Subgroups — C₂×#26019647 #36536785

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 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×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
4	#26019647	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151345	C₂	1	2	yes	maximal
6	C₂×#335×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	C₈×Dic₁₇₈	—	2	2	no	—
15	C₈×Dic₁₇₈	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×C₈×Dic₈₉	—	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	#335×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	C₈×Dic₈₉	—	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	1	no	—
60	C₂×Dic₁₇₈	—	2	2	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	D₈	1	4	yes	—
66	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	4	2	no	—
70	Dic₃₅₆	—	4	16	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	4	2	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	—	2	4	no	—
75	C₄⋊Dic₈₉	—	2	4	no	—
76	C₄⋊Dic₈₉	—	4	2	no	—
77	C₄⋊Dic₈₉	—	4	2	no	—
78	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
79	#13083553	—	2	2	no	—
80	#13083553	—	2	2	no	—
81	C₄×Dic₈₉	—	2	2	no	—
82	C₄×Dic₈₉	—	2	2	no	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	4	4	no	—
89	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
90	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	#41	1	1	yes	characteristic
95	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₈₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₈₉	#135	1	2	yes	—
104	C₂×C₄×C₈₉	#137	1	2	yes	—
105	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₈₉	—	2	2	no	—
109	C₂×Dic₈₉	—	2	4	no	—
110	C₂×Dic₈₉	—	2	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	4	4	no	—
113	C₂×Dic₈₉	—	8	4	no	—
114	Dic₁₇₈	—	2	2	no	—
115	Dic₁₇₈	—	2	4	no	—
116	Dic₁₇₈	—	4	4	no	—
117	Dic₁₇₈	—	4	4	no	—
118	Dic₁₇₈	—	8	4	no	—
119	Dic₁₇₈	C₂×D₈	1	4	yes	—
120	C₈×C₈₉	—	2	2	no	—
121	C₈×C₈₉	—	4	4	no	—
122	C₈×C₈₉	C₂×Q₁₆	1	4	yes	complemented
123	#10623494	—	4	2	no	—
124	C₂²×C₈₉	#462	1	2	yes	—
125	C₂²×C₈₉	#466	1	2	yes	—
126	C₂²×C₈₉	#473	1	2	yes	—
127	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	#462	1	2	yes	—
131	C₄×C₈₉	#466	1	2	yes	—
132	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
134	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
135	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
136	Dic₈₉	—	2	4	no	—
137	Dic₈₉	—	4	2	no	—
138	Dic₈₉	—	8	4	no	—
139	C₂×#1674	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	#1674	1	4	yes	complemented
141	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
142	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
143	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
144	C₂×#500	—	89	1	no	—
145	C₂×#462	—	89	2	no	—
146	C₂×C₈×Q₈	—	89	1	no	—
147	#1674	—	89	8	no	—
148	C₂×#335	D₈₉	1	1	yes	characteristic
149	C₂×#332	—	89	2	no	—
150	C₈₉	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	178	2	no	—
152	C₂×C₄⋊Q₈	—	89	2	no	—
153	C₂×C₄×Q₈	—	89	1	no	—
154	#500	—	89	4	no	—
155	#500	—	178	2	no	—
156	#462	—	89	8	no	—
157	#462	—	178	4	no	—
158	C₈×Q₈	—	89	4	no	—
159	C₈×Q₈	—	178	2	no	—
160	C₂×#107	—	178	1	no	—
161	C₂×#107	—	178	1	no	—
162	C₂×C₄⋊C₈	—	89	1	no	—
163	C₂×C₄⋊C₈	—	178	1	no	—
164	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
165	C₂×#103	—	178	2	no	—
166	C₂×C₄×C₈	—	178	1	no	—
167	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
168	#335	C₂×D₈₉	1	4	yes	—
169	#332	—	89	8	no	—
170	C₂²×Q₈	—	89	1	no	—
171	C₂²×Q₈	—	178	2	no	—
172	C₂×Q₁₆	—	356	8	no	—
173	C₂×Q₁₆	—	178	8	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	178	1	no	—
176	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
177	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
