Subgroups — C₂×#24909690 #35559692

Order 21248.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128106	C₂	1	1	yes	maximal, characteristic
2	#24909690	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19127505	C₂	1	4	yes	maximal
4	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	—	2	1	no	—
7	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
8	#19128106	—	2	2	no	—
9	#19128106	C₂²	1	4	yes	—
10	C₂×#14581726	—	2	1	no	—
11	C₂×#14581726	C₂²	1	2	yes	—
12	#19127505	C₂²	1	16	yes	—
13	C₄×C₈₃⋊C₁₆	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₈₃	—	2	1	no	—
15	C₂²×Q₁₆×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₈₃	—	2	1	no	—
17	#494×C₈₃	—	2	2	no	—
18	#494×C₈₃	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₆₆	—	4	1	no	—
21	#14581792	—	2	4	no	—
22	#14581792	—	4	2	no	—
23	C₂×C₄⋊Dic₈₃	—	2	2	no	—
24	C₂×C₄⋊Dic₈₃	—	4	1	no	—
25	C₂×C₈₃⋊C₁₆	—	2	4	no	—
26	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
27	#14581726	—	2	2	no	—
28	#14581726	—	2	4	no	—
29	#14581726	C₂³	1	4	yes	—
30	C₂²×Q₈×C₈₃	—	2	2	no	—
31	C₂×Q₁₆×C₈₃	—	2	2	no	—
32	C₂×Q₁₆×C₈₃	—	2	4	no	—
33	C₂×Q₁₆×C₈₃	—	2	8	no	—
34	C₂×Q₁₆×C₈₃	—	4	4	no	—
35	C₂×Q₁₆×C₈₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₈₃	—	2	4	no	—
39	C₄⋊Q₈×C₈₃	—	4	2	no	—
40	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
41	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
42	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	2	yes	—
44	C₄×C₈×C₈₃	C₂³	1	2	yes	—
45	C₂×Dic₁₆₆	—	4	4	no	—
46	C₂×Dic₁₆₆	—	8	4	no	—
47	C₂²×Dic₈₃	—	4	2	no	—
48	C₄⋊Dic₈₃	—	2	4	no	—
49	C₄⋊Dic₈₃	—	4	4	no	—
50	C₄⋊Dic₈₃	—	4	2	no	—
51	C₈₃⋊C₁₆	—	2	4	no	—
52	C₂×Q₈×C₈₃	—	2	4	no	—
53	C₂×Q₈×C₈₃	—	4	2	no	—
54	C₂×Q₈×C₈₃	—	4	4	no	—
55	C₂×Q₈×C₈₃	—	4	4	no	—
56	C₂²×C₄×C₈₃	—	4	2	no	—
57	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₈₃	—	2	8	no	—
61	Q₁₆×C₈₃	—	4	4	no	—
62	Q₁₆×C₈₃	—	4	8	no	—
63	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
64	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₈₃	#41	1	4	yes	—
67	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₈₃	—	4	4	no	—
69	C₄⋊C₄×C₈₃	—	4	4	no	—
70	C₄²×C₈₃	D₈	1	2	yes	—
71	C₄²×C₈₃	C₂×D₄	1	2	yes	—
72	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₈₃	—	4	4	no	—
74	Q₈×C₈₃	—	4	4	no	—
75	Q₈×C₈₃	—	8	4	no	—
76	C₂×C₄×C₈₃	—	4	4	no	—
77	C₂×C₄×C₈₃	—	8	4	no	—
78	C₂×C₄×C₈₃	SD₃₂	1	4	yes	—
79	C₂×C₄×C₈₃	Q₃₂	1	4	yes	—
80	C₂×C₄×C₈₃	#124	1	2	yes	—
81	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₈₃	—	4	4	no	—
87	C₂×Dic₈₃	—	8	4	no	—
88	Dic₁₆₆	—	8	8	no	—
89	C₈×C₈₃	#124	1	2	yes	—
90	C₈×C₈₃	C₂×SD₁₆	1	4	yes	—
91	C₈×C₈₃	C₂×#41	1	2	yes	—
92	C₂²×C₈₃	#486	1	2	yes	—
93	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₈₃	C₂×Q₃₂	1	2	yes	—
95	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₈₃	—	8	4	no	—
97	C₄×C₈₃	#486	1	2	yes	—
98	C₄×C₈₃	C₂×SD₃₂	1	4	yes	—
99	C₄×C₈₃	C₂×Q₃₂	1	4	yes	—
100	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
101	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
102	Dic₈₃	—	8	4	no	—
103	C₂×#2241	—	83	1	no	maximal, Sylow
104	C₂×C₈₃	#2241	1	4	yes	complemented
105	C₂×C₈₃	C₂×#486	1	1	yes	characteristic
106	C₂×C₈₃	C₂²×SD₃₂	1	1	yes	characteristic
107	C₂×C₈₃	C₂²×Q₃₂	1	1	yes	characteristic
108	C₂×#500	—	83	1	no	—
109	C₂×#494	D₈₃	1	1	yes	characteristic
110	#2241	—	83	8	no	—
111	C₂×#358	—	83	4	no	—
112	C₂×C₄×C₁₆	—	83	1	no	—
