Subgroups — C₂×#26022297 #36708658

Order 22784.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667956	C₂	1	2	yes	maximal
2	C₂×#19151978	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022297	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151401	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	—	2	1	no	—
8	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
9	C₂×#15668012	—	2	1	no	—
10	C₂×#15667956	—	2	8	no	—
11	C₂×#15667956	C₂²	1	8	yes	—
12	C₂²×#13083549	C₂²	1	2	yes	—
13	#19151978	—	2	2	no	—
14	#19151978	C₂²	1	4	yes	complemented
15	C₄⋊#13083549	C₂²	1	4	yes	complemented
16	#19151401	C₂²	1	8	yes	complemented
17	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₈₉	—	2	1	no	—
19	C₄×D₈×C₈₉	—	2	2	no	—
20	C₄×D₈×C₈₉	C₂²	1	4	yes	—
21	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#102×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₇₈	—	2	2	no	—
25	C₂×Dic₃₅₆	—	2	4	no	—
26	C₂×Dic₃₅₆	—	2	2	no	—
27	C₂×Dic₃₅₆	—	2	4	no	—
28	C₂×Dic₃₅₆	—	2	4	no	—
29	C₂×Dic₃₅₆	—	4	4	no	—
30	C₂×Dic₃₅₆	C₂³	1	4	yes	—
31	#15668012	—	2	4	no	—
32	#15668012	—	4	2	no	—
33	C₂×C₄⋊Dic₈₉	—	4	1	no	—
34	C₂×C₄⋊Dic₈₉	—	4	1	no	—
35	#15667956	—	2	16	no	—
36	C₂×#13083549	—	2	4	no	—
37	C₂×#13083549	C₂³	1	4	yes	—
38	C₂²×D₄×C₈₉	—	2	1	no	—
39	C₂³×C₄×C₈₉	—	4	1	no	—
40	C₂×D₈×C₈₉	—	2	1	no	—
41	C₂×D₈×C₈₉	—	2	2	no	—
42	C₂×D₈×C₈₉	D₄	1	4	yes	—
43	C₂×D₈×C₈₉	C₂³	1	2	yes	—
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₈₉	—	2	4	no	—
47	C₄×D₄×C₈₉	—	4	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	#107×C₈₉	—	2	1	no	—
52	#107×C₈₉	C₂³	1	2	yes	—
53	#102×C₈₉	—	2	4	no	—
54	C₄×C₈×C₈₉	D₄	1	2	yes	—
55	C₄×C₈×C₈₉	C₂³	1	2	yes	—
56	C₂×Dic₁₇₈	—	2	4	no	—
57	C₂×Dic₁₇₈	—	4	4	no	—
58	C₂×Dic₁₇₈	—	4	4	no	—
59	C₂×Dic₁₇₈	—	4	2	no	—
60	C₂²×Dic₈₉	—	4	2	no	—
61	Dic₃₅₆	—	2	4	no	—
62	Dic₃₅₆	—	2	4	no	—
63	Dic₃₅₆	—	4	8	no	—
64	Dic₃₅₆	—	4	4	no	—
65	C₄⋊Dic₈₉	—	4	4	no	—
66	C₄⋊Dic₈₉	—	4	4	no	—
67	#13083549	—	2	4	no	—
68	C₂⁴×C₈₉	—	4	1	no	—
69	C₂×D₄×C₈₉	—	2	4	no	—
70	C₂×D₄×C₈₉	—	2	2	no	—
71	C₂×D₄×C₈₉	—	4	1	no	—
72	C₂×D₄×C₈₉	—	4	2	no	—
73	C₂²×C₄×C₈₉	—	4	1	no	—
74	C₂²×C₄×C₈₉	—	4	4	no	—
75	C₂²×C₄×C₈₉	—	8	4	no	—
76	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	D₈×C₈₉	—	2	4	no	—
80	D₈×C₈₉	—	2	2	no	—
81	D₈×C₈₉	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₈₉	#41	1	2	yes	—
87	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	4	1	no	—
90	C₂²⋊C₄×C₈₉	—	4	4	no	—
91	C₄²×C₈₉	D₈	1	2	yes	—
92	C₄²×C₈₉	C₂×D₄	1	2	yes	—
93	C₂³×C₈₉	—	4	2	no	—
94	C₂³×C₈₉	—	4	4	no	—
95	C₂³×C₈₉	—	8	2	no	—
96	C₂³×C₈₉	—	8	2	no	—
97	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
98	D₄×C₈₉	—	2	4	no	—
99	D₄×C₈₉	—	4	2	no	—
100	D₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	—	8	2	no	—
104	C₂×C₄×C₈₉	—	8	8	no	—
105	C₂×C₄×C₈₉	SD₃₂	1	4	yes	complemented
