Subgroups — C₂×#26019614 #36536660

Order 22784.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151875	C₂	1	2	yes	maximal
2	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019614	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151344	C₂	1	2	yes	maximal, complemented
6	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₉	—	2	1	no	—
8	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	#19151875	—	2	4	no	—
10	#19151875	C₂²	1	8	yes	—
11	C₈×D₃₅₆	—	2	2	no	—
12	C₈×D₃₅₆	C₂²	1	4	yes	—
13	C₂×#15667945	—	2	2	no	—
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂×#15667932	—	2	2	no	—
17	C₂×#15667924	—	2	1	no	—
18	#19151351	C₂²	1	4	yes	complemented
19	#19151344	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₈₉	—	2	1	no	—
21	C₂²×Q₁₆×C₈₉	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
23	#494×C₈₉	—	2	2	no	—
24	#494×C₈₉	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	—	2	1	no	—
28	C₂²×#10623494	—	2	2	no	—
29	C₂²×#10623494	—	2	1	no	—
30	C₂×C₈×D₈₉	—	2	4	no	—
31	C₂×C₈×D₈₉	—	4	4	no	—
32	C₄×D₃₅₆	—	2	1	no	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₄×D₃₅₆	D₄	1	2	yes	—
35	C₄×D₃₅₆	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
37	C₂×#13083551	—	2	1	no	—
38	#15667945	—	2	8	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	—	2	1	no	—
41	C₄⋊#10623494	C₂³	1	4	yes	—
42	C₄⋊#10623494	C₂³	1	2	yes	—
43	#15667932	—	2	8	no	—
44	#15667924	—	2	4	no	—
45	C₂²×Q₈×C₈₉	—	2	2	no	—
46	C₂×Q₁₆×C₈₉	—	2	4	no	—
47	C₂×Q₁₆×C₈₉	—	2	4	no	—
48	C₂×Q₁₆×C₈₉	—	4	4	no	—
49	C₂×Q₁₆×C₈₉	—	4	4	no	—
50	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₉	—	2	2	no	—
53	C₄⋊Q₈×C₈₉	—	2	2	no	—
54	C₄⋊Q₈×C₈₉	—	2	2	no	—
55	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	2	yes	—
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
61	C₂³×D₈₉	—	2	1	no	—
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	—	2	2	no	—
64	C₂×D₃₅₆	Q₁₆	1	4	yes	complemented
65	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	2	no	—
68	C₂×C₄×D₈₉	—	2	2	no	—
69	C₂×C₄×D₈₉	—	4	2	no	—
70	C₂×C₄×D₈₉	—	4	2	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	2	4	no	—
73	C₂×#10623494	—	4	2	no	—
74	C₂×#10623494	—	4	4	no	—
75	C₈×D₈₉	—	4	8	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
78	#13083551	—	2	2	no	—
79	#13083551	—	2	2	no	—
80	C₂×Q₈×C₈₉	—	2	4	no	—
81	C₂×Q₈×C₈₉	—	4	2	no	—
82	C₂×Q₈×C₈₉	—	4	4	no	—
83	C₂×Q₈×C₈₉	—	4	4	no	—
84	C₂²×C₄×C₈₉	—	4	2	no	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
88	Q₁₆×C₈₉	—	4	8	no	—
89	Q₁₆×C₈₉	—	4	8	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	D₈	1	4	yes	—
92	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₈₉	—	2	4	no	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₄⋊C₄×C₈₉	—	4	2	no	—
97	C₄⋊C₄×C₈₉	—	4	2	no	—
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	#41	1	1	yes	characteristic
101	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂³×C₈₉	#121	1	1	yes	characteristic
103	C₂²×D₈₉	—	2	2	no	—
104	C₂²×D₈₉	—	2	4	no	—
105	C₂²×D₈₉	—	4	2	no	—
106	C₂²×D₈₉	—	4	2	no	—
107	Q₈×C₈₉	—	4	4	no	—
108	Q₈×C₈₉	—	4	4	no	—
