Subgroups — C₂×#26019618 #36536664

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×#19151879	C₂	1	2	yes	maximal
3	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#26019618	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151343	C₂	1	2	yes	maximal, complemented
6	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₃₅₆	—	2	1	no	—
8	C₂²×Dic₃₅₆	—	2	1	no	—
9	C₂²×C₈×D₈₉	—	2	1	no	—
10	C₂×#15668012	C₂²	1	2	yes	—
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	#19151978	—	2	2	no	—
13	#19151978	C₂²	1	4	yes	complemented
14	#19151879	—	2	4	no	—
15	#19151879	C₂²	1	8	yes	—
16	C₈×D₃₅₆	—	2	2	no	—
17	C₈×D₃₅₆	C₂²	1	4	yes	—
18	C₂×#15667946	—	2	2	no	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667934	—	2	2	no	—
21	C₂×#15667924	—	2	1	no	—
22	#19151343	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#335×C₈₉	C₂²	1	4	yes	complemented
26	C₂²×Dic₁₇₈	—	2	2	no	—
27	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₈₉	—	2	1	no	—
29	C₂×Dic₃₅₆	—	2	4	no	—
30	C₂×Dic₃₅₆	—	2	4	no	—
31	C₂×Dic₃₅₆	—	4	4	no	—
32	C₂×Dic₃₅₆	—	4	4	no	—
33	C₂²×#10623494	—	2	1	no	—
34	C₂×C₈×D₈₉	—	2	4	no	—
35	C₂×C₈×D₈₉	—	4	4	no	—
36	#15668012	—	2	2	no	—
37	#15668012	—	2	2	no	—
38	#15668012	—	2	2	no	—
39	#15668012	C₂³	1	4	yes	—
40	C₄×D₃₅₆	—	2	1	no	—
41	C₄×D₃₅₆	—	2	1	no	—
42	C₄×D₃₅₆	D₄	1	2	yes	—
43	C₄×D₃₅₆	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₈₉	—	2	2	no	—
45	C₂×C₄⋊Dic₈₉	—	2	2	no	—
46	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
47	C₂×#13083551	—	2	1	no	—
48	#15667946	—	2	8	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	C₂³	1	2	yes	—
51	#15667934	—	2	8	no	—
52	#15667924	—	2	4	no	—
53	C₂²×C₈×C₈₉	—	2	2	no	—
54	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₉	—	2	2	no	—
58	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
59	C₄×C₈×C₈₉	D₄	1	2	yes	—
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
61	C₂³×D₈₉	—	2	1	no	—
62	C₂×Dic₁₇₈	—	2	4	no	—
63	C₂×Dic₁₇₈	—	4	4	no	—
64	C₂×Dic₁₇₈	—	4	4	no	—
65	C₂×Dic₁₇₈	—	4	2	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	2	no	—
68	C₂×D₃₅₆	Q₁₆	1	4	yes	complemented
69	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
70	C₂²×Dic₈₉	—	2	1	no	—
71	C₂²×Dic₈₉	—	4	2	no	—
72	C₂×C₄×D₈₉	—	2	2	no	—
73	C₂×C₄×D₈₉	—	2	2	no	—
74	C₂×C₄×D₈₉	—	4	2	no	—
75	C₂×C₄×D₈₉	—	4	2	no	—
76	Dic₃₅₆	—	4	8	no	—
77	Dic₃₅₆	—	4	8	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	4	2	no	—
80	C₈×D₈₉	—	4	8	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	—	2	4	no	—
83	C₄⋊Dic₈₉	—	2	4	no	—
84	C₄⋊Dic₈₉	—	4	2	no	—
85	C₄⋊Dic₈₉	—	4	2	no	—
86	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
87	#13083551	—	2	2	no	—
88	#13083551	—	2	2	no	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	4	4	no	—
95	C₂×C₈×C₈₉	D₈	1	4	yes	—
96	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
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	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
113	C₂×C₄×C₈₉	#135	1	2	yes	—
114	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₈₉	—	2	2	no	—
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	—	4	4	no	—
121	C₂×Dic₈₉	—	8	4	no	—
122	D₃₅₆	—	2	2	no	—
123	D₃₅₆	—	2	4	no	—
124	D₃₅₆	C₂×Q₁₆	1	4	yes	complemented
125	C₄×D₈₉	—	4	2	no	—
126	C₄×D₈₉	—	4	2	no	—
127	C₄×D₈₉	—	4	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	4	no	—
133	C₈×C₈₉	C₂×D₈	1	4	yes	—
134	#10623494	—	4	2	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	#459	1	2	yes	—
146	C₄×C₈₉	#467	1	2	yes	—
147	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
149	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
150	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	Dic₈₉	—	4	2	no	—
152	Dic₈₉	—	8	4	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	—	89	1	no	—
160	C₂×#474	—	89	2	no	—
161	C₂×C₈×D₄	—	89	1	no	—
162	#1669	—	89	8	no	—
163	C₂×#335	D₈₉	1	1	yes	characteristic, complemented
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	—	178	1	no	—
169	C₂×C₄⋊Q₈	—	89	2	no	—
170	C₂×C₄×D₄	—	89	1	no	—
