Subgroups — C₂×#26022396 #36709034

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26022396	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151425	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151398	C₂	1	2	yes	maximal
6	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₅₆	—	2	1	no	—
9	C₂²×#13083549	C₂²	1	2	yes	—
10	C₄×D₇₁₂	—	2	2	no	—
11	C₄×D₇₁₂	C₂²	1	4	yes	—
12	C₂×#15667946	C₂²	1	1	yes	characteristic
13	C₂×#15667934	—	2	1	no	—
14	#19151425	C₂²	1	8	yes	complemented
15	C₄⋊#13083549	C₂²	1	4	yes	complemented
16	#19151398	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
18	#500×C₈₉	—	2	2	no	—
19	#500×C₈₉	C₂²	1	4	yes	complemented
20	C₂×#107×C₈₉	—	2	1	no	—
21	C₂×#107×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×D₃₅₆	—	2	1	no	—
24	C₂²×C₄×D₈₉	—	4	1	no	—
25	C₂×D₇₁₂	—	2	1	no	—
26	C₂×D₇₁₂	—	2	2	no	—
27	C₂×D₇₁₂	Q₈	1	4	yes	complemented
28	C₂×D₇₁₂	C₂³	1	2	yes	—
29	C₄×D₃₅₆	—	2	4	no	—
30	C₄×D₃₅₆	—	4	2	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×#13083551	—	4	1	no	—
33	C₂×#13083549	—	2	4	no	—
34	C₂×#13083549	C₂³	1	4	yes	—
35	#15667946	—	2	1	no	—
36	#15667946	C₂³	1	2	yes	—
37	#15667934	—	2	4	no	—
38	C₂²×Q₈×C₈₉	—	4	1	no	—
39	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₈₉	—	2	4	no	—
42	C₄⋊Q₈×C₈₉	—	4	2	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	#107×C₈₉	—	2	4	no	—
47	#107×C₈₉	—	2	2	no	—
48	#107×C₈₉	C₂³	1	4	yes	—
49	C₄×C₈×C₈₉	D₄	1	2	yes	—
50	C₄×C₈×C₈₉	C₂³	1	2	yes	—
51	C₂³×D₈₉	—	4	1	no	—
52	C₂×D₃₅₆	—	2	4	no	—
53	C₂×D₃₅₆	—	2	2	no	—
54	C₂×D₃₅₆	—	4	1	no	—
55	C₂×D₃₅₆	—	4	2	no	—
56	C₂²×Dic₈₉	—	4	1	no	—
57	C₂×C₄×D₈₉	—	4	4	no	—
58	C₂×C₄×D₈₉	—	8	4	no	—
59	D₇₁₂	—	2	4	no	—
60	D₇₁₂	—	2	2	no	—
61	D₇₁₂	C₂×Q₈	1	4	yes	complemented
62	C₄⋊Dic₈₉	—	2	2	no	—
63	C₄⋊Dic₈₉	—	4	1	no	—
64	#13083551	—	4	4	no	—
65	#13083549	—	2	4	no	—
66	C₂×Q₈×C₈₉	—	4	4	no	—
67	C₂×Q₈×C₈₉	—	8	4	no	—
68	C₂²×C₄×C₈₉	—	4	2	no	—
69	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₈₉	#41	1	2	yes	—
77	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₈₉	—	2	4	no	—
79	C₄⋊C₄×C₈₉	—	4	2	no	—
80	C₄⋊C₄×C₈₉	—	4	4	no	—
81	C₄²×C₈₉	D₈	1	2	yes	—
82	C₄²×C₈₉	C₂×D₄	1	2	yes	—
83	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₈₉	—	4	2	no	—
85	C₂²×D₈₉	—	4	4	no	—
86	C₂²×D₈₉	—	8	2	no	—
87	C₂²×D₈₉	—	8	2	no	—
88	Q₈×C₈₉	—	8	8	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	4	4	no	—
91	C₂×C₄×C₈₉	—	8	4	no	—
92	C₂×C₄×C₈₉	SD₃₂	1	4	yes	—
93	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₈₉	#137	1	2	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	8	2	no	—
101	D₃₅₆	—	2	4	no	—
102	D₃₅₆	—	4	2	no	—
103	D₃₅₆	—	4	4	no	—
104	C₄×D₈₉	—	8	8	no	—
105	C₈×C₈₉	—	2	2	no	—
106	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₈₉	#474	1	2	yes	—
110	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
111	C₂²×C₈₉	#509	1	2	yes	—
112	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₈₉	—	4	4	no	—
114	C₂×D₈₉	—	8	2	no	—
115	C₂×D₈₉	—	8	2	no	—
116	C₂×D₈₉	—	8	8	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	8	4	no	—
119	C₄×C₈₉	#474	1	2	yes	—
120	C₄×C₈₉	C₂×SD₃₂	1	4	yes	—
121	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
123	Dic₈₉	—	8	2	no	—
124	C₂×#2226	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	#2226	1	4	yes	complemented
126	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
127	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
129	D₈₉	—	8	4	no	—
130	C₂×#500	D₈₉	1	1	yes	characteristic, complemented
131	C₂×C₄×D₈	—	89	1	no	—
132	#2226	—	89	8	no	—
133	C₂×#367	—	89	2	no	—
134	C₂×C₄⋊C₁₆	—	89	1	no	—
135	C₂×#357	—	89	2	no	—
136	C₈₉	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
137	C₂²×D₈	—	89	1	no	—
138	C₂×C₄⋊Q₈	—	2	1	no	—
139	C₂×C₄×D₄	—	178	1	no	—
140	C₂²×C₁₆	—	89	2	no	—
141	#500	—	2	2	no	—
142	#500	C₂×D₈₉	1	4	yes	complemented
143	C₄×D₈	—	89	4	no	—
