Subgroups — C₂×#26019771 #36537057

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×#19152003	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151809	C₂	1	2	yes	maximal
3	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
4	#26019771	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151345	C₂	1	2	yes	maximal
6	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	—	2	2	no	—
8	C₂×#15668012	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#19152003	—	2	2	no	—
11	#19152003	C₂²	1	4	yes	—
12	#19151809	—	2	4	no	—
13	#19151809	C₂²	1	8	yes	—
14	#19151728	—	2	2	no	—
15	#19151728	C₂²	1	4	yes	—
16	C₂×#15667946	—	2	1	no	—
17	C₂×#15667946	—	2	1	no	—
18	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
19	C₂×#15667941	—	2	1	no	—
20	C₂×#15667937	—	2	2	no	—
21	C₂×#15667922	—	2	1	no	—
22	#19151345	C₂²	1	8	yes	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#334×C₈₉	C₂²	1	4	yes	—
26	C₂²×Dic₁₇₈	—	2	2	no	—
27	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
28	C₂×Dic₃₅₆	—	2	8	no	—
29	C₂×Dic₃₅₆	—	4	8	no	—
30	C₂²×#10623494	—	2	1	no	—
31	#15668012	—	2	2	no	—
32	#15668012	—	2	2	no	—
33	#15668012	—	2	2	no	—
34	#15668012	C₂³	1	4	yes	—
35	C₄×Dic₁₇₈	—	2	1	no	—
36	C₄×Dic₁₇₈	—	2	1	no	—
37	C₄×Dic₁₇₈	D₄	1	2	yes	—
38	C₄×Dic₁₇₈	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₈₉	—	2	2	no	—
40	C₂×C₄⋊Dic₈₉	—	2	2	no	—
41	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
42	C₂×#13083553	—	2	1	no	—
43	C₂×C₄×Dic₈₉	—	2	1	no	—
44	#15667946	—	2	4	no	—
45	#15667946	—	2	4	no	—
46	C₄⋊#10623494	—	2	1	no	—
47	C₄⋊#10623494	C₂³	1	2	yes	—
48	#15667941	—	2	4	no	—
49	#15667937	—	2	8	no	—
50	#15667922	—	2	4	no	—
51	C₂²×C₈×C₈₉	—	2	2	no	—
52	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	—	2	2	no	—
56	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
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	—	2	4	no	—
62	C₂×Dic₁₇₈	—	4	4	no	—
63	C₂×Dic₁₇₈	—	4	4	no	—
64	C₂×Dic₁₇₈	—	4	2	no	—
65	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	4	2	no	—
69	Dic₃₅₆	—	4	16	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	4	2	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	4	no	—
74	C₄⋊Dic₈₉	—	2	4	no	—
75	C₄⋊Dic₈₉	—	4	2	no	—
76	C₄⋊Dic₈₉	—	4	2	no	—
77	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
78	#13083553	—	2	2	no	—
79	#13083553	—	2	2	no	—
80	C₄×Dic₈₉	—	2	2	no	—
81	C₄×Dic₈₉	—	2	2	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	—	4	4	no	—
88	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
89	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
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	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
102	C₂×C₄×C₈₉	#137	1	2	yes	—
103	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
106	C₂×Dic₈₉	—	2	2	no	—
107	C₂×Dic₈₉	—	2	2	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	4	4	no	—
111	C₂×Dic₈₉	—	8	4	no	—
112	Dic₁₇₈	—	2	4	no	—
113	Dic₁₇₈	—	4	2	no	—
114	Dic₁₇₈	—	4	4	no	—
115	Dic₁₇₈	—	4	4	no	—
116	Dic₁₇₈	—	8	4	no	—
117	C₈×C₈₉	—	2	2	no	—
118	C₈×C₈₉	—	4	4	no	—
119	C₈×C₈₉	C₂×Q₁₆	1	4	yes	complemented
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	#462	1	2	yes	—
128	C₄×C₈₉	#469	1	2	yes	—
129	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
130	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
132	Dic₈₉	—	4	2	no	—
133	Dic₈₉	—	4	2	no	—
134	Dic₈₉	—	8	4	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	—	89	1	no	—
141	C₂×#462	—	89	2	no	—
142	C₂×#446	—	89	1	no	—
143	#1694	—	89	8	no	—
144	C₂×#334	D₈₉	1	1	yes	characteristic
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	—	89	2	no	—
149	C₂×C₄×Q₈	—	89	1	no	—
150	#500	—	89	4	no	—
151	#500	—	178	2	no	—
152	#462	—	89	8	no	—
153	#462	—	178	4	no	—
154	#446	—	89	4	no	—
155	#446	—	178	2	no	—
156	C₂×#107	—	178	1	no	—
