Subgroups — C₂×#26022377 #36708991

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151873	C₂	1	2	yes	maximal
3	#26022377	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151415	C₂	1	2	yes	maximal, complemented
6	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668042	—	2	2	no	—
8	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
9	C₂×Dic₇₁₂	—	2	1	no	—
10	C₂×Dic₇₁₂	—	2	1	no	—
11	C₂×Dic₇₁₂	—	2	2	no	—
12	C₂×Dic₇₁₂	C₂²	1	4	yes	complemented
13	#19151873	—	2	4	no	—
14	#19151873	C₂²	1	8	yes	—
15	C₂×#15667948	C₂²	1	2	yes	—
16	C₂×#15667930	—	2	2	no	—
17	#19151428	C₂²	1	4	yes	complemented
18	#19151415	C₂²	1	8	yes	complemented
19	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
20	#504×C₈₉	—	2	2	no	—
21	#504×C₈₉	C₂²	1	4	yes	—
22	C₂×M₃₂×C₈₉	—	2	1	no	—
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₉	—	2	1	no	—
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	—	2	2	no	—
27	#15668042	—	2	8	no	—
28	#15668042	—	4	4	no	—
29	C₂×Dic₃₅₆	—	2	2	no	—
30	C₂×Dic₃₅₆	—	2	2	no	—
31	C₂×Dic₃₅₆	—	2	2	no	—
32	C₂×Dic₃₅₆	—	2	2	no	—
33	C₂×Dic₃₅₆	—	2	2	no	—
34	C₂×Dic₃₅₆	C₂³	1	4	yes	—
35	C₂×#13083564	—	4	2	no	—
36	C₂×#13083562	—	4	2	no	—
37	C₂×#13083558	—	2	2	no	—
38	Dic₇₁₂	—	2	4	no	—
39	Dic₇₁₂	—	2	4	no	—
40	#15667948	—	2	2	no	—
41	#15667948	C₂³	1	4	yes	—
42	#15667930	—	2	8	no	—
43	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
44	#131×C₈₉	—	2	1	no	—
45	#131×C₈₉	—	2	1	no	—
46	#131×C₈₉	D₄	1	2	yes	—
47	#131×C₈₉	C₂³	1	2	yes	—
48	C₂×M₁₆×C₈₉	—	2	1	no	—
49	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	—	2	4	no	—
54	M₃₂×C₈₉	C₂³	1	2	yes	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	—	2	4	no	—
57	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×Dic₁₇₈	—	2	4	no	—
62	C₂×Dic₁₇₈	—	4	2	no	—
63	C₂×Dic₁₇₈	—	4	2	no	—
64	C₂×Dic₁₇₈	—	4	2	no	—
65	C₂×Dic₁₇₈	—	4	2	no	—
66	C₂×Dic₁₇₈	—	4	2	no	—
67	C₂²×Dic₈₉	—	4	2	no	—
68	Dic₃₅₆	—	2	4	no	—
69	Dic₃₅₆	—	2	4	no	—
70	Dic₃₅₆	—	4	2	no	—
71	Dic₃₅₆	—	4	2	no	—
72	#13083564	—	4	8	no	—
73	#13083562	—	4	8	no	—
74	#13083558	—	2	4	no	—
75	#13083558	—	4	2	no	—
76	C₂×#10623494	—	4	2	no	—
77	#41×C₈₉	—	2	1	no	—
78	#41×C₈₉	—	2	1	no	—
79	#41×C₈₉	D₈	1	2	yes	—
80	#41×C₈₉	C₂×D₄	1	2	yes	—
81	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
82	C₂×D₄×C₈₉	—	2	1	no	—
83	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	—	2	1	no	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	M₁₆×C₈₉	—	2	1	no	—
87	M₁₆×C₈₉	—	2	2	no	—
88	M₁₆×C₈₉	—	2	2	no	—
89	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₁₆×C₈₉	—	2	2	no	—
100	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
101	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
102	C₂³×C₈₉	—	2	1	no	—
103	C₂³×C₈₉	#135	1	1	yes	characteristic
104	Q₈×C₈₉	—	2	1	no	—
105	Q₈×C₈₉	C₂×D₈	1	2	yes	—
106	D₄×C₈₉	—	2	2	no	—
107	D₄×C₈₉	—	2	2	no	—
108	D₄×C₈₉	—	2	1	no	—
109	D₄×C₈₉	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₈₉	—	4	4	no	—
120	C₂×Dic₈₉	—	8	2	no	—
121	C₂×Dic₈₉	—	8	2	no	—
122	Dic₁₇₈	—	4	4	no	—
123	Dic₁₇₈	—	4	4	no	—
124	Dic₁₇₈	—	8	2	no	—
125	Dic₁₇₈	—	8	2	no	—
126	C₈×C₈₉	—	2	2	no	—
127	C₈×C₈₉	#121	1	1	yes	characteristic
128	C₈×C₈₉	#121	1	1	yes	characteristic
129	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
131	#10623494	—	4	4	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	4	2	no	—
135	C₂²×C₈₉	#466	1	1	yes	characteristic
136	C₂²×C₈₉	#466	1	1	yes	characteristic
137	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	#466	1	1	yes	characteristic
140	C₄×C₈₉	#466	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
142	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
143	Dic₈₉	—	8	4	no	—
144	C₂×#2222	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	—	4	2	no	—
148	C₂×C₈₉	#2222	1	2	yes	complemented
149	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
150	C₂²×Q₃₂	—	89	1	no	—
151	C₂×#504	D₈₉	1	1	yes	characteristic
152	C₂×#473	—	89	2	no	—
153	#2222	—	89	8	no	—
154	C₂×#368	—	89	1	no	—
155	C₂×#362	—	89	2	no	—
156	C₈₉	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
157	C₂×#137	—	178	2	no	—
158	C₂²×Q₁₆	—	89	2	no	—
159	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
160	C₂×Q₃₂	—	89	4	no	—
161	C₂×Q₃₂	—	178	1	no	—
162	C₂×Q₃₂	—	178	1	no	—
163	C₂×Q₃₂	—	178	2	no	—
164	#504	—	2	2	no	—