106	C₂×C₄×C₈₉	#121	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₈₉	—	4	4	no	—
113	C₂×Dic₈₉	—	8	4	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	#121	1	2	yes	—
119	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
121	C₂²×C₈₉	—	4	4	no	—
122	C₂²×C₈₉	—	8	2	no	—
123	C₂²×C₈₉	—	8	2	no	—
124	C₂²×C₈₉	—	8	8	no	—
125	C₂²×C₈₉	#459	1	2	yes	—
126	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
127	C₂²×C₈₉	#510	1	2	yes	—
128	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	8	2	no	—
131	C₄×C₈₉	#459	1	2	yes	—
132	C₄×C₈₉	C₂×SD₃₂	1	4	yes	complemented
133	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
134	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
135	Dic₈₉	—	8	4	no	—
136	C₂×#2208	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	8	4	no	—
138	C₂×C₈₉	#2208	1	4	yes	complemented
139	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
140	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
141	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
142	C₂²×SD₃₂	—	89	2	no	—
143	C₂×#494	—	89	1	no	—
144	C₂×C₄×D₈	D₈₉	1	1	yes	characteristic
145	#2208	—	89	8	no	—
146	C₂×C₄⋊C₁₆	—	89	1	no	—
147	C₂×#358	—	89	2	no	—
148	C₈₉	C₂×#2208	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₁₆	—	89	2	no	—
150	C₂²×Q₁₆	—	178	1	no	—
151	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	—	178	1	no	—
153	C₂×C₄×D₄	—	2	1	no	—
154	C₂×SD₃₂	—	89	8	no	—
155	C₂×SD₃₂	—	178	8	no	—
156	C₂²×C₁₆	—	89	2	no	—
157	#494	—	89	4	no	—
158	#494	—	178	2	no	—
159	C₄×D₈	—	2	2	no	—
160	C₄×D₈	C₂×D₈₉	1	4	yes	—
161	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
162	C₂×#102	—	2	1	no	—
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₄⋊C₁₆	—	89	4	no	—
165	#358	—	89	8	no	—
166	C₂²×Q₈	—	178	2	no	—
167	C₂²×D₄	—	2	1	no	—
168	C₂³×C₄	—	4	1	no	—
169	C₂×Q₁₆	—	89	4	no	—
170	C₂×Q₁₆	—	356	4	no	—
171	C₂×Q₁₆	—	178	2	no	—
172	C₂×Q₁₆	—	178	4	no	—
173	C₂×Q₁₆	—	178	4	no	—
174	C₂×Q₁₆	—	178	4	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	D₃₅₆	1	4	yes	—
178	C₂×D₈	C₂²×D₈₉	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
181	C₄⋊Q₈	—	356	2	no	—
182	C₄⋊Q₈	—	178	4	no	—
183	C₄×D₄	—	2	4	no	—
184	C₄×D₄	—	4	2	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	356	1	no	—
187	C₂×C₄⋊C₄	—	356	1	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
190	SD₃₂	—	178	16	no	—
191	C₂×C₁₆	—	89	4	no	—
192	C₂×C₁₆	—	178	4	no	—
193	#107	—	2	1	no	—
194	#107	C₂²×D₈₉	1	2	yes	—
195	#102	—	2	4	no	—
196	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
197	C₄×C₈	C₂²×D₈₉	1	2	yes	—
198	C₂⁴	—	4	1	no	—
199	C₂×Q₈	—	356	2	no	—
200	C₂×Q₈	—	356	4	no	—
201	C₂×Q₈	—	356	4	no	—
202	C₂×Q₈	—	178	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	8	4	no	—
210	C₂²×C₄	—	356	2	no	—
211	C₂²×C₄	#13083559	1	1	yes	characteristic
