Subgroups — C₂×#26022426 #36709162

Order 22784.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151998	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151715	C₂	1	1	yes	maximal, characteristic
3	#26022426	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151421	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151408	C₂	1	1	yes	maximal, characteristic
8	C₂×#360×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15668019	—	2	1	no	—
10	C₂×#15668003	—	2	1	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	#19151998	—	2	2	no	—
13	#19151998	C₂²	1	4	yes	complemented
14	#19151715	—	2	2	no	—
15	#19151715	C₂²	1	4	yes	—
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	C₂×#15667945	—	2	1	no	—
18	C₂×#15667944	C₂²	1	1	yes	characteristic
19	C₂×#15667939	—	2	1	no	—
20	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
21	#19151421	C₂²	1	4	yes	complemented
22	#19151420	C₂²	1	4	yes	complemented
23	#19151418	C₂²	1	4	yes	complemented
24	#19151408	C₂²	1	4	yes	—
25	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
28	#360×C₈₉	C₂²	1	4	yes	—
29	#15668019	—	2	4	no	—
30	#15668019	—	4	2	no	—
31	C₂×#13083558	—	2	1	no	—
32	C₂×#13083558	—	4	1	no	—
33	C₂²×#10623494	—	4	1	no	—
34	C₂²×#10623494	D₄	1	1	yes	characteristic
35	#15668003	—	2	4	no	—
36	#15668003	—	4	2	no	—
37	C₂×C₄⋊Dic₈₉	—	2	1	no	—
38	C₂×C₄⋊Dic₈₉	—	4	1	no	—
39	C₂×#13083553	—	4	1	no	—
40	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
41	#15667952	—	2	1	no	—
42	#15667952	C₂³	1	2	yes	—
43	C₂×#13083549	—	2	1	no	—
44	#15667948	—	2	1	no	—
45	#15667948	C₂³	1	2	yes	—
46	#15667945	—	2	4	no	—
47	#15667944	—	2	1	no	—
48	#15667944	C₂³	1	2	yes	—
49	#15667939	—	2	4	no	—
50	C₈×Dic₈₉	D₄	1	2	yes	—
51	C₈×Dic₈₉	C₂³	1	2	yes	—
52	C₂×#41×C₈₉	—	2	1	no	—
53	#135×C₈₉	—	2	1	no	—
54	#135×C₈₉	—	2	1	no	—
55	#135×C₈₉	Q₈	1	2	yes	—
56	#135×C₈₉	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic
58	C₂×SD₁₆×C₈₉	—	2	1	no	—
59	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
62	M₃₂×C₈₉	—	2	1	no	—
63	M₃₂×C₈₉	C₂³	1	2	yes	—
64	C₂×C₁₆×C₈₉	—	2	1	no	—
65	#106×C₈₉	—	2	1	no	—
66	#106×C₈₉	C₂³	1	2	yes	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	4	1	no	—
69	#13083558	—	2	2	no	—
70	#13083558	—	4	1	no	—
71	#13083558	—	4	4	no	—
72	C₂×#10623494	—	4	1	no	—
73	C₂×#10623494	—	4	4	no	—
74	C₂×#10623494	SD₁₆	1	1	yes	characteristic
75	C₂×#10623494	SD₁₆	1	1	yes	characteristic
76	C₂×#10623494	SD₁₆	1	2	yes	—
77	C₂×#10623494	C₂×D₄	1	2	yes	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	4	4	no	—
80	C₄⋊Dic₈₉	—	4	1	no	—
81	#13083553	—	4	4	no	—
82	C₄×Dic₈₉	—	2	1	no	—
83	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
84	#13083549	—	2	2	no	—
85	#41×C₈₉	—	2	2	no	—
86	#41×C₈₉	—	2	2	no	—
87	#41×C₈₉	—	4	1	no	—
88	#41×C₈₉	—	4	1	no	—
89	C₂×Q₈×C₈₉	—	2	1	no	—
90	C₂×D₄×C₈₉	—	2	1	no	—
91	C₂×D₄×C₈₉	—	4	1	no	—
92	C₂²×C₄×C₈₉	—	4	1	no	—
93	C₂²×C₄×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₈₉	—	2	1	no	—
96	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
