41	C₂×C₄²×C₈₃	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₈₃	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₈₃	C₂³	1	6	yes	—
44	C₂×C₄²×C₈₃	C₂³	1	6	yes	—
45	C₂×C₈₃⋊C₈	—	2	24	no	—
46	C₂×C₈₃⋊C₈	—	2	24	no	—
47	C₂×C₈₃⋊C₈	C₂⁴	1	8	yes	—
48	C₂×C₈₃⋊C₈	C₂⁴	1	8	yes	—
49	C₂⁴×C₈₃	—	2	4	no	—
50	C₂⁴×C₈₃	—	2	8	no	—
51	C₂⁴×C₈₃	C₂²×C₄	1	6	yes	—
52	C₂⁴×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	—	2	4	no	—
54	C₂²×C₄×C₈₃	—	2	8	no	—
55	C₂²×C₄×C₈₃	—	2	8	no	—
56	C₂²×C₄×C₈₃	—	2	24	no	—
57	C₂²×C₄×C₈₃	—	2	24	no	—
58	C₂²×C₄×C₈₃	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₈₃	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₈₃	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₈₃	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₈₃	#41	1	8	yes	—
64	C₂²×C₄×C₈₃	C₂⁴	1	4	yes	—
65	C₂²×C₄×C₈₃	C₂⁴	1	3	yes	—
66	C₄⋊C₄×C₈₃	C₂²×C₄	1	32	yes	—
67	C₂²⋊C₄×C₈₃	C₂²×C₄	1	32	yes	—
68	C₄²×C₈₃	C₂²×C₄	1	12	yes	—
69	C₄²×C₈₃	C₂²×C₄	1	12	yes	—
70	C₄²×C₈₃	C₂⁴	1	4	yes	—
71	C₄²×C₈₃	C₂⁴	1	4	yes	—
72	C₂³×C₈₃	—	2	12	no	—
73	C₂³×C₈₃	—	2	24	no	—
74	C₂³×C₈₃	—	4	16	no	—
75	C₂³×C₈₃	#117	1	4	yes	—
76	C₂³×C₈₃	#131	1	8	yes	—
77	C₂³×C₈₃	C₂³×C₄	1	4	yes	—
78	C₂³×C₈₃	C₂³×C₄	1	3	yes	—
79	C₂×C₄×C₈₃	—	2	12	no	—
80	C₂×C₄×C₈₃	—	2	24	no	—
81	C₂×C₄×C₈₃	—	2	24	no	—
82	C₂×C₄×C₈₃	—	2	24	no	—
83	C₂×C₄×C₈₃	—	2	8	no	—
84	C₂×C₄×C₈₃	#117	1	12	yes	—
85	C₂×C₄×C₈₃	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₈₃	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₈₃	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₈₃	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₈₃	C₂×#41	1	12	yes	—
90	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
91	C₈₃⋊C₈	—	2	8	no	—
92	C₈₃⋊C₈	—	2	8	no	—
93	C₂²×C₈₃	—	2	4	no	—
94	C₂²×C₈₃	—	2	8	no	—
95	C₂²×C₈₃	—	4	24	no	—
96	C₂²×C₈₃	#433	1	16	yes	complemented
97	C₂²×C₈₃	C₂×#117	1	6	yes	—
98	C₂²×C₈₃	C₂×#131	1	12	yes	—
99	C₂²×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
100	C₄×C₈₃	—	2	4	no	—
101	C₄×C₈₃	—	2	8	no	—
102	C₄×C₈₃	—	2	8	no	—
103	C₄×C₈₃	C₂×#117	1	6	yes	—
104	C₄×C₈₃	C₂²×#41	1	2	yes	—
105	C₂²×#433	—	83	1	no	maximal, Sylow
106	C₂×C₈₃	—	4	4	no	—