109	Q₈×C₈₉	—	8	4	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	4	4	no	—
112	C₂×C₄×C₈₉	—	8	4	no	—
113	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₈₉	#135	1	2	yes	—
119	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₈₉	—	2	2	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	D₃₅₆	—	2	2	no	—
126	D₃₅₆	—	2	4	no	—
127	D₃₅₆	C₂×Q₁₆	1	4	yes	complemented
128	C₄×D₈₉	—	4	2	no	—
129	C₄×D₈₉	—	4	2	no	—
130	C₄×D₈₉	—	4	4	no	—
131	C₈×C₈₉	—	2	2	no	—
132	C₈×C₈₉	C₂×D₈	1	4	yes	—
133	#10623494	—	4	2	no	—
134	#10623494	—	4	4	no	—
135	C₂²×C₈₉	#459	1	2	yes	—
136	C₂²×C₈₉	#467	1	2	yes	—
137	C₂²×C₈₉	#473	1	2	yes	—
138	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
139	C₂×D₈₉	—	2	4	no	—
140	C₂×D₈₉	—	4	8	no	—
141	C₂×D₈₉	—	4	2	no	—
142	C₂×D₈₉	—	4	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	8	4	no	—
146	C₄×C₈₉	#459	1	2	yes	—
147	C₄×C₈₉	#467	1	2	yes	—
148	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
150	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
151	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	Dic₈₉	—	4	2	no	—
153	C₂×#1669	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	#1669	1	4	yes	complemented
155	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
156	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
158	D₈₉	—	4	4	no	—
159	C₂×#494	D₈₉	1	1	yes	characteristic, complemented
160	C₂×#474	—	89	2	no	—
161	C₂×C₈×D₄	—	89	1	no	—
162	#1669	—	89	8	no	—
163	C₂×#335	—	89	1	no	—
164	C₂×#332	—	89	2	no	—
165	C₈₉	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	—	2	1	no	—
167	C₂²×Q₁₆	—	2	1	no	—
168	C₂³×C₈	—	178	1	no	—
169	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
170	C₂×C₄×D₄	—	89	1	no	—
171	#494	—	2	2	no	—
172	#494	C₂×D₈₉	1	4	yes	complemented
173	#474	—	89	8	no	—
174	#474	—	178	4	no	—
175	C₈×D₄	—	89	4	no	—
176	C₈×D₄	—	178	2	no	—
177	C₂×#107	—	178	2	no	—
178	C₂×C₄⋊C₈	—	89	1	no	—
179	C₂×C₄⋊C₈	—	89	2	no	—
180	C₂×#102	—	178	2	no	—
181	C₂×C₂²⋊C₈	—	178	1	no	—
182	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
183	#335	—	89	4	no	—
184	#332	—	89	8	no	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×D₄	—	89	1	no	—
187	C₂³×C₄	—	178	1	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	2	4	no	—
190	C₂×Q₁₆	—	4	4	no	—
191	C₂×Q₁₆	—	4	4	no	—
192	C₂²×C₈	—	356	4	no	—
193	C₂²×C₈	—	178	2	no	—
194	C₂²×C₈	—	178	1	no	—
195	C₂²×C₈	—	178	4	no	—
196	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
202	C₄×D₄	—	89	2	no	—
203	C₄×D₄	—	89	2	no	—
204	C₄×D₄	—	178	1	no	—
205	C₄×D₄	—	178	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	89	1	no	—
209	C₂×C₂²⋊C₄	—	178	1	no	—
210	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
211	#107	—	178	8	no	—
212	C₄⋊C₈	—	89	2	no	—
213	C₄⋊C₈	—	89	4	no	—
214	C₄⋊C₈	—	178	2	no	—
215	C₄⋊C₈	—	178	1	no	—
216	#102	—	178	8	no	—
217	C₂²⋊C₈	—	178	4	no	—
218	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
219	C₄×C₈	C₂²×D₈₉	1	2	yes	—
220	C₂⁴	—	178	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×D₄	—	89	4	no	—
226	C₂×D₄	—	89	2	no	—
227	C₂×D₄	—	178	1	no	—
