Subgroups — C₂×#26022423 #36709158

Order 22784.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151996	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022423	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151422	C₂	1	2	yes	maximal, complemented
4	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151408	C₂	1	2	yes	maximal
6	C₂×#443×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668012	—	2	1	no	—
8	C₂×#15667952	C₂²	1	2	yes	—
9	#19151996	—	2	2	no	—
10	#19151996	C₂²	1	4	yes	complemented
11	C₂×#15667944	—	2	1	no	—
12	C₂×#15667944	C₂²	1	2	yes	—
13	#19151422	C₂²	1	8	yes	complemented
14	#19151419	C₂²	1	4	yes	complemented
15	#19151408	C₂²	1	8	yes	—
16	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#131×C₈₉	—	2	1	no	—
18	#443×C₈₉	—	2	2	no	—
19	#443×C₈₉	C₂²	1	4	yes	—
20	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#104×C₈₉	—	2	1	no	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₇₈	—	4	1	no	—
24	#15668012	—	2	4	no	—
25	#15668012	—	4	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	4	1	no	—
28	#15667952	—	2	2	no	—
29	#15667952	C₂³	1	4	yes	—
30	C₂×#13083549	—	2	2	no	—
31	#15667944	—	2	4	no	—
32	#15667944	—	2	2	no	—
33	#15667944	C₂³	1	4	yes	—
34	C₂×#41×C₈₉	—	2	1	no	—
35	#135×C₈₉	—	2	1	no	—
36	#135×C₈₉	—	2	1	no	—
37	#135×C₈₉	Q₈	1	2	yes	complemented
38	#135×C₈₉	C₂³	1	2	yes	—
39	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic, complemented
40	C₂×SD₁₆×C₈₉	—	2	1	no	—
41	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic, complemented
42	#131×C₈₉	—	2	4	no	—
43	#131×C₈₉	—	4	2	no	—
44	C₂×M₁₆×C₈₉	—	2	1	no	—
45	C₂×M₁₆×C₈₉	—	4	1	no	—
46	C₂²×C₈×C₈₉	—	4	1	no	—
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	#108×C₈₉	—	2	1	no	—
51	#108×C₈₉	C₂³	1	2	yes	—
52	#104×C₈₉	—	2	4	no	—
53	C₄×C₈×C₈₉	D₄	1	2	yes	—
54	C₄×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂×Dic₁₇₈	—	4	4	no	—
56	C₂×Dic₁₇₈	—	8	4	no	—
57	C₂²×Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	4	4	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	#13083549	—	2	4	no	—
62	#41×C₈₉	—	2	2	no	—
63	#41×C₈₉	—	2	2	no	—
64	#41×C₈₉	—	4	1	no	—
65	#41×C₈₉	—	4	1	no	—
66	C₂×Q₈×C₈₉	—	2	1	no	—
67	C₂×D₄×C₈₉	—	2	1	no	—
68	C₂×D₄×C₈₉	—	4	1	no	—
69	C₂²×C₄×C₈₉	—	2	1	no	—
70	C₂²×C₄×C₈₉	—	4	1	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₈₉	—	2	1	no	—
73	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
74	SD₁₆×C₈₉	—	2	2	no	—
75	SD₁₆×C₈₉	—	2	2	no	—
76	D₈×C₈₉	—	2	1	no	—
77	D₈×C₈₉	C₂×Q₈	1	2	yes	complemented
78	M₁₆×C₈₉	—	2	2	no	—
79	M₁₆×C₈₉	—	4	1	no	—
80	M₁₆×C₈₉	—	4	4	no	—
81	C₂×C₈×C₈₉	—	4	1	no	—
82	C₂×C₈×C₈₉	—	4	4	no	—
83	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
84	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
85	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₄²×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	C₂×D₄	1	2	yes	—
95	C₂³×C₈₉	—	4	1	no	—
96	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	Q₈×C₈₉	—	2	2	no	—
98	Q₈×C₈₉	—	4	1	no	—
99	D₄×C₈₉	—	2	2	no	—
100	D₄×C₈₉	—	4	2	no	—
101	D₄×C₈₉	—	4	2	no	—
102	D₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₈₉	—	4	4	no	—
115	C₂×Dic₈₉	—	8	4	no	—
116	Dic₁₇₈	—	8	8	no	—
117	C₈×C₈₉	—	4	2	no	—
118	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
121	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
122	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
124	C₂²×C₈₉	—	2	2	no	—
125	C₂²×C₈₉	—	4	2	no	—
126	C₂²×C₈₉	—	8	2	no	—
127	C₂²×C₈₉	#476	1	1	yes	characteristic
128	C₂²×C₈₉	#476	1	1	yes	characteristic
129	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₄×C₈₉	—	2	4	no	—
131	C₄×C₈₉	—	4	2	no	—
132	C₄×C₈₉	#476	1	1	yes	characteristic
133	C₄×C₈₉	#476	1	1	yes	characteristic
134	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
135	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₈₉	—	8	4	no	—
137	C₂×#2230	—	89	1	no	maximal, Sylow
138	C₂×C₈₉	—	2	1	no	—
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	8	2	no	—
141	C₂×C₈₉	#2230	1	2	yes	complemented
142	C₂×C₈₉	C₂×#476	1	1	yes	characteristic
143	C₂×#498	—	89	1	no	—
144	C₂×#443	D₈₉	1	1	yes	characteristic
145	#2230	—	89	8	no	—
146	C₂×#365	—	89	2	no	—
147	C₂×#364	—	89	1	no	—
148	C₂×#360	—	89	2	no	—
149	C₈₉	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
150	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
151	C₂×#131	—	2	1	no	—
152	C₂×C₄⋊Q₈	—	178	1	no	—
153	C₂×M₃₂	—	89	2	no	—
154	#498	—	89	4	no	—
155	#498	—	178	2	no	—
