Subgroups — C₂×#26019767 #36537053

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151805	C₂	1	2	yes	maximal
2	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
3	#26019767	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151344	C₂	1	2	yes	maximal
6	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083564	—	2	2	no	—
8	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	#19151805	—	2	4	no	—
10	#19151805	C₂²	1	8	yes	—
11	#19151728	—	2	2	no	—
12	#19151728	C₂²	1	4	yes	—
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
15	C₂×#15667941	—	2	1	no	—
16	C₂×#15667936	—	2	2	no	—
17	C₂×#15667922	—	2	1	no	—
18	#19151349	C₂²	1	4	yes	—
19	#19151344	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	#500×C₈₉	—	2	2	no	—
22	#500×C₈₉	C₂²	1	4	yes	—
23	C₂×#107×C₈₉	—	2	1	no	—
24	C₂×#107×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
27	C₂×#13083564	—	2	8	no	—
28	C₂×#13083564	—	4	8	no	—
29	C₂²×#10623494	—	2	1	no	—
30	C₂²×#10623494	—	2	2	no	—
31	C₄×Dic₁₇₈	—	2	1	no	—
32	C₄×Dic₁₇₈	—	2	1	no	—
33	C₄×Dic₁₇₈	D₄	1	2	yes	—
34	C₄×Dic₁₇₈	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
36	C₂×#13083553	—	2	1	no	—
37	C₂×C₄×Dic₈₉	—	2	1	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	C₂³	1	2	yes	—
41	C₄⋊#10623494	C₂³	1	4	yes	—
42	#15667941	—	2	4	no	—
43	#15667936	—	2	8	no	—
44	#15667922	—	2	4	no	—
45	C₂²×Q₈×C₈₉	—	2	2	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	—	2	2	no	—
49	C₄⋊Q₈×C₈₉	—	2	2	no	—
50	C₄⋊Q₈×C₈₉	—	2	2	no	—
51	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	#107×C₈₉	—	2	4	no	—
56	#107×C₈₉	—	2	4	no	—
57	C₄×C₈×C₈₉	D₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	2	yes	—
59	C₂×Dic₁₇₈	—	2	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂²×Dic₈₉	—	2	1	no	—
64	#13083564	—	4	16	no	—
65	C₂×#10623494	—	2	4	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	4	4	no	—
68	C₂×#10623494	—	4	2	no	—
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
71	#13083553	—	2	2	no	—
72	#13083553	—	2	2	no	—
73	C₄×Dic₈₉	—	2	2	no	—
74	C₄×Dic₈₉	—	2	2	no	—
75	C₂×Q₈×C₈₉	—	2	4	no	—
76	C₂×Q₈×C₈₉	—	4	2	no	—
77	C₂×Q₈×C₈₉	—	4	4	no	—
78	C₂×Q₈×C₈₉	—	4	4	no	—
79	C₂²×C₄×C₈₉	—	4	2	no	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
85	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₄⋊C₄×C₈₉	—	2	4	no	—
89	C₄⋊C₄×C₈₉	—	4	2	no	—
90	C₄⋊C₄×C₈₉	—	4	2	no	—
91	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	#41	1	1	yes	characteristic
94	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
95	C₂³×C₈₉	#121	1	1	yes	characteristic
96	Q₈×C₈₉	—	4	4	no	—
97	Q₈×C₈₉	—	4	4	no	—
98	Q₈×C₈₉	—	8	4	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	8	4	no	—
102	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₈₉	#137	1	2	yes	—
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	2	no	—
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	Dic₁₇₈	—	2	4	no	—
116	Dic₁₇₈	—	4	2	no	—
117	C₈×C₈₉	—	2	2	no	—
118	C₈×C₈₉	C₂×Q₁₆	1	4	yes	complemented
119	#10623494	—	4	4	no	—
120	#10623494	—	4	2	no	—
121	C₂²×C₈₉	#462	1	2	yes	—
122	C₂²×C₈₉	#469	1	2	yes	—
123	C₂²×C₈₉	#473	1	2	yes	—
124	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	8	4	no	—
128	C₄×C₈₉	#462	1	2	yes	—
129	C₄×C₈₉	#469	1	2	yes	—
130	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
131	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
132	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
133	Dic₈₉	—	4	2	no	—
134	Dic₈₉	—	4	2	no	—
135	C₂×#1694	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	#1694	1	4	yes	complemented
137	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
138	C₂×C₈₉	C₂×#469	1	1	yes	characteristic
139	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
140	C₂×#500	D₈₉	1	1	yes	characteristic
141	C₂×#462	—	89	2	no	—
142	C₂×#446	—	89	1	no	—
143	#1694	—	89	8	no	—
144	C₂×#334	—	89	1	no	—
145	C₂×#332	—	89	2	no	—
146	C₈₉	C₂×#1694	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₁₆	—	178	2	no	—