178	C₄⋊Q₈	—	89	4	no	—
179	C₄⋊Q₈	—	178	2	no	—
180	C₄⋊Q₈	—	178	2	no	—
181	C₄⋊Q₈	—	178	2	no	—
182	C₄×Q₈	—	89	2	no	—
183	C₄×Q₈	—	89	2	no	—
184	C₄×Q₈	—	178	1	no	—
185	C₄×Q₈	—	178	1	no	—
186	C₂×C₄⋊C₄	—	89	1	no	—
187	C₂×C₄⋊C₄	—	178	2	no	—
188	C₂×C₄⋊C₄	—	178	2	no	—
189	C₂×C₄⋊C₄	—	178	1	no	—
190	C₂×C₄²	—	178	1	no	—
191	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
192	#107	—	178	4	no	—
193	#107	—	178	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	89	2	no	—
196	C₄⋊C₈	—	178	4	no	—
197	C₄⋊C₈	—	178	1	no	—
198	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
199	#103	—	178	8	no	—
200	C₄×C₈	—	178	4	no	—
201	C₄×C₈	D₃₅₆	1	2	yes	—
202	C₄×C₈	C₂²×D₈₉	1	2	yes	—
203	C₂×Q₈	—	89	4	no	—
204	C₂×Q₈	—	89	2	no	—
205	C₂×Q₈	—	356	2	no	—
206	C₂×Q₈	—	356	4	no	—
207	C₂×Q₈	—	356	4	no	—
208	C₂×Q₈	—	178	4	no	—
209	C₂×Q₈	—	178	1	no	—
210	C₂×Q₈	—	178	2	no	—
211	C₂²×C₄	—	356	2	no	—
212	C₂²×C₄	—	178	1	no	—
213	C₂²×C₄	—	178	1	no	—
214	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂²×C₄	#13083575	1	1	yes	characteristic
217	Q₁₆	—	356	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	356	2	no	—
222	C₂×C₈	—	178	2	no	—
223	C₂×C₈	Dic₃₅₆	1	4	yes	complemented
224	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
225	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
226	C₄⋊C₄	—	89	2	no	—
227	C₄⋊C₄	—	356	2	no	—
228	C₄⋊C₄	—	356	2	no	—
229	C₄⋊C₄	—	178	4	no	—
230	C₄⋊C₄	—	178	4	no	—
231	C₄⋊C₄	—	178	1	no	—
232	C₄⋊C₄	—	178	2	no	—
233	C₄⋊C₄	—	178	2	no	—
234	C₄²	—	178	2	no	—
235	C₄²	—	178	2	no	—
236	C₄²	D₄×D₈₉	1	1	yes	characteristic
237	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
238	C₄²	#13083575	1	1	yes	characteristic
239	C₄²	C₂³×D₈₉	1	1	yes	characteristic
240	C₂³	#15667987	1	1	yes	characteristic
241	Q₈	—	89	4	no	—
242	Q₈	—	356	4	no	—
243	Q₈	—	356	4	no	—
244	Q₈	—	178	4	no	—
245	Q₈	—	178	2	no	—
246	Q₈	—	712	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	356	2	no	—
249	C₂×C₄	—	356	4	no	—
250	C₂×C₄	—	178	2	no	—
251	C₂×C₄	—	178	4	no	—
252	C₂×C₄	—	178	2	no	—
253	C₂×C₄	—	712	4	no	—
254	C₂×C₄	#15667987	1	1	yes	characteristic
255	C₂×C₄	#15667987	1	1	yes	characteristic
256	C₂×C₄	#15667987	1	1	yes	characteristic
257	C₂×C₄	D₈×D₈₉	1	2	yes	—
258	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
259	C₂×C₄	#15668035	1	2	yes	—
260	C₂×C₄	#15668045	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
263	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	4	no	—
266	C₈	—	356	2	no	—
267	C₈	C₂×Dic₃₅₆	1	4	yes	complemented
268	C₂²	#19151809	1	2	yes	—
269	C₂²	#19151835	1	2	yes	—
270	C₂²	#19151874	1	2	yes	—
271	C₂²	C₂×#15667987	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	356	2	no	—
275	C₄	—	178	4	no	—
276	C₄	—	712	4	no	—
277	C₄	#19151809	1	2	yes	—
278	C₄	#19151835	1	2	yes	—
279	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
280	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
281	C₄	C₂×#15668035	1	1	yes	characteristic
282	C₄	C₂×#15668045	1	1	yes	characteristic
283	C₂	#26019647	1	4	yes	direct product factor
284	C₂	C₂×#19151809	1	1	yes	characteristic
285	C₂	C₂×#19151835	1	1	yes	characteristic
286	C₂	C₂×#19151874	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	C₂×#1674
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #26019647	C₂	direct (N × H)
89. C₂×C₈×C₈₉	Q₁₆	semidirect
122. C₈×C₈₉	C₂×Q₁₆	semidirect
140. C₂×C₈₉	#1674	semidirect
150. C₈₉	C₂×#1674	semidirect
223. C₂×C₈	Dic₃₅₆	semidirect
267. C₈	C₂×Dic₃₅₆	semidirect
283. C₂	#26019647	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×C₈×Dic₁₇₈	2
#26019647	2
C₂×#19151345	2
C₂×#335×C₈₉	2
C₂×#1674	89

Subgroups — C2×#26019647 #36536785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019647 #36536785