171	#494	—	89	4	no	—
172	#494	—	178	2	no	—
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	C₂×D₈₉	1	2	yes	—
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	C₂×D₈₉	1	4	yes	complemented
184	#332	—	89	8	no	—
185	C₂²×Q₈	—	178	2	no	—
186	C₂²×D₄	—	89	1	no	—
187	C₂³×C₄	—	178	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	2	no	—
193	C₂²×C₈	—	356	4	no	—
194	C₂²×C₈	—	178	1	no	—
195	C₂²×C₈	—	178	4	no	—
196	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
197	C₂²×C₈	D₃₅₆	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	—	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	—	89	1	no	—
207	C₂×C₄⋊C₄	—	178	2	no	—
208	C₂×C₄⋊C₄	—	178	2	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	—	2	2	no	—
213	C₄⋊C₈	—	89	2	no	—
214	C₄⋊C₈	—	178	1	no	—
215	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
216	#102	—	178	8	no	—
217	C₂²⋊C₈	—	178	4	no	—
218	C₄×C₈	D₃₅₆	1	2	yes	—
219	C₄×C₈	C₂²×D₈₉	1	2	yes	—
220	C₂⁴	—	178	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	—	89	4	no	—
226	C₂×D₄	—	89	2	no	—
227	C₂×D₄	—	178	1	no	—
228	C₂×D₄	—	178	2	no	—
229	C₂²×C₄	—	356	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×D₃₅₆	1	1	yes	characteristic
237	C₂²×C₄	#13083575	1	1	yes	characteristic
238	Q₁₆	—	356	8	no	—
239	Q₁₆	—	356	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	356	2	no	—
244	C₂×C₈	—	356	8	no	—
245	C₂×C₈	—	178	2	no	—
246	C₂×C₈	D₇₁₂	1	4	yes	—
247	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
248	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
249	C₄⋊C₄	—	89	2	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	—	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×D₃₅₆	1	1	yes	characteristic
259	C₄²	#13083575	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	#15667987	1	1	yes	characteristic
266	Q₈	—	356	4	no	—
267	Q₈	—	356	4	no	—
268	Q₈	—	712	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	—	356	2	no	—
274	C₂×C₄	—	356	2	no	—
275	C₂×C₄	—	356	4	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	4	no	—
278	C₂×C₄	—	178	2	no	—
279	C₂×C₄	—	712	4	no	—
280	C₂×C₄	#15667987	1	1	yes	characteristic
281	C₂×C₄	#15667987	1	1	yes	characteristic
282	C₂×C₄	#15667987	1	1	yes	characteristic
283	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
284	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
285	C₂×C₄	#15668031	1	2	yes	—
286	C₂×C₄	#15668040	1	2	yes	—
287	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
289	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	4	no	—
292	C₈	—	356	2	no	—
293	C₈	C₂×D₇₁₂	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	#19151794	1	2	yes	—
299	C₂²	#19151840	1	2	yes	—
300	C₂²	#19151874	1	2	yes	—
301	C₂²	C₂×#15667987	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	356	2	no	—
305	C₄	—	712	4	no	—
306	C₄	#19151794	1	2	yes	—
307	C₄	#19151840	1	2	yes	—
308	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
309	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
310	C₄	C₂×#15668031	1	1	yes	characteristic
311	C₄	C₂×#15668040	1	1	yes	characteristic
312	C₂	—	356	4	no	—
313	C₂	#26019618	1	4	yes	direct product factor
314	C₂	C₂×#19151794	1	1	yes	characteristic
315	C₂	C₂×#19151840	1	1	yes	characteristic
316	C₂	C₂×#19151874	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	C₂×#1669
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. C₂×#19151978	C₂	semidirect
4. #26019618	C₂	direct (N × H)
5. C₂×#19151343	C₂	semidirect
6. C₂×#335×C₈₉	C₂	semidirect
13. #19151978	C₂²	semidirect
22. #19151343	C₂²	semidirect
25. #335×C₈₉	C₂²	semidirect
68. C₂×D₃₅₆	Q₁₆	semidirect
124. D₃₅₆	C₂×Q₁₆	semidirect
154. C₂×C₈₉	#1669	semidirect
163. C₂×#335	D₈₉	semidirect
165. C₈₉	C₂×#1669	semidirect
183. #335	C₂×D₈₉	semidirect
313. C₂	#26019618	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×#19151879	2
C₂×C₈×D₃₅₆	2
#26019618	2
C₂×#19151343	2
C₂×#335×C₈₉	2
C₂×#1669	89

Subgroups — C2×#26019618 #36536664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019618 #36536664