148	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
149	C₂×C₄×Q₈	—	89	1	no	—
150	#500	—	2	2	no	—
151	#500	C₂×D₈₉	1	4	yes	—
152	#462	—	89	8	no	—
153	#462	—	178	4	no	—
154	#446	—	89	4	no	—
155	#446	—	178	2	no	—
156	C₂×#107	—	2	1	no	—
157	C₂×#107	—	2	1	no	—
158	C₂×C₄⋊C₈	—	89	1	no	—
159	C₂×C₄⋊C₈	—	89	2	no	—
160	C₂×C₄⋊C₈	—	178	1	no	—
161	C₂×#103	—	178	2	no	—
162	C₂×#97	—	178	1	no	—
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	#334	—	89	4	no	—
165	#332	—	89	8	no	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	—	89	1	no	—
168	C₂×Q₁₆	—	356	8	no	—
169	C₂×Q₁₆	—	178	8	no	—
170	C₂²×C₈	—	178	2	no	—
171	C₂²×C₈	—	178	1	no	—
172	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
178	C₄×Q₈	—	89	2	no	—
179	C₄×Q₈	—	89	2	no	—
180	C₄×Q₈	—	178	1	no	—
181	C₄×Q₈	—	178	1	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	89	1	no	—
185	C₂×C₄⋊C₄	—	178	1	no	—
186	C₂×C₄²	—	178	1	no	—
187	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#107	—	2	4	no	—
190	C₄⋊C₈	—	89	2	no	—
191	C₄⋊C₈	—	89	4	no	—
192	C₄⋊C₈	—	178	4	no	—
193	C₄⋊C₈	—	178	2	no	—
194	C₄⋊C₈	—	178	1	no	—
195	#103	—	178	8	no	—
196	#97	—	178	4	no	—
197	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
198	C₄×C₈	C₂²×D₈₉	1	2	yes	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	89	2	no	—
204	C₂×Q₈	—	178	4	no	—
205	C₂×Q₈	—	178	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	178	1	no	—
208	C₂²×C₄	—	178	1	no	—
209	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
211	C₂²×C₄	#13083573	1	1	yes	characteristic
212	Q₁₆	—	356	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	356	4	no	—
215	C₂×C₈	—	356	2	no	—
216	C₂×C₈	—	178	4	no	—
217	C₂×C₈	—	178	2	no	—
218	C₂×C₈	#13083564	1	4	yes	complemented
219	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
220	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	89	2	no	—
226	C₄⋊C₄	—	178	1	no	—
227	C₄⋊C₄	—	178	2	no	—
228	C₄⋊C₄	—	178	2	no	—
229	C₄²	—	178	2	no	—
230	C₄²	—	178	2	no	—
231	C₄²	D₄×D₈₉	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
233	C₄²	#13083573	1	1	yes	characteristic
234	C₄²	C₂³×D₈₉	1	1	yes	characteristic
235	C₂³	#15667983	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	4	4	no	—
238	Q₈	—	8	4	no	—
239	Q₈	—	356	2	no	—
240	Q₈	—	178	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	356	2	no	—
245	C₂×C₄	—	356	2	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	—	178	2	no	—
248	C₂×C₄	#15667983	1	1	yes	characteristic
249	C₂×C₄	#15667983	1	1	yes	characteristic
250	C₂×C₄	#15667983	1	1	yes	characteristic
251	C₂×C₄	C₂×#13083564	1	2	yes	—
252	C₂×C₄	#15668036	1	4	yes	—
253	C₂×C₄	#15668043	1	2	yes	—
254	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	356	2	no	—
259	C₈	—	356	4	no	—
260	C₈	C₂×#13083564	1	4	yes	complemented
261	C₂²	#19151805	1	2	yes	—
262	C₂²	#19151848	1	2	yes	—
263	C₂²	#19151870	1	2	yes	—
264	C₂²	C₂×#15667983	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	8	4	no	—
268	C₄	—	356	2	no	—
269	C₄	—	356	2	no	—
270	C₄	#19151805	1	2	yes	—
271	C₄	#19151848	1	2	yes	—
272	C₄	C₂²×#13083564	1	1	yes	characteristic
273	C₄	C₂×#15668036	1	2	yes	—
274	C₄	C₂×#15668043	1	1	yes	characteristic
275	C₂	#26019767	1	4	yes	direct product factor
276	C₂	C₂×#19151805	1	1	yes	characteristic
277	C₂	C₂×#19151848	1	1	yes	characteristic
278	C₂	C₂×#19151870	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	C₂×#1694
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #26019767	C₂	direct (N × H)
84. C₂×C₈×C₈₉	Q₁₆	semidirect
118. C₈×C₈₉	C₂×Q₁₆	semidirect
136. C₂×C₈₉	#1694	semidirect
146. C₈₉	C₂×#1694	semidirect
218. C₂×C₈	#13083564	semidirect
260. C₈	C₂×#13083564	semidirect
275. C₂	#26019767	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×#19151805	2
C₂×#19151728	2
#26019767	2
C₂×#19151349	2
C₂×#19151344	2
C₂×#500×C₈₉	2
C₂×#1694	89

Subgroups — C2×#26019767 #36537053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019767 #36537053