212	C₂²×C₄	#13083559	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
214	Q₁₆	—	356	8	no	—
215	Q₁₆	—	356	4	no	—
216	Q₁₆	—	178	4	no	—
217	Q₁₆	—	178	4	no	—
218	D₈	—	2	4	no	—
219	D₈	—	2	2	no	—
220	D₈	C₂×D₃₅₆	1	4	yes	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
224	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
225	C₂×C₈	#13083574	1	2	yes	—
226	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	356	4	no	—
230	C₄⋊C₄	—	356	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	#13083559	1	2	yes	—
233	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
234	C₁₆	—	178	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×#13083559	1	1	yes	characteristic
240	Q₈	—	356	4	no	—
241	Q₈	—	356	4	no	—
242	Q₈	—	712	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	8	8	no	—
250	C₂×C₄	—	356	4	no	—
251	C₂×C₄	—	712	4	no	—
252	C₂×C₄	#15667956	1	4	yes	complemented
253	C₂×C₄	#15667986	1	2	yes	—
254	C₂×C₄	C₂×#13083559	1	2	yes	—
255	C₂×C₄	C₂×#13083559	1	2	yes	—
256	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
257	C₂×C₄	#15668037	1	2	yes	—
258	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	#15667986	1	2	yes	—
261	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
262	C₈	C₂×#13083574	1	1	yes	characteristic
263	C₂²	—	4	4	no	—
264	C₂²	—	8	8	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	#19151793	1	2	yes	—
268	C₂²	C₂×#15667956	1	2	yes	—
269	C₂²	#19152038	1	2	yes	—
270	C₂²	C₂²×#13083559	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	712	4	no	—
274	C₄	#19151793	1	2	yes	—
275	C₄	C₂×#15667956	1	4	yes	complemented
276	C₄	C₂×#15667986	1	1	yes	characteristic
277	C₄	C₂×#15668037	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	#26022297	1	4	yes	direct product factor
280	C₂	C₂×#19151793	1	1	yes	characteristic
281	C₂	C₂²×#15667956	1	1	yes	characteristic
282	C₂	C₂×#19152038	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×D₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₈×C₈₉	C₂²
hypercenter	C₂×C₄×D₈	D₈₉
hypocenter	C₈₉	C₂×#2208
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19151978	C₂	semidirect
3. #26022297	C₂	direct (N × H)
4. C₂×C₄⋊#13083549	C₂	semidirect
5. C₂×#19151401	C₂	semidirect
14. #19151978	C₂²	semidirect
15. C₄⋊#13083549	C₂²	semidirect
16. #19151401	C₂²	semidirect
105. C₂×C₄×C₈₉	SD₃₂	semidirect
132. C₄×C₈₉	C₂×SD₃₂	semidirect
138. C₂×C₈₉	#2208	semidirect
148. C₈₉	C₂×#2208	semidirect
252. C₂×C₄	#15667956	semidirect
275. C₄	C₂×#15667956	semidirect
279. C₂	#26022297	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#15667956	2
C₂×#19151978	2
#26022297	2
C₂×C₄⋊#13083549	2
C₂×#19151401	2
C₂×C₄×D₈×C₈₉	2
C₂×#2208	89

Subgroups — C2×#26022297 #36708658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022297 #36708658