144	C₄×D₈	—	178	2	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×#107	—	89	1	no	—
147	C₂×#107	C₂×D₈₉	1	2	yes	—
148	C₂×#102	—	178	1	no	—
149	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
150	#367	—	89	8	no	—
151	C₄⋊C₁₆	—	89	4	no	—
152	#357	—	89	8	no	—
153	C₂²×Q₈	—	4	1	no	—
154	C₂²×D₄	—	178	1	no	—
155	C₂³×C₄	—	356	1	no	—
156	C₂×D₈	—	89	4	no	—
157	C₂×D₈	—	89	2	no	—
158	C₂×D₈	—	178	1	no	—
159	C₂×D₈	—	178	2	no	—
160	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₄×D₄	—	356	2	no	—
165	C₄×D₄	—	178	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	178	1	no	—
169	C₂×C₂²⋊C₄	—	356	1	no	—
170	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
171	C₂×C₁₆	—	89	4	no	—
172	C₂×C₁₆	—	178	4	no	—
173	#107	—	2	4	no	—
174	#107	—	2	2	no	—
175	#107	—	89	2	no	—
176	#107	—	178	1	no	—
177	#107	C₂²×D₈₉	1	4	yes	—
178	#102	—	178	4	no	—
179	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
180	C₄×C₈	C₂²×D₈₉	1	2	yes	—
181	C₂⁴	—	356	1	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	8	4	no	—
184	C₂×D₄	—	356	1	no	—
185	C₂×D₄	—	356	2	no	—
186	C₂×D₄	—	178	4	no	—
187	C₂×D₄	—	178	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	356	1	no	—
190	C₂²×C₄	—	356	4	no	—
191	C₂²×C₄	—	712	4	no	—
192	C₂²×C₄	#13083559	1	1	yes	characteristic
193	C₂²×C₄	#13083559	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
195	D₈	—	89	4	no	—
196	D₈	—	178	4	no	—
197	D₈	—	178	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₈₉	1	2	yes	—
202	C₂×C₈	#13083575	1	2	yes	—
203	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	356	1	no	—
208	C₄⋊C₄	—	178	2	no	—
209	C₂²⋊C₄	—	356	4	no	—
210	C₄²	#13083559	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
212	C₁₆	—	178	4	no	—
213	C₂³	—	356	4	no	—
214	C₂³	—	356	2	no	—
215	C₂³	—	712	2	no	—
216	C₂³	—	712	2	no	—
217	C₂³	C₂×#13083559	1	1	yes	characteristic
218	Q₈	—	8	8	no	—
219	D₄	—	356	4	no	—
220	D₄	—	356	2	no	—
221	D₄	—	178	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	—	356	2	no	—
226	C₂×C₄	—	712	2	no	—
227	C₂×C₄	—	712	8	no	—
228	C₂×C₄	#15667955	1	4	yes	—
229	C₂×C₄	#15667988	1	2	yes	—
230	C₂×C₄	C₂×#13083559	1	2	yes	—
231	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
232	C₂×C₄	C₂×#13083559	1	2	yes	—
233	C₂×C₄	#15668043	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#15667988	1	2	yes	—
237	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
238	C₈	C₂×#13083575	1	1	yes	characteristic
239	C₂²	—	356	4	no	—
240	C₂²	—	712	2	no	—
241	C₂²	—	712	2	no	—
242	C₂²	—	712	8	no	—
243	C₂²	#19151875	1	2	yes	—
244	C₂²	C₂×#15667955	1	2	yes	—
245	C₂²	#19152034	1	2	yes	—
246	C₂²	C₂²×#13083559	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	4	no	—
249	C₄	—	712	2	no	—
250	C₄	#19151875	1	2	yes	—
251	C₄	C₂×#15667955	1	4	yes	—
252	C₄	C₂×#15667988	1	1	yes	characteristic
253	C₄	C₂×#15668043	1	1	yes	characteristic
254	C₂	—	712	4	no	—
255	C₂	#26022396	1	4	yes	direct product factor
256	C₂	C₂×#19151875	1	1	yes	characteristic
257	C₂	C₂²×#15667955	1	1	yes	characteristic
258	C₂	C₂×#19152034	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	C₂×#2226
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #26022396	C₂	direct (N × H)
3. C₂×#19151425	C₂	semidirect
4. C₂×C₄⋊#13083549	C₂	semidirect
6. C₂×#500×C₈₉	C₂	semidirect
14. #19151425	C₂²	semidirect
15. C₄⋊#13083549	C₂²	semidirect
19. #500×C₈₉	C₂²	semidirect
27. C₂×D₇₁₂	Q₈	semidirect
61. D₇₁₂	C₂×Q₈	semidirect
125. C₂×C₈₉	#2226	semidirect
130. C₂×#500	D₈₉	semidirect
136. C₈₉	C₂×#2226	semidirect
142. #500	C₂×D₈₉	semidirect
255. C₂	#26022396	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×C₄×D₇₁₂	2
#26022396	2
C₂×#19151425	2
C₂×C₄⋊#13083549	2
C₂×#19151398	2
C₂×#500×C₈₉	2
C₂×#2226	89

Subgroups — C2×#26022396 #36709034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022396 #36709034