Subgroups — C₂×#24906361 #35374147

Order 21248.

670 conjugacy classes of subgroups.

The subgroup lattice has 258 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	2	yes	maximal
2	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#24906361	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
5	C₂×#19127433	C₂	1	2	yes	maximal
6	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₆	C₂²	1	2	yes	—
9	#19127831	—	2	4	no	—
10	#19127831	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×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₈₃⋊C₈	—	2	2	no	—
16	C₂×#14581717	C₄	1	2	yes	—
17	C₂×#14581702	—	2	2	no	—
18	#19127454	C₂²	1	4	yes	—
19	#19127433	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₈²×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	2	no	—
28	C₄×Dic₁₆₆	—	2	2	no	—
29	C₄×Dic₁₆₆	—	2	2	no	—
30	C₄×Dic₁₆₆	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₆₆	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
33	C₂×Dic₈₃⋊C₄	—	2	2	no	—
34	C₂×C₄×Dic₈₃	—	2	2	no	—
35	#14581726	C₂×C₄	1	4	yes	—
36	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
37	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
38	Dic₈₃⋊C₈	—	2	8	no	—
39	#14581717	C₂×C₄	1	8	yes	—
40	#14581702	—	2	8	no	—
41	C₂²×C₈×C₈₃	—	2	1	no	—
42	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₈₃	—	2	1	no	—
48	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₃	D₄	1	2	yes	—
50	C₄×C₈×C₈₃	C₂³	1	2	yes	—
51	C₄×C₈×C₈₃	C₂³	1	2	yes	—
52	C₄×C₈×C₈₃	C₂³	1	2	yes	—
53	C₂×Dic₁₆₆	—	2	4	no	—
54	C₂×Dic₁₆₆	—	2	4	no	—
55	C₂×Dic₁₆₆	—	2	2	no	—
56	C₂×Dic₁₆₆	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₈₃	—	2	2	no	—
58	C₂²×Dic₈₃	—	2	2	no	—
59	Dic₃₃₂	—	2	8	no	—
60	C₂×C₈₃⋊C₈	—	2	4	no	—
61	C₂×C₈₃⋊C₈	—	4	4	no	—
62	C₄⋊Dic₈₃	—	2	2	no	—
63	C₄⋊Dic₈₃	C₂²×C₄	1	4	yes	—
64	Dic₈₃⋊C₄	—	2	4	no	—
65	Dic₈₃⋊C₄	—	2	4	no	—
66	C₄×Dic₈₃	—	2	4	no	—
67	C₄×Dic₈₃	—	2	4	no	—
68	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₈₃	—	2	4	no	—
72	C₂×C₈×C₈₃	—	2	2	no	—
73	C₂×C₈×C₈₃	—	2	2	no	—
74	C₂×C₈×C₈₃	—	2	2	no	—
75	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
76	C₂×C₈×C₈₃	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
81	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₃	#41	1	1	yes	characteristic
84	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
85	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₈₃	#131	1	2	yes	—
92	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₈₃	#135	1	2	yes	—
94	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₈₃	—	2	4	no	—
98	C₂×Dic₈₃	—	2	4	no	—
99	C₂×Dic₈₃	—	4	4	no	—
100	C₂×Dic₈₃	—	4	4	no	—
101	Dic₁₆₆	—	2	4	no	—
102	Dic₁₆₆	—	2	4	no	—
103	Dic₁₆₆	—	4	4	no	—
104	C₈×C₈₃	—	2	4	no	—
105	C₈×C₈₃	—	2	2	no	—
106	C₈×C₈₃	—	2	2	no	—
107	C₈×C₈₃	C₂×M₁₆	1	4	yes	—
108	C₈×C₈₃	C₂×Q₁₆	1	4	yes	complemented
109	C₈₃⋊C₈	—	4	4	no	—
110	C₂²×C₈₃	#436	1	2	yes	—
111	C₂²×C₈₃	C₄×Q₁₆	1	2	yes	—
112	C₂²×C₈₃	#443	1	2	yes	—
113	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₈₃	—	2	1	no	—
115	C₄×C₈₃	—	2	1	no	—
116	C₄×C₈₃	#436	1	2	yes	—
117	C₄×C₈₃	C₄×Q₁₆	1	2	yes	—
118	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
119	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
120	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
121	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
122	Dic₈₃	—	4	4	no	—
123	Dic₈₃	—	4	4	no	—
124	C₂×#1591	—	83	1	no	maximal, Sylow
125	C₂×C₈₃	#1591	1	4	yes	complemented
126	C₂×C₈₃	C₂×#436	1	1	yes	characteristic
127	C₂×C₈₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
128	C₂×C₈₃	C₂×#443	1	1	yes	characteristic
129	C₂×#446	—	83	2	no	—
130	C₂×C₄×Q₁₆	—	83	1	no	—
131	#1591	—	83	8	no	—
132	C₂×#335	—	83	1	no	—
133	C₂×Q₈⋊C₈	—	83	2	no	—
134	C₂×C₈²	D₈₃	1	1	yes	characteristic
135	C₈₃	C₂×#1591	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₁₆	—	83	1	no	—
137	C₂×C₄×Q₈	—	83	2	no	—
138	#446	—	83	8	no	—
139	#446	—	166	4	no	—
140	C₄×Q₁₆	—	83	4	no	—
141	C₄×Q₁₆	—	83	4	no	—
142	C₂×#107	—	83	1	no	—