97	SD₁₆×C₈₉	—	2	2	no	—
98	SD₁₆×C₈₉	—	2	2	no	—
99	D₈×C₈₉	—	2	1	no	—
100	D₈×C₈₉	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₁₆×C₈₉	—	2	2	no	—
110	C₂³×C₈₉	—	4	1	no	—
111	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	Q₈×C₈₉	—	2	2	no	—
113	Q₈×C₈₉	—	4	1	no	—
114	D₄×C₈₉	—	2	2	no	—
115	D₄×C₈₉	—	4	2	no	—
116	D₄×C₈₉	—	4	2	no	—
117	D₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	8	2	no	—
133	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
137	#10623494	—	4	2	no	—
138	#10623494	C₂×SD₁₆	1	2	yes	—
139	#10623494	C₂×SD₁₆	1	2	yes	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	—	4	2	no	—
142	C₂²×C₈₉	—	8	2	no	—
143	C₂²×C₈₉	#476	1	1	yes	characteristic
144	C₂²×C₈₉	#476	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₄×C₈₉	—	4	2	no	—
147	C₄×C₈₉	—	8	2	no	—
148	C₄×C₈₉	#476	1	1	yes	characteristic
149	C₄×C₈₉	#476	1	1	yes	characteristic
150	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
152	Dic₈₉	—	4	2	no	—
153	Dic₈₉	—	8	2	no	—
154	C₂×#2231	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	2	1	no	—
156	C₂×C₈₉	—	2	1	no	—
157	C₂×C₈₉	—	8	2	no	—
158	C₂×C₈₉	#2231	1	2	yes	complemented
159	C₂×C₈₉	C₂×#476	1	1	yes	characteristic
160	C₂×#499	—	89	1	no	—
161	C₂×#443	—	89	1	no	—
162	#2231	—	89	8	no	—
163	C₂×#365	—	89	1	no	—
164	C₂×#365	—	89	1	no	—
165	C₂×#364	—	89	1	no	—
166	C₂×#360	—	89	1	no	—
167	C₂×#360	D₈₉	1	1	yes	characteristic
168	C₈₉	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
169	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
170	C₂×#131	—	178	1	no	—
171	C₂×#125	—	178	1	no	—
172	C₂×M₃₂	—	89	1	no	—
173	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
174	#499	—	89	4	no	—
175	#499	—	178	2	no	—
176	#443	—	89	4	no	—
177	#443	—	178	2	no	—
178	C₂×#108	—	89	1	no	—
179	C₂×#107	—	178	1	no	—
180	C₂×#106	—	89	1	no	—
181	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
182	C₂×#104	—	178	1	no	—
183	C₂×C₄×C₈	—	89	1	no	—
184	#365	—	89	4	no	—
185	#365	—	89	4	no	—
186	#364	—	89	4	no	—
187	#360	—	89	4	no	—
188	#360	C₂×D₈₉	1	4	yes	—
189	C₂×#41	—	2	1	no	—
190	#135	—	2	1	no	—
191	#135	—	2	1	no	—
192	#135	Dic₁₇₈	1	2	yes	—
193	#135	C₂²×D₈₉	1	2	yes	—
194	C₂×Q₁₆	Dic₁₇₈	1	1	yes	characteristic
195	C₂×SD₁₆	—	2	1	no	—
196	C₂×D₈	Dic₁₇₈	1	1	yes	characteristic
197	#131	—	356	2	no	—
198	#131	—	178	4	no	—
199	C₂×M₁₆	—	356	1	no	—
200	C₂×M₁₆	—	178	1	no	—
201	C₂²×C₈	—	89	1	no	—
202	C₂²×C₈	—	356	1	no	—
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	#125	—	356	2	no	—
205	#125	—	178	4	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	356	1	no	—
208	C₂×C₄⋊C₄	—	356	1	no	—
209	C₂×C₄⋊C₄	—	178	1	no	—
210	C₂×C₄²	—	89	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	89	2	no	—
213	M₃₂	—	178	1	no	—
214	M₃₂	C₂²×D₈₉	1	2	yes	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	178	1	no	—
217	#108	—	89	2	no	—
218	#108	—	178	1	no	—
219	#107	—	178	4	no	—
220	#106	—	2	1	no	—
221	#106	—	89	2	no	—
222	#106	—	178	1	no	—
223	#106	C₂²×D₈₉	1	2	yes	—
224	#104	—	178	4	no	—
