Subgroups — C₂×#24905444 #35314218

Order 21248.

346 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127831	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127477	C₂	1	1	yes	maximal, characteristic
3	#24905444	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₆₆	C₄	1	1	yes	characteristic
6	C₂²×C₈₃⋊C₁₆	—	2	1	no	—
7	#19127831	—	2	1	no	—
8	#19127831	—	2	1	no	—
9	#19127831	C₄	1	2	yes	—
10	#19127831	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×Dic₈₃⋊C₈	—	2	1	no	—
13	C₂×#14581702	—	2	1	no	—
14	#19127477	—	2	1	no	—
15	#19127477	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₈₃	—	2	1	no	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₃	—	2	1	no	—
19	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
20	C₂²×Dic₁₆₆	C₈	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	—	2	1	no	—
22	C₄×Dic₁₆₆	—	2	1	no	—
23	C₄×Dic₁₆₆	—	2	1	no	—
24	C₄×Dic₁₆₆	C₈	1	2	yes	—
25	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
27	C₂×Dic₈₃⋊C₄	—	2	1	no	—
28	C₂×C₄×Dic₈₃	—	2	1	no	—
29	C₂×C₈₃⋊C₁₆	—	2	4	no	—
30	C₂×C₈₃⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
32	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
33	Dic₈₃⋊C₈	—	2	2	no	—
34	Dic₈₃⋊C₈	—	2	2	no	—
35	#14581702	—	2	2	no	—
36	#14581702	—	2	2	no	—
37	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₈₃	—	2	2	no	—
41	C₂×C₁₆×C₈₃	—	4	2	no	—
42	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₆₆	—	2	1	no	—
47	C₂×Dic₁₆₆	—	4	2	no	—
48	C₂×Dic₁₆₆	C₂×C₈	1	2	yes	—
49	C₂²×Dic₈₃	—	2	1	no	—
50	C₂²×Dic₈₃	—	2	1	no	—
51	C₂×C₈₃⋊C₈	—	2	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₄⋊Dic₈₃	—	2	1	no	—
54	C₄⋊Dic₈₃	C₂×C₈	1	2	yes	—
55	Dic₈₃⋊C₄	—	2	2	no	—
56	Dic₈₃⋊C₄	—	2	2	no	—
57	C₄×Dic₈₃	—	2	2	no	—
58	C₄×Dic₈₃	—	2	2	no	—
59	C₈₃⋊C₁₆	—	2	4	no	—
60	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
63	C₂×C₈×C₈₃	—	2	2	no	—
64	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
68	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
73	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
74	C₁₆×C₈₃	—	4	2	no	—
75	C₂³×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₈₃	—	2	2	no	—
77	C₂×C₄×C₈₃	—	2	2	no	—
78	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#103	1	2	yes	—
82	C₂×C₄×C₈₃	#104	1	2	yes	—
83	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₈₃	—	2	2	no	—
87	C₂×Dic₈₃	—	2	2	no	—
88	C₂×Dic₈₃	—	4	2	no	—
89	C₂×Dic₈₃	—	4	2	no	—
90	Dic₁₆₆	—	4	2	no	—
91	Dic₁₆₆	—	4	2	no	—
92	C₈×C₈₃	—	2	1	no	—
93	C₈×C₈₃	—	2	1	no	—
94	C₈×C₈₃	#103	1	2	yes	—
95	C₈×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
97	C₈₃⋊C₈	—	4	2	no	—
98	C₂²×C₈₃	Q₈⋊C₈	1	2	yes	—
99	C₂²×C₈₃	#349	1	2	yes	—
100	C₂²×C₈₃	#350	1	2	yes	—
101	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₈₃	—	2	1	no	—
103	C₄×C₈₃	—	2	1	no	—
104	C₄×C₈₃	—	2	1	no	—
105	C₄×C₈₃	—	2	1	no	—
106	C₄×C₈₃	Q₈⋊C₈	1	2	yes	—
107	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
108	C₄×C₈₃	C₂×#104	1	1	yes	characteristic
109	Dic₈₃	—	4	2	no	—
110	Dic₈₃	—	4	2	no	—
111	C₂×#1338	—	83	1	no	maximal, Sylow
112	C₂×C₈₃	#1338	1	4	yes	complemented
113	C₂×C₈₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
114	C₂×C₈₃	C₂×#349	1	1	yes	characteristic
115	C₂×C₈₃	C₂×#350	1	1	yes	characteristic
116	C₂×#446	—	83	1	no	—
117	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic
118	C₂×#346	—	83	1	no	—
119	#1338	—	83	4	no	—
120	C₈₃	C₂×#1338	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	83	1	no	—
122	C₂²×C₁₆	—	2	1	no	—
123	C₂²×C₁₆	—	166	1	no	—
124	#446	—	83	2	no	—
