Subgroups — C₂×#26019617 #36536663

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×#19151978	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151878	C₂	1	2	yes	maximal
3	#26019617	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151345	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	—	2	1	no	—
8	C₂²×Dic₃₅₆	—	2	1	no	—
9	C₂×#15668012	C₂²	1	2	yes	—
10	#19151978	—	2	2	no	—
11	#19151978	C₂²	1	4	yes	complemented
12	#19151878	—	2	4	no	—
13	#19151878	C₂²	1	8	yes	—
14	C₂×#15667945	—	2	2	no	—
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂×#15667933	—	2	2	no	—
17	#19151351	C₂²	1	4	yes	complemented
18	#19151345	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₈₉	—	2	1	no	—
20	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₈₉	—	2	2	no	—
22	C₈×D₄×C₈₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	—	2	2	no	—
27	C₂×Dic₃₅₆	—	2	4	no	—
28	C₂×Dic₃₅₆	—	2	4	no	—
29	C₂×Dic₃₅₆	—	4	4	no	—
30	C₂×Dic₃₅₆	—	4	4	no	—
31	C₂²×#10623494	—	2	2	no	—
32	#15668012	—	2	2	no	—
33	#15668012	—	2	2	no	—
34	#15668012	—	2	2	no	—
35	#15668012	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₈₉	—	2	2	no	—
37	C₂×C₄⋊Dic₈₉	—	2	2	no	—
38	#15667945	—	2	8	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	C₂³	1	4	yes	—
41	#15667933	—	2	8	no	—
42	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₈₉	—	2	1	no	—
44	C₂²×C₈×C₈₉	—	2	1	no	—
45	C₂²×C₈×C₈₉	—	2	4	no	—
46	C₂²×C₈×C₈₉	—	4	4	no	—
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₄×D₄×C₈₉	—	2	1	no	—
50	C₄×D₄×C₈₉	—	2	1	no	—
51	C₄×D₄×C₈₉	D₄	1	2	yes	—
52	C₄×D₄×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	—	2	1	no	—
57	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
58	C₂²⋊C₈×C₈₉	—	2	4	no	—
59	C₄×C₈×C₈₉	D₄	1	2	yes	—
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
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	C₂²×Dic₈₉	—	4	2	no	—
66	Dic₃₅₆	—	4	8	no	—
67	Dic₃₅₆	—	4	8	no	—
68	C₂×#10623494	—	2	4	no	—
69	C₂×#10623494	—	4	4	no	—
70	C₄⋊Dic₈₉	—	2	4	no	—
71	C₄⋊Dic₈₉	—	2	4	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	C₄⋊Dic₈₉	—	4	2	no	—
74	C₂⁴×C₈₉	—	2	1	no	—
75	C₂×D₄×C₈₉	—	2	1	no	—
76	C₂×D₄×C₈₉	—	2	2	no	—
77	C₂×D₄×C₈₉	Q₁₆	1	4	yes	complemented
78	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₈₉	—	2	1	no	—
80	C₂²×C₄×C₈₉	—	2	2	no	—
81	C₂²×C₄×C₈₉	—	2	2	no	—
82	C₂²×C₄×C₈₉	—	4	2	no	—
83	C₂²×C₄×C₈₉	—	4	2	no	—
84	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	—	4	8	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	1	no	—
95	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
96	C₂²⋊C₄×C₈₉	—	2	2	no	—
97	C₂²⋊C₄×C₈₉	—	2	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	—	2	2	no	—
103	C₂³×C₈₉	—	2	4	no	—
104	C₂³×C₈₉	—	4	2	no	—
105	C₂³×C₈₉	—	4	2	no	—
106	C₂³×C₈₉	#121	1	1	yes	characteristic
107	D₄×C₈₉	—	2	2	no	—
108	D₄×C₈₉	—	2	4	no	—
109	D₄×C₈₉	C₂×Q₁₆	1	4	yes	complemented
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	—	4	4	no	—
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₈₉	#135	1	2	yes	—
122	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₈₉	—	4	4	no	—
127	C₂×Dic₈₉	—	8	4	no	—
128	Dic₁₇₈	—	4	4	no	—
129	Dic₁₇₈	—	4	4	no	—
130	Dic₁₇₈	—	8	4	no	—
131	C₈×C₈₉	—	2	2	no	—
132	C₈×C₈₉	—	4	2	no	—
133	C₈×C₈₉	C₂×D₈	1	4	yes	—
134	#10623494	—	4	4	no	—
135	C₂²×C₈₉	—	2	4	no	—
136	C₂²×C₈₉	—	4	8	no	—
137	C₂²×C₈₉	—	4	2	no	—
138	C₂²×C₈₉	—	4	2	no	—
139	C₂²×C₈₉	#459	1	2	yes	—
140	C₂²×C₈₉	#467	1	2	yes	—
141	C₂²×C₈₉	#473	1	2	yes	—
142	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	4	2	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	—	8	4	no	—
153	C₂×#1669	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	4	4	no	—
155	C₂×C₈₉	#1669	1	4	yes	complemented
156	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
158	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
159	C₂×#494	—	89	1	no	—
160	C₂×#474	—	89	2	no	—
161	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
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	—	178	1	no	—
167	C₂²×Q₁₆	—	178	1	no	—
168	C₂³×C₈	—	2	1	no	—
169	C₂×C₄⋊Q₈	—	89	2	no	—
170	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
171	#494	—	89	4	no	—
172	#494	—	178	2	no	—
173	#474	—	89	8	no	—
174	#474	—	178	4	no	—