225	C₄×C₈	—	89	2	no	—
226	C₄×C₈	—	89	2	no	—
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	356	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
239	Q₁₆	—	2	1	no	—
240	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	2	2	no	—
243	D₈	—	2	1	no	—
244	D₈	C₂×Dic₁₇₈	1	2	yes	—
245	M₁₆	—	356	4	no	—
246	M₁₆	—	356	1	no	—
247	M₁₆	—	178	2	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	—	89	2	no	—
252	C₂×C₈	—	356	4	no	—
253	C₂×C₈	—	356	1	no	—
254	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
256	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
257	C₂×C₈	#13083573	1	1	yes	characteristic
258	C₂×C₈	#13083573	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	356	4	no	—
263	C₄⋊C₄	—	356	4	no	—
264	C₄⋊C₄	—	356	1	no	—
265	C₄⋊C₄	—	178	2	no	—
266	C₄²	—	89	2	no	—
267	C₄²	—	178	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	178	2	no	—
270	C₂³	—	4	1	no	—
271	C₂³	#15668036	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	2	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	356	2	no	—
284	C₂×C₄	—	356	2	no	—
285	C₂×C₄	—	178	2	no	—
286	C₂×C₄	—	712	2	no	—
287	C₂×C₄	#15667989	1	1	yes	characteristic
288	C₂×C₄	#15667989	1	1	yes	characteristic
289	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
291	C₂×C₄	#15668036	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
293	C₈	—	89	2	no	—
294	C₈	—	89	2	no	—
295	C₈	—	356	2	no	—
296	C₈	#15667989	1	1	yes	characteristic
297	C₈	#15667989	1	1	yes	characteristic
298	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
299	C₈	C₂×#13083573	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#19151886	1	1	yes	characteristic
304	C₂²	#19151886	1	1	yes	characteristic
305	C₂²	C₂×#15668036	1	1	yes	characteristic
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	356	2	no	—
309	C₄	—	712	2	no	—
310	C₄	#19151886	1	1	yes	characteristic
311	C₄	#19151886	1	1	yes	characteristic
312	C₄	C₂×#15667989	1	1	yes	characteristic
313	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	8	2	no	—
317	C₂	#26022426	1	2	yes	direct product factor
318	C₂	C₂×#19151886	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151886
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#360×C₈₉	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	C₂×#360	D₈₉
hypocenter	C₈₉	C₂×#2231
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#19151998	C₂	semidirect
3. #26022426	C₂	direct (N × H)
4. C₂×#19151421	C₂	semidirect
5. C₂×#19151420	C₂	semidirect
6. C₂×#19151418	C₂	semidirect
13. #19151998	C₂²	semidirect
21. #19151421	C₂²	semidirect
22. #19151420	C₂²	semidirect
23. #19151418	C₂²	semidirect
158. C₂×C₈₉	#2231	semidirect
168. C₈₉	C₂×#2231	semidirect
317. C₂	#26022426	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×#19151998	2
C₂×#19151715	2
#26022426	2
C₂×#19151421	2
C₂×#19151420	2
C₂×#19151418	2
C₂×#19151408	2
C₂×#360×C₈₉	2
C₂×#2231	89

Subgroups — C2×#26022426 #36709162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022426 #36709162