228	C₂×D₄	—	178	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	356	2	no	—
231	C₂²×C₄	—	356	2	no	—
232	C₂²×C₄	—	178	1	no	—
233	C₂²×C₄	—	178	2	no	—
234	C₂²×C₄	—	178	2	no	—
235	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
237	C₂²×C₄	#13083573	1	1	yes	characteristic
238	Q₁₆	—	4	8	no	—
239	Q₁₆	—	4	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	356	2	no	—
242	C₂×C₈	—	356	4	no	—
243	C₂×C₈	—	356	8	no	—
244	C₂×C₈	—	178	4	no	—
245	C₂×C₈	—	178	2	no	—
246	C₂×C₈	#13083559	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	89	2	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₂²⋊C₄	—	178	2	no	—
256	C₂²⋊C₄	—	178	2	no	—
257	C₄²	D₄×D₈₉	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₄²	#13083573	1	1	yes	characteristic
260	C₄²	C₂³×D₈₉	1	1	yes	characteristic
261	C₂³	—	356	2	no	—
262	C₂³	—	356	2	no	—
263	C₂³	—	178	2	no	—
264	C₂³	—	178	4	no	—
265	C₂³	#15667983	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	8	4	no	—
269	D₄	—	89	4	no	—
270	D₄	—	178	4	no	—
271	D₄	—	178	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	4	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	178	2	no	—
280	C₂×C₄	#15667983	1	1	yes	characteristic
281	C₂×C₄	#15667983	1	1	yes	characteristic
282	C₂×C₄	#15667983	1	1	yes	characteristic
283	C₂×C₄	C₂×#13083559	1	2	yes	—
284	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
285	C₂×C₄	#15668035	1	2	yes	—
286	C₂×C₄	#15668038	1	2	yes	—
287	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	356	2	no	—
292	C₈	—	356	4	no	—
293	C₈	C₂×#13083559	1	4	yes	—
294	C₂²	—	356	8	no	—
295	C₂²	—	356	2	no	—
296	C₂²	—	356	2	no	—
297	C₂²	—	178	4	no	—
298	C₂²	#19151790	1	2	yes	—
299	C₂²	#19151836	1	2	yes	—
300	C₂²	#19151870	1	2	yes	—
301	C₂²	C₂×#15667983	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	4	no	—
305	C₄	—	356	2	no	—
306	C₄	#19151790	1	2	yes	—
307	C₄	#19151836	1	2	yes	—
308	C₄	C₂²×#13083559	1	1	yes	characteristic
309	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
310	C₄	C₂×#15668035	1	1	yes	characteristic
311	C₄	C₂×#15668038	1	1	yes	characteristic
312	C₂	—	356	4	no	—
313	C₂	#26019614	1	4	yes	direct product factor
314	C₂	C₂×#19151790	1	1	yes	characteristic
315	C₂	C₂×#19151836	1	1	yes	characteristic
316	C₂	C₂×#19151870	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#494×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	C₂×#494	D₈₉
hypocenter	C₈₉	C₂×#1669
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #26019614	C₂	direct (N × H)
4. C₂×#19151351	C₂	semidirect
5. C₂×#19151344	C₂	semidirect
6. C₂×#494×C₈₉	C₂	semidirect
18. #19151351	C₂²	semidirect
19. #19151344	C₂²	semidirect
24. #494×C₈₉	C₂²	semidirect
64. C₂×D₃₅₆	Q₁₆	semidirect
127. D₃₅₆	C₂×Q₁₆	semidirect
154. C₂×C₈₉	#1669	semidirect
159. C₂×#494	D₈₉	semidirect
165. C₈₉	C₂×#1669	semidirect
172. #494	C₂×D₈₉	semidirect
313. C₂	#26019614	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×#19151875	2
C₂×C₈×D₃₅₆	2
#26019614	2
C₂×#19151351	2
C₂×#19151344	2
C₂×#494×C₈₉	2
C₂×#1669	89

Subgroups — C2×#26019614 #36536660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019614 #36536660