125	#446	—	83	2	no	—
126	#446	—	166	1	no	—
127	#446	—	166	1	no	—
128	C₂×C₄⋊C₈	—	83	1	no	—
129	C₂×C₄⋊C₈	—	166	1	no	—
130	C₂×#97	—	166	1	no	—
131	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
134	#346	—	83	2	no	—
135	#346	—	166	1	no	—
136	C₂²×Q₈	—	83	1	no	—
137	C₂²×C₈	—	166	1	no	—
138	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
140	C₄×Q₈	—	83	2	no	—
141	C₄×Q₈	—	83	2	no	—
142	C₄×Q₈	—	166	1	no	—
143	C₄×Q₈	—	166	1	no	—
144	C₂×C₄⋊C₄	—	83	1	no	—
145	C₂×C₄⋊C₄	—	166	1	no	—
146	C₂×C₄²	—	166	1	no	—
147	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	4	2	no	—
150	C₂×C₁₆	—	166	4	no	—
151	C₂×C₁₆	—	166	2	no	—
152	C₄⋊C₈	—	83	2	no	—
153	C₄⋊C₈	—	166	2	no	—
154	C₄⋊C₈	—	166	2	no	—
155	C₄⋊C₈	—	166	1	no	—
156	#97	—	166	2	no	—
157	#97	—	166	2	no	—
158	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
159	C₄×C₈	D₃₃₂	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
162	C₂×Q₈	—	332	2	no	—
163	C₂×Q₈	—	83	2	no	—
164	C₂×Q₈	—	166	1	no	—
165	C₂²×C₄	—	166	1	no	—
166	C₂²×C₄	—	166	1	no	—
167	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
169	C₂²×C₄	#13050573	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	332	2	no	—
172	C₂×C₈	—	166	2	no	—
173	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	#13050578	1	2	yes	—
176	C₂×C₈	#13050580	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
178	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
179	C₄⋊C₄	—	83	2	no	—
180	C₄⋊C₄	—	166	2	no	—
181	C₄⋊C₄	—	166	1	no	—
182	C₄⋊C₄	—	166	2	no	—
183	C₄²	—	166	2	no	—
184	C₄²	—	166	2	no	—
185	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
186	C₄²	C₈×D₈₃	1	1	yes	characteristic
187	C₄²	#13050573	1	1	yes	characteristic
188	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₁₆	—	166	4	no	—
191	C₂³	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
192	Q₈	—	332	2	no	—
193	Q₈	—	332	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	332	2	no	—
197	C₂×C₄	—	332	2	no	—
198	C₂×C₄	—	166	2	no	—
199	C₂×C₄	—	166	2	no	—
200	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
201	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
202	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	#14581715	1	2	yes	—
204	C₂×C₄	#14581718	1	2	yes	—
205	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
207	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	332	2	no	—
211	C₈	#14581715	1	2	yes	—
212	C₈	C₂×#13050578	1	1	yes	characteristic
213	C₈	C₂×#13050580	1	1	yes	characteristic
214	C₂²	#19127431	1	2	yes	—
215	C₂²	#19127482	1	2	yes	—
216	C₂²	#19127484	1	2	yes	—
217	C₂²	C₂×(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	332	2	no	—
223	C₄	—	332	2	no	—
224	C₄	#19127431	1	2	yes	—
225	C₄	C₂×#14581715	1	1	yes	characteristic
226	C₄	C₂×#14581718	1	1	yes	characteristic
227	C₂	#24905444	1	4	yes	direct product factor
228	C₂	C₂×#19127431	1	1	yes	characteristic
229	C₂	C₂×#19127482	1	1	yes	characteristic
230	C₂	C₂×#19127484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×C₄⋊C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²⋊C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	C₂×#1338
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #24905444	C₂	direct (N × H)
112. C₂×C₈₃	#1338	semidirect
120. C₈₃	C₂×#1338	semidirect
227. C₂	#24905444	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×#19127831	2
C₂×#19127477	2
#24905444	2
C₂×C₄⋊C₁₆×C₈₃	2
C₂×#1338	83

Subgroups — C2×#24905444 #35314218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24905444 #35314218