157	C₂×#107	—	178	1	no	—
158	C₂×C₄⋊C₈	—	89	1	no	—
159	C₂×C₄⋊C₈	—	178	1	no	—
160	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
161	C₂×#103	—	178	2	no	—
162	C₂×#97	—	178	1	no	—
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	#334	C₂×D₈₉	1	4	yes	—
165	#332	—	89	8	no	—
166	C₂²×Q₈	—	89	1	no	—
167	C₂²×Q₈	—	178	2	no	—
168	C₂×Q₁₆	—	356	8	no	—
169	C₂×Q₁₆	—	178	8	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	178	1	no	—
172	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
173	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
174	C₄⋊Q₈	—	89	4	no	—
175	C₄⋊Q₈	—	178	2	no	—
176	C₄⋊Q₈	—	178	2	no	—
177	C₄⋊Q₈	—	178	2	no	—
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	—	89	1	no	—
183	C₂×C₄⋊C₄	—	178	2	no	—
184	C₂×C₄⋊C₄	—	178	2	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	—	178	4	no	—
189	#107	—	178	4	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	89	2	no	—
192	C₄⋊C₈	—	178	4	no	—
193	C₄⋊C₈	—	178	1	no	—
194	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
195	#103	—	178	8	no	—
196	#97	—	178	4	no	—
197	C₄×C₈	D₃₅₆	1	2	yes	—
198	C₄×C₈	C₂²×D₈₉	1	2	yes	—
199	C₂×Q₈	—	89	2	no	—
200	C₂×Q₈	—	356	2	no	—
201	C₂×Q₈	—	356	4	no	—
202	C₂×Q₈	—	356	4	no	—
203	C₂×Q₈	—	178	4	no	—
204	C₂×Q₈	—	178	4	no	—
205	C₂×Q₈	—	178	1	no	—
206	C₂²×C₄	—	356	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×D₃₅₆	1	1	yes	characteristic
211	C₂²×C₄	#13083575	1	1	yes	characteristic
212	Q₁₆	—	356	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	—	178	2	no	—
218	C₂×C₈	Dic₃₅₆	1	4	yes	complemented
219	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
220	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
221	C₄⋊C₄	—	89	2	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	356	2	no	—
224	C₄⋊C₄	—	178	4	no	—
225	C₄⋊C₄	—	178	4	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×D₃₅₆	1	1	yes	characteristic
233	C₄²	#13083575	1	1	yes	characteristic
234	C₄²	C₂³×D₈₉	1	1	yes	characteristic
235	C₂³	#15667987	1	1	yes	characteristic
236	Q₈	—	356	2	no	—
237	Q₈	—	356	4	no	—
238	Q₈	—	356	4	no	—
239	Q₈	—	178	4	no	—
240	Q₈	—	712	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	356	2	no	—
243	C₂×C₄	—	356	2	no	—
244	C₂×C₄	—	356	4	no	—
245	C₂×C₄	—	178	2	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	—	712	4	no	—
248	C₂×C₄	#15667987	1	1	yes	characteristic
249	C₂×C₄	#15667987	1	1	yes	characteristic
250	C₂×C₄	#15667987	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
252	C₂×C₄	#15668039	1	4	yes	—
253	C₂×C₄	#15668045	1	2	yes	—
254	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	—	356	2	no	—
260	C₈	C₂×Dic₃₅₆	1	4	yes	complemented
261	C₂²	#19151809	1	2	yes	—
262	C₂²	#19151852	1	2	yes	—
263	C₂²	#19151874	1	2	yes	—
264	C₂²	C₂×#15667987	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	356	2	no	—
268	C₄	—	356	2	no	—
269	C₄	—	712	4	no	—
270	C₄	#19151809	1	2	yes	—
271	C₄	#19151852	1	2	yes	—
272	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
273	C₄	C₂×#15668039	1	2	yes	—
274	C₄	C₂×#15668045	1	1	yes	characteristic
275	C₂	#26019771	1	4	yes	direct product factor
276	C₂	C₂×#19151809	1	1	yes	characteristic
277	C₂	C₂×#19151852	1	1	yes	characteristic
278	C₂	C₂×#19151874	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	C₂×#1694
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #26019771	C₂	direct (N × H)
88. C₂×C₈×C₈₉	Q₁₆	semidirect
119. C₈×C₈₉	C₂×Q₁₆	semidirect
136. C₂×C₈₉	#1694	semidirect
146. C₈₉	C₂×#1694	semidirect
218. C₂×C₈	Dic₃₅₆	semidirect
260. C₈	C₂×Dic₃₅₆	semidirect
275. C₂	#26019771	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×#19152003	2
C₂×#19151809	2
C₂×#19151728	2
#26019771	2
C₂×#19151345	2
C₂×#334×C₈₉	2
C₂×#1694	89

Subgroups — C2×#26019771 #36537057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019771 #36537057