107	C₂×C₈₃	C₂×#433	1	12	yes	complemented
108	C₂×C₈₃	C₂²×#117	1	1	yes	characteristic
109	C₂×C₈₃	C₂²×#131	1	2	yes	—
110	C₂²×#117	D₈₃	1	1	yes	characteristic
111	C₂×#433	—	83	24	no	—
112	C₂²×C₄⋊C₈	—	83	2	no	—
113	C₂²×C₂²⋊C₈	—	83	2	no	—
114	C₂²×#97	—	83	1	no	—
115	C₂²×C₄×C₈	—	83	1	no	—
116	C₈₃	C₂²×#433	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴×C₄	C₂×D₈₃	1	1	yes	characteristic
118	C₂³×C₈	—	83	2	no	—
119	C₂³×C₈	—	83	2	no	—
120	C₂×#117	Dic₈₃	1	12	yes	—
121	C₂×#117	C₂×D₈₃	1	12	yes	—
122	C₂²×C₄⋊C₄	Dic₈₃	1	2	yes	—
123	C₂²×C₂²⋊C₄	Dic₈₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₈₃	1	1	yes	characteristic
125	C₂²×C₄²	C₂×D₈₃	1	1	yes	characteristic
126	#433	—	83	64	no	—
127	C₂×C₄⋊C₈	—	83	24	no	—
128	C₂×C₂²⋊C₈	—	83	24	no	—
129	C₂×#97	—	83	12	no	—
130	C₂×C₄×C₈	—	83	12	no	—
131	C₂⁵	C₂×Dic₈₃	1	1	yes	characteristic
132	C₂³×C₄	—	2	8	no	—
133	C₂³×C₄	C₂×Dic₈₃	1	2	yes	—
134	C₂³×C₄	C₂×Dic₈₃	1	2	yes	—
135	C₂³×C₄	C₂×Dic₈₃	1	1	yes	characteristic
136	C₂³×C₄	C₂×Dic₈₃	1	6	yes	—
137	C₂³×C₄	C₂²×D₈₃	1	6	yes	—
138	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
139	C₂²×C₈	—	83	12	no	—
140	C₂²×C₈	—	83	12	no	—
141	C₂²×C₈	—	166	8	no	—
142	C₂²×C₈	—	166	8	no	—
143	#117	C₂×Dic₈₃	1	48	yes	—
144	#117	C₂²×D₈₃	1	16	yes	—
145	C₂×C₄⋊C₄	C₂×Dic₈₃	1	24	yes	—
146	C₂×C₂²⋊C₄	C₂×Dic₈₃	1	24	yes	—
147	C₂×C₄²	C₂×Dic₈₃	1	6	yes	—
148	C₂×C₄²	C₂×Dic₈₃	1	6	yes	—
149	C₂×C₄²	C₂²×D₈₃	1	6	yes	—
150	C₂×C₄²	C₂²×D₈₃	1	6	yes	—
151	C₄⋊C₈	—	83	32	no	—
152	C₂²⋊C₈	—	83	32	no	—
153	#97	—	83	16	no	—
154	C₄×C₈	—	83	16	no	—
155	C₂⁴	—	2	8	no	—
156	C₂⁴	—	2	4	no	—
157	C₂⁴	C₂²×Dic₈₃	1	6	yes	—
158	C₂⁴	C₂²×Dic₈₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	C₂²×Dic₈₃	1	12	yes	—
165	C₂²×C₄	C₂²×Dic₈₃	1	12	yes	—
166	C₂²×C₄	C₂²×Dic₈₃	1	6	yes	—
167	C₂²×C₄	C₂²×Dic₈₃	1	12	yes	—
168	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
169	C₂²×C₄	#13050590	1	8	yes	—
170	C₂²×C₄	C₂³×D₈₃	1	4	yes	—
171	C₂²×C₄	C₂³×D₈₃	1	3	yes	—
172	C₂×C₈	—	83	8	no	—
173	C₂×C₈	—	83	8	no	—
174	C₂×C₈	—	166	24	no	—
175	C₂×C₈	—	166	24	no	—
176	C₄⋊C₄	C₂²×Dic₈₃	1	32	yes	—
177	C₂²⋊C₄	C₂²×Dic₈₃	1	32	yes	—
178	C₄²	C₂²×Dic₈₃	1	12	yes	—
179	C₄²	C₂²×Dic₈₃	1	12	yes	—
180	C₄²	C₂³×D₈₃	1	4	yes	—
