Subgroups — C₂×#26018265 #36464652

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151839	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151361	C₂	1	1	yes	maximal, characteristic
3	#26018265	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic, complemented
6	C₂×#15667991	—	2	1	no	—
7	C₂²×C₄⋊Dic₈₉	—	2	1	no	—
8	#19151839	—	2	1	no	—
9	#19151839	—	2	1	no	—
10	#19151839	C₄	1	2	yes	complemented
11	#19151839	C₂²	1	2	yes	—
12	C₂×#15667946	C₄	1	1	yes	characteristic, complemented
13	C₂×#15667937	—	2	1	no	—
14	#19151361	—	2	1	no	—
15	#19151361	C₂²	1	2	yes	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₈₉	—	2	1	no	—
18	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
19	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂³×Dic₈₉	—	4	1	no	—
22	C₂×Dic₃₅₆	—	2	1	no	—
23	C₂×Dic₃₅₆	—	4	2	no	—
24	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	complemented
25	#15667991	—	2	2	no	—
26	#15667991	—	2	2	no	—
27	#15667991	—	4	1	no	—
28	#15667991	—	4	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	4	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	2	no	—
32	C₂×C₄⋊Dic₈₉	—	4	2	no	—
33	C₂×C₄⋊Dic₈₉	—	4	1	no	—
34	C₂×#13083553	—	4	1	no	—
35	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
36	#15667946	—	2	1	no	—
37	#15667946	C₂×C₄	1	2	yes	complemented
38	#15667937	—	2	2	no	—
39	#15667937	—	2	2	no	—
40	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	—	4	2	no	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	—	2	2	no	—
49	C₂×C₁₆×C₈₉	—	4	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂×Dic₁₇₈	—	4	1	no	—
54	C₂²×Dic₈₉	—	4	4	no	—
55	C₂²×Dic₈₉	—	4	1	no	—
56	C₂²×Dic₈₉	—	4	1	no	—
57	C₂²×Dic₈₉	—	4	2	no	—
58	C₂²×Dic₈₉	—	8	2	no	—
59	C₂²×Dic₈₉	—	8	2	no	—
60	Dic₃₅₆	—	4	2	no	—
61	Dic₃₅₆	—	4	2	no	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	—	2	2	no	—
64	C₄⋊Dic₈₉	—	4	4	no	—
65	C₄⋊Dic₈₉	—	4	2	no	—
66	C₄⋊Dic₈₉	—	4	1	no	—
67	#13083553	—	4	2	no	—
68	#13083553	—	4	2	no	—
69	C₂²⋊Dic₈₉	—	4	2	no	—
70	C₂²⋊Dic₈₉	—	4	2	no	—
71	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	—	4	1	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₂×C₈×C₈₉	—	4	1	no	—
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₁₆×C₈₉	—	4	2	no	—
91	C₂³×C₈₉	—	2	2	no	—
92	C₂³×C₈₉	—	2	2	no	—
93	C₂³×C₈₉	#102	1	1	yes	characteristic
94	C₂³×C₈₉	#102	1	1	yes	characteristic
95	C₂³×C₈₉	SD₃₂	1	2	yes	—
96	C₂³×C₈₉	Q₃₂	1	2	yes	—
97	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	#99	1	2	yes	—
106	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#104	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×Dic₈₉	—	4	4	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	8	8	no	—
115	C₂×Dic₈₉	—	8	2	no	—
116	C₂×Dic₈₉	—	8	2	no	—
117	C₂×Dic₈₉	—	8	2	no	—
118	C₂×Dic₈₉	—	8	2	no	—
119	Dic₁₇₈	—	4	2	no	—
120	Dic₁₇₈	—	4	2	no	—
121	Dic₁₇₈	—	8	2	no	—
122	C₈×C₈₉	—	2	1	no	—
123	C₈×C₈₉	—	2	1	no	—
124	C₈×C₈₉	—	2	1	no	—
125	C₈×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	—	2	4	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	—	2	1	no	—
131	C₂²×C₈₉	—	2	1	no	—
132	C₂²×C₈₉	#327	1	2	yes	—
133	C₂²×C₈₉	#358	1	2	yes	—
134	C₂²×C₈₉	#358	1	2	yes	—
135	C₂²×C₈₉	#360	1	2	yes	—
136	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
137	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
138	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	#327	1	2	yes	—
142	C₄×C₈₉	C₂×#99	1	1	yes	characteristic
143	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
144	Dic₈₉	—	8	4	no	—
145	Dic₈₉	—	8	2	no	—
146	Dic₈₉	—	8	2	no	—
147	C₂×#1348	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	#1348	1	4	yes	complemented
152	C₂×C₈₉	C₂×#327	1	1	yes	characteristic
153	C₂×C₈₉	C₂×#358	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#360	1	1	yes	characteristic
155	C₂×#467	—	89	1	no	—
156	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
157	C₂×#340	—	89	1	no	—
158	#1348	—	89	4	no	—
159	C₈₉	C₂×#1348	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₁₆	—	89	1	no	—
161	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
162	C₂×C₂²⋊Q₈	—	178	1	no	—
163	C₂²×C₄⋊C₄	—	178	1	no	—
164	C₂²×C₁₆	—	2	1	no	—