165	#504	C₂×D₈₉	1	4	yes	—
166	C₂×M₃₂	—	2	1	no	—
167	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
168	C₂²×C₁₆	—	2	1	no	—
169	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
170	#473	—	89	8	no	—
171	#473	—	178	4	no	—
172	C₂×#108	—	89	2	no	—
173	C₂×#101	—	178	2	no	—
174	#368	—	89	4	no	—
175	#362	—	89	8	no	—
176	C₂×#41	D₃₅₆	1	1	yes	characteristic
177	C₂²×Q₈	—	178	2	no	—
178	#137	—	356	4	no	—
179	#137	—	178	8	no	—
180	C₂×Q₁₆	—	89	4	no	—
181	C₂×Q₁₆	—	356	2	no	—
182	C₂×Q₁₆	—	178	2	no	—
183	C₂×Q₁₆	—	178	2	no	—
184	C₂×Q₁₆	—	178	2	no	—
185	C₂×Q₁₆	—	178	2	no	—
186	C₂×Q₁₆	—	178	2	no	—
187	C₂×SD₁₆	—	356	2	no	—
188	#131	—	2	1	no	—
189	#131	—	2	1	no	—
190	#131	D₃₅₆	1	2	yes	—
191	#131	C₂²×D₈₉	1	2	yes	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	178	2	no	—
194	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
197	Q₃₂	—	178	4	no	—
198	Q₃₂	—	178	4	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	4	no	—
201	M₃₂	C₂²×D₈₉	1	2	yes	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
207	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
208	#108	—	89	4	no	—
209	#108	—	178	2	no	—
210	#101	—	178	8	no	—
211	#41	—	2	1	no	—
212	#41	—	2	1	no	—
213	#41	D₇₁₂	1	2	yes	—
214	#41	C₂×D₃₅₆	1	2	yes	—
215	C₂×Q₈	—	356	2	no	—
216	C₂×Q₈	—	356	2	no	—
217	C₂×Q₈	—	356	2	no	—
218	C₂×Q₈	—	356	2	no	—
219	C₂×Q₈	—	356	2	no	—
220	C₂×Q₈	—	178	4	no	—
221	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	D₇₁₂	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	356	2	no	—
226	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
227	Q₁₆	—	356	8	no	—
228	Q₁₆	—	356	2	no	—
229	Q₁₆	—	356	2	no	—
230	Q₁₆	—	178	4	no	—
231	Q₁₆	—	178	4	no	—
232	SD₁₆	—	356	8	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	356	2	no	—
237	M₁₆	—	178	4	no	—
238	M₁₆	C₂×D₃₅₆	1	2	yes	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	356	2	no	—
243	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
246	C₂×C₈	#13083574	1	1	yes	characteristic
247	C₂×C₈	#13083574	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
249	C₁₆	—	2	2	no	—
250	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
251	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
252	C₂³	—	2	1	no	—
253	C₂³	#15668034	1	1	yes	characteristic
254	Q₈	—	2	1	no	—
255	Q₈	—	356	4	no	—
256	Q₈	—	356	4	no	—
257	Q₈	—	712	2	no	—
258	Q₈	—	712	2	no	—
259	Q₈	C₂×D₇₁₂	1	2	yes	—
260	D₄	—	2	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	2	1	no	—
263	D₄	C₂×D₇₁₂	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	356	4	no	—
268	C₂×C₄	—	712	2	no	—
269	C₂×C₄	—	712	2	no	—
270	C₂×C₄	#15667986	1	1	yes	characteristic
271	C₂×C₄	#15667986	1	1	yes	characteristic
272	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
273	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
274	C₂×C₄	#15668034	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	356	4	no	—
278	C₈	#15667986	1	1	yes	characteristic
279	C₈	#15667986	1	1	yes	characteristic
280	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
281	C₈	C₂×#13083574	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	#19151834	1	1	yes	characteristic
286	C₂²	#19151834	1	1	yes	characteristic
287	C₂²	C₂×#15668034	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	712	4	no	—
290	C₄	#19151834	1	1	yes	characteristic
291	C₄	#19151834	1	1	yes	characteristic
292	C₄	C₂×#15667986	1	1	yes	characteristic
293	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	#26022377	1	2	yes	direct product factor
298	C₂	C₂×#19151834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151834
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂×#41×C₈₉	D₄
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	C₂×#2222
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₇₁₂	C₂	semidirect
3. #26022377	C₂	direct (N × H)
4. C₂×#19151428	C₂	semidirect
5. C₂×#19151415	C₂	semidirect
12. C₂×Dic₇₁₂	C₂²	semidirect
17. #19151428	C₂²	semidirect
18. #19151415	C₂²	semidirect
148. C₂×C₈₉	#2222	semidirect
156. C₈₉	C₂×#2222	semidirect
297. C₂	#26022377	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×Dic₇₁₂	2
C₂×#19151873	2
#26022377	2
C₂×#19151428	2
C₂×#19151415	2
C₂×#504×C₈₉	2
C₂×#2222	89

Subgroups — C2×#26022377 #36708991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022377 #36708991