Subgroups — C₂×#24906371 #35374167

Order 21248.

654 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127828	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#24906371	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127431	C₂	1	2	yes	maximal
5	C₂×#19127422	C₂	1	1	yes	maximal, characteristic
6	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₆	C₂²	1	2	yes	—
9	#19127828	—	2	4	no	—
10	#19127828	C₂²	1	8	yes	—
11	C₄×Dic₃₃₂	C₄	1	4	yes	—
12	C₄×Dic₃₃₂	C₂²	1	4	yes	—
13	C₂×#14581726	C₄	1	1	yes	characteristic
14	C₂×Dic₈₃⋊C₈	—	2	2	no	—
15	C₂×#14581717	C₄	1	2	yes	—
16	C₂×#14581702	—	2	2	no	—
17	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	#19127431	C₂²	1	8	yes	—
20	#19127422	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	#334×C₈₃	C₂²	1	4	yes	—
24	C₂²×Dic₁₆₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₃₃₂	—	2	4	no	—
26	C₂×Dic₃₃₂	C₂×C₄	1	4	yes	—
27	C₂²×C₈₃⋊C₈	—	2	1	no	—
28	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
29	C₄×Dic₁₆₆	—	2	2	no	—
30	C₄×Dic₁₆₆	—	2	2	no	—
31	C₄×Dic₁₆₆	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₆₆	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
34	C₂×Dic₈₃⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₈₃	—	2	2	no	—
36	#14581726	C₂×C₄	1	4	yes	—
37	Dic₈₃⋊C₈	—	2	8	no	—
38	#14581717	C₂×C₄	1	8	yes	—
39	#14581702	—	2	8	no	—
40	C₄×C₈₃⋊C₈	—	2	1	no	—
41	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
42	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
43	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₈×C₈₃	—	2	2	no	—
45	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₃	—	2	2	no	—
49	C₄⋊C₈×C₈₃	C₂³	1	4	yes	—
50	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₈₃	C₂³	1	2	yes	—
52	C₂×Dic₁₆₆	—	2	4	no	—
53	C₂×Dic₁₆₆	—	2	4	no	—
54	C₂×Dic₁₆₆	—	2	2	no	—
55	C₂×Dic₁₆₆	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₈₃	—	2	2	no	—
57	C₂²×Dic₈₃	—	2	2	no	—
58	Dic₃₃₂	—	2	8	no	—
59	C₂×C₈₃⋊C₈	—	2	4	no	—
60	C₂×C₈₃⋊C₈	—	2	2	no	—
61	C₂×C₈₃⋊C₈	—	4	2	no	—
62	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
63	C₄⋊Dic₈₃	—	2	2	no	—
64	C₄⋊Dic₈₃	C₂²×C₄	1	4	yes	—
65	Dic₈₃⋊C₄	—	2	4	no	—
66	Dic₈₃⋊C₄	—	2	4	no	—
67	C₄×Dic₈₃	—	2	4	no	—
68	C₄×Dic₈₃	—	2	4	no	—
69	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₈₃	—	2	2	no	—
73	C₂×C₈×C₈₃	—	2	4	no	—
74	C₂×C₈×C₈₃	—	4	4	no	—
75	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
76	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₃	#41	1	1	yes	characteristic
81	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₈₃	—	2	2	no	—
84	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₈₃	#131	1	2	yes	—
89	C₂×C₄×C₈₃	#137	1	4	yes	—
90	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₈₃	—	2	4	no	—
94	C₂×Dic₈₃	—	2	4	no	—
95	C₂×Dic₈₃	—	4	4	no	—
96	C₂×Dic₈₃	—	4	4	no	—
97	Dic₁₆₆	—	2	4	no	—
98	Dic₁₆₆	—	2	4	no	—
99	Dic₁₆₆	—	4	4	no	—
100	C₈×C₈₃	—	2	2	no	—
101	C₈×C₈₃	—	4	4	no	—
102	C₈×C₈₃	C₂×M₁₆	1	4	yes	—
103	C₈₃⋊C₈	—	2	4	no	—
104	C₈₃⋊C₈	—	4	2	no	—
105	C₂²×C₈₃	#436	1	2	yes	—
106	C₂²×C₈₃	#441	1	2	yes	—
107	C₂²×C₈₃	#444	1	2	yes	—
108	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₈₃	—	2	1	no	—
110	C₄×C₈₃	—	2	1	no	—
111	C₄×C₈₃	#436	1	2	yes	—
112	C₄×C₈₃	#441	1	2	yes	—
113	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
114	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
115	C₄×C₈₃	C₂×#137	1	2	yes	—
116	Dic₈₃	—	4	4	no	—
117	Dic₈₃	—	4	4	no	—
118	C₂×#1593	—	83	1	no	maximal, Sylow
119	C₂×C₈₃	#1593	1	4	yes	complemented
120	C₂×C₈₃	C₂×#436	1	1	yes	characteristic
121	C₂×C₈₃	C₂×#441	1	1	yes	characteristic
122	C₂×C₈₃	C₂×#444	1	1	yes	characteristic
123	C₂×#446	—	83	2	no	—
124	C₂×C₄×Q₁₆	—	83	1	no	—
125	#1593	—	83	8	no	—
126	C₂×#334	D₈₃	1	1	yes	characteristic
127	C₂×Q₈⋊C₈	—	83	2	no	—
128	C₂×#322	—	83	1	no	—