165	#467	—	89	2	no	—
166	#467	—	89	2	no	—
167	#467	—	178	1	no	—
168	#467	—	178	1	no	—
169	C₂×#107	—	89	1	no	—
170	C₂×#103	—	178	1	no	—
171	C₂²⋊C₁₆	—	2	1	no	—
172	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
173	#340	—	89	2	no	—
174	#340	—	178	1	no	—
175	C₂²×Q₈	—	178	1	no	—
176	C₂³×C₄	—	356	1	no	—
177	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
178	C₂×Q₁₆	—	89	2	no	—
179	C₂×Q₁₆	—	356	2	no	—
180	C₂×Q₁₆	—	178	1	no	—
181	C₂²×C₈	—	4	2	no	—
182	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
186	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
188	C₂²⋊Q₈	—	356	1	no	—
189	C₂²⋊Q₈	—	356	1	no	—
190	C₂²⋊Q₈	—	178	2	no	—
191	C₂²⋊Q₈	—	178	2	no	—
192	C₂×C₄⋊C₄	—	356	2	no	—
193	C₂×C₄⋊C₄	—	356	1	no	—
194	C₂×C₄⋊C₄	—	356	1	no	—
195	C₂×C₄⋊C₄	—	178	4	no	—
196	C₂×C₄⋊C₄	—	178	1	no	—
197	C₂×C₄⋊C₄	—	178	2	no	—
198	C₂×C₂²⋊C₄	—	356	1	no	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	4	2	no	—
201	#107	—	89	2	no	—
202	#107	—	178	1	no	—
203	#103	—	178	2	no	—
204	#103	—	178	2	no	—
205	C₂⁴	D₇₁₂	1	1	yes	characteristic
206	C₂×Q₈	—	356	1	no	—
207	C₂×Q₈	—	356	2	no	—
208	C₂×Q₈	—	356	2	no	—
209	C₂×Q₈	—	178	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	356	4	no	—
213	C₂²×C₄	—	356	2	no	—
214	C₂²×C₄	—	356	1	no	—
215	C₂²×C₄	—	356	1	no	—
216	C₂²×C₄	—	712	2	no	—
217	C₂²×C₄	—	712	2	no	—
218	C₂²×C₄	#13083551	1	1	yes	characteristic
219	C₂²×C₄	#13083551	1	1	yes	characteristic
220	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
221	C₂²×C₄	#13083563	1	1	yes	characteristic
222	C₂²×C₄	#13083563	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
224	Q₁₆	—	356	2	no	—
225	Q₁₆	—	356	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	#13083551	1	1	yes	characteristic
233	C₂×C₈	#13083551	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
236	C₄⋊C₄	—	356	2	no	—
237	C₄⋊C₄	—	356	4	no	—
238	C₄⋊C₄	—	356	2	no	—
239	C₄⋊C₄	—	356	2	no	—
240	C₄⋊C₄	—	356	1	no	—
241	C₄⋊C₄	—	178	4	no	—
242	C₄⋊C₄	—	178	2	no	—
243	C₂²⋊C₄	—	356	2	no	—
244	C₂²⋊C₄	—	356	2	no	—
245	C₁₆	—	4	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	#15667934	1	1	yes	characteristic
249	C₂³	#15667934	1	1	yes	characteristic
250	C₂³	#15667957	1	2	yes	—
251	C₂³	Dic₇₁₂	1	2	yes	—
252	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
253	Q₈	—	356	2	no	—
254	Q₈	—	356	2	no	—
255	Q₈	—	712	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	356	4	no	—
264	C₂×C₄	—	356	2	no	—
265	C₂×C₄	—	356	2	no	—
266	C₂×C₄	—	712	2	no	—
267	C₂×C₄	—	712	8	no	—
268	C₂×C₄	—	712	2	no	—
269	C₂×C₄	—	712	2	no	—
270	C₂×C₄	—	712	2	no	—
271	C₂×C₄	#15667926	1	2	yes	—
272	C₂×C₄	#15667934	1	1	yes	characteristic
273	C₂×C₄	#15667934	1	1	yes	characteristic
274	C₂×C₄	#15667938	1	2	yes	—
275	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
276	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	#19151333	1	2	yes	—
288	C₂²	#19151403	1	2	yes	—
289	C₂²	#19151403	1	2	yes	—
290	C₂²	#19151409	1	2	yes	—
291	C₂²	C₂×#15667934	1	1	yes	characteristic
292	C₂²	C₂×#15667957	1	1	yes	characteristic
293	C₂²	C₂×Dic₇₁₂	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	712	4	no	—
297	C₄	—	712	2	no	—
298	C₄	—	712	2	no	—
299	C₄	#19151333	1	2	yes	—
300	C₄	C₂×#15667926	1	1	yes	characteristic
301	C₄	C₂×#15667938	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	#26018265	1	4	yes	direct product factor
306	C₂	C₂×#19151333	1	1	yes	characteristic
307	C₂	C₂×#19151403	1	1	yes	characteristic
308	C₂	C₂×#19151409	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	#102
prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	C₂×#1348
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #26018265	C₂	direct (N × H)
5. C₂²×Dic₃₅₆	C₄	semidirect
10. #19151839	C₄	semidirect
12. C₂×#15667946	C₄	semidirect
24. C₂×Dic₃₅₆	C₂×C₄	semidirect
37. #15667946	C₂×C₄	semidirect
151. C₂×C₈₉	#1348	semidirect
159. C₈₉	C₂×#1348	semidirect
305. C₂	#26018265	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×#19151839	2
C₂×#19151361	2
#26018265	2
C₂×C₂²⋊C₁₆×C₈₉	2
C₂×#1348	89

Subgroups — C2×#26018265 #36464652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018265 #36464652