143	C₂×C₄⋊C₈	—	83	2	no	—
144	C₂×C₄⋊C₈	—	166	2	no	—
145	C₂×#103	—	83	2	no	—
146	C₂×#97	—	166	2	no	—
147	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
150	#335	—	83	4	no	—
151	Q₈⋊C₈	—	83	8	no	—
152	C₈²	C₂×D₈₃	1	4	yes	—
153	C₂²×Q₈	—	83	2	no	—
154	C₂×Q₁₆	—	83	4	no	—
155	C₂×Q₁₆	—	166	4	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	166	2	no	—
158	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
160	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
161	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
162	C₄×Q₈	—	83	4	no	—
163	C₄×Q₈	—	83	4	no	—
164	C₄×Q₈	—	166	2	no	—
165	C₄×Q₈	—	166	2	no	—
166	C₂×C₄⋊C₄	—	83	2	no	—
167	C₂×C₄⋊C₄	—	166	2	no	—
168	C₂×C₄²	—	166	2	no	—
169	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
170	#107	—	83	4	no	—
171	C₄⋊C₈	—	83	4	no	—
172	C₄⋊C₈	—	166	8	no	—
173	C₄⋊C₈	—	166	2	no	—
174	#103	—	83	8	no	—
175	#97	—	166	8	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	C₄×D₈₃	1	2	yes	—
178	C₄×C₈	D₃₃₂	1	2	yes	—
179	C₄×C₈	C₂²×D₈₃	1	2	yes	—
180	C₄×C₈	C₂²×D₈₃	1	2	yes	—
181	C₄×C₈	C₂²×D₈₃	1	2	yes	—
182	C₂×Q₈	—	83	4	no	—
183	C₂×Q₈	—	166	4	no	—
184	C₂×Q₈	—	166	4	no	—
185	C₂×Q₈	—	166	2	no	—
186	C₂²×C₄	—	166	2	no	—
187	C₂²×C₄	—	166	2	no	—
188	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
190	C₂²×C₄	#13050590	1	1	yes	characteristic
191	Q₁₆	—	166	8	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	332	4	no	—
197	C₂×C₈	—	166	4	no	—
198	C₂×C₈	#13050573	1	4	yes	—
199	C₂×C₈	Dic₃₃₂	1	4	yes	complemented
200	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
202	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
203	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
204	C₄⋊C₄	—	83	4	no	—
205	C₄⋊C₄	—	166	4	no	—
206	C₄⋊C₄	—	166	2	no	—
207	C₄⋊C₄	—	166	4	no	—
208	C₄²	—	166	4	no	—
209	C₄²	—	166	4	no	—
210	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
211	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
212	C₄²	#13050590	1	1	yes	characteristic
213	C₄²	C₂³×D₈₃	1	1	yes	characteristic
214	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
215	Q₈	—	332	4	no	—
216	Q₈	—	166	4	no	—
217	Q₈	—	166	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	332	4	no	—
220	C₂×C₄	—	332	4	no	—
221	C₂×C₄	—	166	4	no	—
222	C₂×C₄	—	166	4	no	—
223	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
224	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
225	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
226	C₂×C₄	C₂×#13050573	1	2	yes	—
227	C₂×C₄	#14581798	1	2	yes	—
228	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
229	C₂×C₄	#14581814	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
232	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	332	4	no	—
237	C₈	C₂×#13050573	1	4	yes	—
238	C₈	C₂×Dic₃₃₂	1	4	yes	complemented
239	C₂²	#19127780	1	2	yes	—
240	C₂²	C₄×Dic₃₃₂	1	2	yes	—
241	C₂²	#19127819	1	2	yes	—
242	C₂²	C₂×C₄×D₃₃₂	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	332	4	no	—
246	C₄	—	332	4	no	—
247	C₄	#19127780	1	2	yes	—
248	C₄	C₄×Dic₃₃₂	1	2	yes	—
249	C₄	C₂²×#13050573	1	1	yes	characteristic
250	C₄	C₂×#14581798	1	1	yes	characteristic
251	C₄	C₂²×Dic₃₃₂	1	1	yes	characteristic
252	C₄	C₂×#14581814	1	1	yes	characteristic
253	C₂	#24906361	1	4	yes	direct product factor
254	C₂	C₂×#19127780	1	1	yes	characteristic
255	C₂	C₂×C₄×Dic₃₃₂	1	1	yes	characteristic
256	C₂	C₂×#19127819	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	C₂×#1591
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #24906361	C₂	direct (N × H)
76. C₂×C₈×C₈₃	Q₁₆	semidirect
108. C₈×C₈₃	C₂×Q₁₆	semidirect
125. C₂×C₈₃	#1591	semidirect
135. C₈₃	C₂×#1591	semidirect
199. C₂×C₈	Dic₃₃₂	semidirect
238. C₈	C₂×Dic₃₃₂	semidirect
253. C₂	#24906361	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×C₄×Dic₃₃₂	2
#24906361	2
C₂×#19127454	2
C₂×#19127433	2
C₂×C₈²×C₈₃	2
C₂×#1591	83

Subgroups — C2×#24906361 #35374147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24906361 #35374147