175	C₈×D₄	—	2	2	no	—
176	C₈×D₄	C₂×D₈₉	1	4	yes	—
177	C₂×#107	—	178	2	no	—
178	C₂×C₄⋊C₈	—	89	2	no	—
179	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
180	C₂×#102	—	178	2	no	—
181	C₂×C₂²⋊C₈	—	2	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	—	178	2	no	—
186	C₂²×D₄	D₃₅₆	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂×Q₁₆	—	356	4	no	—
189	C₂×Q₁₆	—	356	4	no	—
190	C₂×Q₁₆	—	178	4	no	—
191	C₂×Q₁₆	—	178	4	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	4	4	no	—
195	C₂²×C₈	—	178	2	no	—
196	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	89	4	no	—
199	C₄⋊Q₈	—	178	2	no	—
200	C₄⋊Q₈	—	178	2	no	—
201	C₄⋊Q₈	—	178	2	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₃₅₆	1	2	yes	—
205	C₄×D₄	C₂²×D₈₉	1	2	yes	—
206	C₂×C₄⋊C₄	—	178	2	no	—
207	C₂×C₄⋊C₄	—	178	2	no	—
208	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
211	#107	—	178	8	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	89	4	no	—
214	C₄⋊C₈	—	178	2	no	—
215	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
216	#102	—	178	8	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
219	C₄×C₈	C₂²×D₈₉	1	2	yes	—
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	356	2	no	—
222	C₂×Q₈	—	356	4	no	—
223	C₂×Q₈	—	356	4	no	—
224	C₂×Q₈	—	178	4	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	Dic₃₅₆	1	4	yes	complemented
228	C₂×D₄	C₂×D₃₅₆	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	356	2	no	—
235	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
237	C₂²×C₄	#13083574	1	1	yes	characteristic
238	Q₁₆	—	356	8	no	—
239	Q₁₆	—	356	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	—	356	4	no	—
245	C₂×C₈	—	178	4	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	1	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	178	4	no	—
253	C₄⋊C₄	—	178	4	no	—
254	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
259	C₄²	#13083574	1	1	yes	characteristic
260	C₄²	C₂³×D₈₉	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	#15667986	1	1	yes	characteristic
266	Q₈	—	356	4	no	—
267	Q₈	—	356	4	no	—
268	Q₈	—	712	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	D₄	C₂×Dic₃₅₆	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	356	4	no	—
279	C₂×C₄	—	712	4	no	—
280	C₂×C₄	#15667986	1	1	yes	characteristic
281	C₂×C₄	#15667986	1	1	yes	characteristic
282	C₂×C₄	#15667986	1	1	yes	characteristic
283	C₂×C₄	C₂×#13083559	1	2	yes	—
284	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
285	C₂×C₄	#15668034	1	2	yes	—
286	C₂×C₄	#15668037	1	2	yes	—
287	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
289	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	356	4	no	—
293	C₈	C₂×#13083559	1	4	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#19151793	1	2	yes	—
299	C₂²	#19151839	1	2	yes	—
300	C₂²	#19151873	1	2	yes	—
301	C₂²	C₂×#15667986	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	712	4	no	—
306	C₄	#19151793	1	2	yes	—
307	C₄	#19151839	1	2	yes	—
308	C₄	C₂²×#13083559	1	1	yes	characteristic
309	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
310	C₄	C₂×#15668034	1	1	yes	characteristic
311	C₄	C₂×#15668037	1	1	yes	characteristic
312	C₂	—	4	4	no	—
313	C₂	#26019617	1	4	yes	direct product factor
314	C₂	C₂×#19151793	1	1	yes	characteristic
315	C₂	C₂×#19151839	1	1	yes	characteristic
316	C₂	C₂×#19151873	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈×D₄×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₄×C₈₉	D₄
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	C₂×#1669
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19151978	C₂	semidirect
3. #26019617	C₂	direct (N × H)
4. C₂×#19151351	C₂	semidirect
5. C₂×#19151345	C₂	semidirect
11. #19151978	C₂²	semidirect
17. #19151351	C₂²	semidirect
18. #19151345	C₂²	semidirect
77. C₂×D₄×C₈₉	Q₁₆	semidirect
109. D₄×C₈₉	C₂×Q₁₆	semidirect
155. C₂×C₈₉	#1669	semidirect
165. C₈₉	C₂×#1669	semidirect
227. C₂×D₄	Dic₃₅₆	semidirect
271. D₄	C₂×Dic₃₅₆	semidirect
313. C₂	#26019617	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×#19151978	2
C₂×#19151878	2
#26019617	2
C₂×#19151351	2
C₂×#19151345	2
C₂×C₈×D₄×C₈₉	2
C₂×#1669	89

Subgroups — C2×#26019617 #36536663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019617 #36536663