156	#443	—	2	2	no	—
157	#443	C₂×D₈₉	1	4	yes	—
158	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
159	C₂×#106	—	89	2	no	—
160	C₂×#106	—	178	1	no	—
161	C₂×#104	—	2	1	no	—
162	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
163	#365	—	89	8	no	—
164	#364	—	89	4	no	—
165	#360	—	89	8	no	—
166	C₂×#41	—	2	1	no	—
167	C₂²×Q₈	—	356	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	Dic₁₇₈	1	2	yes	complemented
171	#135	C₂²×D₈₉	1	2	yes	—
172	C₂×Q₁₆	Dic₁₇₈	1	1	yes	characteristic, complemented
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	Dic₁₇₈	1	1	yes	characteristic, complemented
175	#131	—	2	4	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	4	1	no	—
179	C₂²×C₈	—	4	1	no	—
180	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
182	C₄⋊Q₈	—	356	2	no	—
183	C₄⋊Q₈	—	178	4	no	—
184	C₂×C₄⋊C₄	—	356	1	no	—
185	C₂×C₄⋊C₄	—	178	2	no	—
186	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
187	M₃₂	—	89	4	no	—
188	M₃₂	—	178	2	no	—
189	C₂×C₁₆	—	178	2	no	—
190	#108	—	2	1	no	—
191	#108	C₂²×D₈₉	1	2	yes	—
192	#106	—	89	4	no	—
193	#106	—	178	2	no	—
194	#106	—	178	4	no	—
195	#104	—	2	4	no	—
196	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
197	C₄×C₈	C₂²×D₈₉	1	2	yes	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	356	4	no	—
204	C₂×Q₈	—	712	4	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	356	2	no	—
210	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
211	Q₁₆	—	2	1	no	—
212	Q₁₆	C₂×Dic₁₇₈	1	2	yes	complemented
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	2	no	—
215	D₈	—	2	1	no	—
216	D₈	C₂×Dic₁₇₈	1	2	yes	complemented
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	#13083562	1	1	yes	characteristic
223	C₂×C₈	#13083562	1	1	yes	characteristic
224	C₂×C₈	#13083562	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
227	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
228	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
229	C₂×C₈	#13083574	1	1	yes	characteristic
230	C₂×C₈	#13083574	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
232	C₄⋊C₄	—	356	4	no	—
233	C₄⋊C₄	—	356	2	no	—
234	C₄⋊C₄	—	178	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
237	C₁₆	—	178	4	no	—
238	C₂³	—	4	1	no	—
239	C₂³	#15668037	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	—	712	8	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	356	4	no	—
253	C₂×C₄	—	712	4	no	—
254	C₂×C₄	#15667991	1	1	yes	characteristic
255	C₂×C₄	#15667991	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
257	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
258	C₂×C₄	#15668037	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	#15667991	1	1	yes	characteristic
262	C₈	#15667991	1	1	yes	characteristic
263	C₈	C₂×#13083562	1	2	yes	—
264	C₈	C₂×#13083562	1	2	yes	—
265	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
266	C₈	C₂×#13083574	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	#19151888	1	1	yes	characteristic
271	C₂²	#19151888	1	1	yes	characteristic
272	C₂²	C₂×#15668037	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	712	4	no	—
276	C₄	#19151888	1	1	yes	characteristic
277	C₄	#19151888	1	1	yes	characteristic
278	C₄	C₂×#15667991	1	1	yes	characteristic
279	C₄	C₂²×#13083562	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	#26022423	1	2	yes	direct product factor
284	C₂	C₂×#19151888	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151888
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#443×C₈₉	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	C₂×#443	D₈₉
hypocenter	C₈₉	C₂×#2230
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#19151996	C₂	semidirect
2. #26022423	C₂	direct (N × H)
3. C₂×#19151422	C₂	semidirect
4. C₂×#19151419	C₂	semidirect
10. #19151996	C₂²	semidirect
13. #19151422	C₂²	semidirect
14. #19151419	C₂²	semidirect
37. #135×C₈₉	Q₈	semidirect
39. C₂×Q₁₆×C₈₉	Q₈	semidirect
41. C₂×D₈×C₈₉	Q₈	semidirect
73. Q₁₆×C₈₉	C₂×Q₈	semidirect
77. D₈×C₈₉	C₂×Q₈	semidirect
141. C₂×C₈₉	#2230	semidirect
149. C₈₉	C₂×#2230	semidirect
170. #135	Dic₁₇₈	semidirect
172. C₂×Q₁₆	Dic₁₇₈	semidirect
174. C₂×D₈	Dic₁₇₈	semidirect
212. Q₁₆	C₂×Dic₁₇₈	semidirect
216. D₈	C₂×Dic₁₇₈	semidirect
283. C₂	#26022423	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×#19151996	2
#26022423	2
C₂×#19151422	2
C₂×#19151419	2
C₂×#19151408	2
C₂×#443×C₈₉	2
C₂×#2230	89

Subgroups — C2×#26022423 #36709158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022423 #36709158