129	C₈₃	C₂×#1593	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	83	1	no	—
131	C₂×C₄×Q₈	—	83	2	no	—
132	#446	—	83	8	no	—
133	#446	—	166	4	no	—
134	C₄×Q₁₆	—	83	4	no	—
135	C₄×Q₁₆	—	83	4	no	—
136	C₂×#107	—	83	1	no	—
137	C₂×C₄⋊C₈	—	166	2	no	—
138	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
139	C₂×#103	—	83	2	no	—
140	C₂×#97	—	166	2	no	—
141	C₂×C₄×C₈	—	83	1	no	—
142	C₂×C₄×C₈	—	83	1	no	—
143	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
144	#334	C₂×D₈₃	1	4	yes	—
145	Q₈⋊C₈	—	83	8	no	—
146	#322	—	83	4	no	—
147	C₂²×Q₈	—	83	2	no	—
148	C₂×Q₁₆	—	83	4	no	—
149	C₂×Q₁₆	—	166	4	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	83	2	no	—
152	C₂²×C₈	—	166	1	no	—
153	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
154	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
155	C₄×Q₈	—	83	4	no	—
156	C₄×Q₈	—	83	4	no	—
157	C₄×Q₈	—	166	2	no	—
158	C₄×Q₈	—	166	2	no	—
159	C₂×C₄⋊C₄	—	83	2	no	—
160	C₂×C₄⋊C₄	—	166	2	no	—
161	C₂×C₄²	—	166	2	no	—
162	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
163	#107	—	83	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	166	8	no	—
166	C₄⋊C₈	C₂²×D₈₃	1	4	yes	—
167	#103	—	83	8	no	—
168	#97	—	166	8	no	—
169	C₄×C₈	—	83	2	no	—
170	C₄×C₈	—	83	2	no	—
171	C₄×C₈	—	83	2	no	—
172	C₄×C₈	—	166	1	no	—
173	C₄×C₈	C₄×D₈₃	1	2	yes	—
174	C₄×C₈	C₂²×D₈₃	1	2	yes	—
175	C₂×Q₈	—	83	4	no	—
176	C₂×Q₈	—	166	4	no	—
177	C₂×Q₈	—	166	4	no	—
178	C₂×Q₈	—	166	2	no	—
179	C₂²×C₄	—	166	2	no	—
180	C₂²×C₄	—	166	2	no	—
181	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
183	C₂²×C₄	#13050591	1	1	yes	characteristic
184	Q₁₆	—	166	8	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	332	2	no	—
189	C₂×C₈	—	83	4	no	—
190	C₂×C₈	—	166	2	no	—
191	C₂×C₈	—	166	4	no	—
192	C₂×C₈	#13050573	1	4	yes	—
193	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
195	C₄⋊C₄	—	83	4	no	—
196	C₄⋊C₄	—	166	4	no	—
197	C₄⋊C₄	—	166	2	no	—
198	C₄⋊C₄	—	166	4	no	—
199	C₄²	—	166	4	no	—
200	C₄²	—	166	4	no	—
201	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
202	C₄²	D₄×D₈₃	1	1	yes	characteristic
203	C₄²	#13050591	1	1	yes	characteristic
204	C₄²	C₂³×D₈₃	1	1	yes	characteristic
205	C₂³	#14581754	1	1	yes	characteristic
206	Q₈	—	332	4	no	—
207	Q₈	—	166	4	no	—
208	Q₈	—	166	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	332	4	no	—
211	C₂×C₄	—	332	4	no	—
212	C₂×C₄	—	166	4	no	—
213	C₂×C₄	—	166	4	no	—
214	C₂×C₄	#14581754	1	1	yes	characteristic
215	C₂×C₄	#14581754	1	1	yes	characteristic
216	C₂×C₄	#14581754	1	1	yes	characteristic
217	C₂×C₄	C₂×#13050573	1	2	yes	—
218	C₂×C₄	#14581800	1	2	yes	—
219	C₂×C₄	#14581821	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
222	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₈	—	332	2	no	—
226	C₈	—	166	4	no	—
227	C₈	C₂×#13050573	1	4	yes	—
228	C₂²	#19127778	1	2	yes	—
229	C₂²	#19127807	1	2	yes	—
230	C₂²	#19127822	1	2	yes	—
231	C₂²	C₂×#14581754	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	332	4	no	—
235	C₄	—	332	4	no	—
236	C₄	#19127778	1	2	yes	—
237	C₄	#19127807	1	2	yes	—
238	C₄	C₂²×#13050573	1	1	yes	characteristic
239	C₄	C₂×#14581800	1	1	yes	characteristic
240	C₄	C₂×#14581821	1	2	yes	—
241	C₂	#24906371	1	4	yes	direct product factor
242	C₂	C₂×#19127778	1	1	yes	characteristic
243	C₂	C₂×#19127807	1	1	yes	characteristic
244	C₂	C₂×#19127822	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	C₂×#1593
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #24906371	C₂	direct (N × H)
119. C₂×C₈₃	#1593	semidirect
129. C₈₃	C₂×#1593	semidirect
241. C₂	#24906371	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×#19127828	2
C₂×C₄×Dic₃₃₂	2
#24906371	2
C₂×#19127431	2
C₂×#19127422	2
C₂×#334×C₈₃	2
C₂×#1593	83

Subgroups — C2×#24906371 #35374167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24906371 #35374167