181	C₄²	C₂³×D₈₃	1	4	yes	—
182	C₂³	—	2	12	no	—
183	C₂³	—	2	24	no	—
184	C₂³	—	4	16	no	—
185	C₂³	#14581751	1	4	yes	—
186	C₂³	#14581799	1	8	yes	—
187	C₂³	C₂³×Dic₈₃	1	4	yes	—
188	C₂³	C₂³×Dic₈₃	1	3	yes	—
189	C₂×C₄	—	2	12	no	—
190	C₂×C₄	—	2	24	no	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	24	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	#14581751	1	12	yes	—
195	C₂×C₄	C₂³×Dic₈₃	1	8	yes	—
196	C₂×C₄	C₂³×Dic₈₃	1	8	yes	—
197	C₂×C₄	C₂³×Dic₈₃	1	4	yes	—
198	C₂×C₄	C₂³×Dic₈₃	1	3	yes	—
199	C₂×C₄	C₂×#13050590	1	12	yes	—
200	C₂×C₄	C₂⁴×D₈₃	1	1	yes	characteristic
201	C₈	—	166	8	no	—
202	C₈	—	166	8	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	8	no	—
205	C₂²	—	4	24	no	—
206	C₂²	#19127762	1	16	yes	direct product factor
207	C₂²	C₂×#14581751	1	6	yes	—
208	C₂²	C₂×#14581799	1	12	yes	—
209	C₂²	C₂⁴×Dic₈₃	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	2	8	no	—
212	C₄	—	2	8	no	—
213	C₄	C₂×#14581751	1	6	yes	—
214	C₄	C₂²×#13050590	1	2	yes	—
215	C₂	—	4	4	no	—
216	C₂	C₂×#19127762	1	12	yes	direct product factor
217	C₂	C₂²×#14581751	1	1	yes	characteristic
218	C₂	C₂²×#14581799	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₈₃
Fitting	C₂²×#117×C₈₃	C₂
socle	C₂⁴×C₈₃	C₂²×C₄
prime radical	C₂⁵×C₈₃	C₂×C₄
cosocle	C₂×C₄×C₈₃	C₂⁵
hypercenter	C₂²×#117	D₈₃
hypocenter	C₈₃	C₂²×#433
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂×#19127762	C₂	direct (N × H)
2. C₂²×C₄⋊(C₈₃⋊C₈)	C₂	semidirect
4. C₂²×#14581703	C₂	semidirect
5. C₂²×C₄×C₈₃⋊C₈	C₂	semidirect
9. #19127762	C₂²	direct (N × H)
10. C₂×C₄⋊(C₈₃⋊C₈)	C₂²	semidirect
12. C₂×#14581703	C₂²	semidirect
13. C₂×C₄×C₈₃⋊C₈	C₂²	semidirect
25. C₄⋊(C₈₃⋊C₈)	C₂³	semidirect
27. #14581703	C₂³	semidirect
28. C₄×C₈₃⋊C₈	C₂³	semidirect
96. C₂²×C₈₃	#433	semidirect
107. C₂×C₈₃	C₂×#433	semidirect
116. C₈₃	C₂²×#433	semidirect
206. C₂²	#19127762	direct (N × H)
216. C₂	C₂×#19127762	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19127762	2
C₂²×C₄⋊(C₈₃⋊C₈)	2
C₂²×C₂²⋊(C₈₃⋊C₈)	2
C₂²×#14581703	2
C₂²×C₄×C₈₃⋊C₈	2
C₂²×#117×C₈₃	2
C₂²×#433	83

Subgroups — C22×#19127762 #36268121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19127762 #36268121