113	C₈₃	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₁₆	—	2	1	no	—
115	C₂²×Q₁₆	C₂×D₈₃	1	2	yes	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×C₄⋊Q₈	—	166	1	no	—
118	C₂²×C₁₆	—	83	2	no	—
119	#500	—	83	4	no	—
120	#500	—	166	2	no	—
121	#494	—	2	2	no	—
122	#494	C₂×D₈₃	1	4	yes	—
123	C₂×#107	—	83	2	no	—
124	C₂×#107	—	166	1	no	—
125	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
126	#358	—	83	16	no	—
127	C₄×C₁₆	—	83	4	no	—
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×Q₈	—	332	1	no	—
130	C₂×Q₁₆	—	2	4	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	8	no	—
133	C₂×Q₁₆	—	4	4	no	—
134	C₂×Q₁₆	C₂²×D₈₃	1	4	yes	—
135	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	4	2	no	—
139	C₄⋊Q₈	—	332	2	no	—
140	C₄⋊Q₈	—	166	4	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	332	1	no	—
144	C₂×C₄⋊C₄	—	166	2	no	—
145	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
146	C₂×C₁₆	—	83	4	no	—
147	C₂×C₁₆	—	166	4	no	—
148	#107	—	83	4	no	—
149	#107	—	166	2	no	—
150	#107	—	166	4	no	—
151	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
152	C₄×C₈	C₂²×D₈₃	1	2	yes	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	332	4	no	—
158	C₂×Q₈	—	664	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	332	2	no	—
161	C₂²×C₄	#13050575	1	1	yes	characteristic
162	C₂²×C₄	#13050575	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
164	Q₁₆	—	2	8	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	8	no	—
167	C₂×C₈	#13050578	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
170	C₂×C₈	#13050589	1	4	yes	—
171	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	332	4	no	—
175	C₄⋊C₄	—	332	2	no	—
176	C₄⋊C₄	—	166	4	no	—
177	C₄²	#13050575	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
179	C₁₆	—	166	4	no	—
180	C₂³	C₂×#13050575	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	Q₈	—	4	4	no	—
183	Q₈	—	8	4	no	—
184	Q₈	—	664	8	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	332	4	no	—
188	C₂×C₄	—	664	4	no	—
189	C₂×C₄	#14581735	1	4	yes	—
190	C₂×C₄	#14581738	1	4	yes	—
191	C₂×C₄	#14581779	1	2	yes	—
192	C₂×C₄	C₂×#13050575	1	2	yes	—
193	C₂×C₄	C₂×#13050575	1	2	yes	—
194	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
195	C₂×C₄	C₂×#13050578	1	2	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
197	C₈	#14581779	1	2	yes	—
198	C₈	C₂×#13050578	1	4	yes	—
199	C₈	C₂×#13050589	1	2	yes	—
200	C₂²	#19128047	1	2	yes	—
201	C₂²	C₂×#14581735	1	2	yes	—
202	C₂²	C₂×#14581738	1	2	yes	—
203	C₂²	C₂²×#13050575	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	664	4	no	—
206	C₄	#19128047	1	2	yes	—
207	C₄	C₂×#14581735	1	4	yes	—
208	C₄	C₂×#14581738	1	4	yes	—
209	C₄	C₂×#14581779	1	1	yes	characteristic
210	C₄	C₂²×#13050578	1	1	yes	characteristic
211	C₂	#24909690	1	4	yes	direct product factor
212	C₂	C₂×#19128047	1	1	yes	characteristic
213	C₂	C₂²×#14581735	1	1	yes	characteristic
214	C₂	C₂²×#14581738	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	C₂×#2241
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #24909690	C₂	direct (N × H)
104. C₂×C₈₃	#2241	semidirect
113. C₈₃	C₂×#2241	semidirect
211. C₂	#24909690	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×#19128106	2
#24909690	2
C₂×#19127505	2
C₂×C₄×C₈₃⋊C₁₆	2
C₂×#494×C₈₃	2
C₂×#2241	83

Subgroups — C2×#24909690 #35559692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909690 #35559692