Subgroups — C₂×#26019032 #36523495

Order 22784.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151655	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151651	C₂	1	1	yes	maximal, characteristic
3	#26019032	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151351	C₂	1	2	yes	maximal, complemented
5	C₂×#19151349	C₂	1	2	yes	maximal, complemented
6	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19151655	—	2	2	no	—
8	#19151655	C₂²	1	4	yes	—
9	#19151651	—	2	2	no	—
10	#19151651	C₂²	1	4	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×C₂²⋊#10623494	—	2	1	no	—
14	C₂×C₂²⋊#10623494	—	2	1	no	—
15	C₂×#15667923	—	2	1	no	—
16	C₂×C₄×#10623494	—	2	1	no	—
17	#19151351	C₂²	1	8	yes	complemented
18	#19151349	C₂²	1	8	yes	complemented
19	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#431×C₈₉	C₄	1	4	yes	—
21	#431×C₈₉	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
23	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×#10623494	—	2	2	no	—
27	C₂²×#10623494	—	2	2	no	—
28	C₄⋊#10623494	—	2	2	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	C₂³	1	4	yes	—
31	C₄⋊#10623494	C₂³	1	4	yes	—
32	C₂²⋊#10623494	—	2	4	no	—
33	C₂²⋊#10623494	—	2	4	no	—
34	#15667923	—	2	4	no	—
35	C₄×#10623494	—	2	4	no	—
36	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₈₉	D₄	1	2	yes	—
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
49	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
50	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₈₉	C₂³	1	2	yes	—
54	C₄×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	4	4	no	—
58	C₂×#10623494	—	4	4	no	—
59	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	—	2	4	no	—
61	C₂²×C₄×C₈₉	—	2	4	no	—
62	C₂²×C₄×C₈₉	—	4	2	no	—
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₈₉	—	2	4	no	—
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
79	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
80	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
81	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
82	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
84	C₄²×C₈₉	C₂×D₄	1	2	yes	—
85	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
87	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
89	C₂³×C₈₉	—	2	2	no	—
90	C₂³×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	—	2	2	no	—
92	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	2	4	no	—
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	—	4	2	no	—
104	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#131	1	4	yes	—
111	C₂×C₄×C₈₉	#135	1	4	yes	—
112	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₈₉	—	2	4	no	—
116	C₈×C₈₉	—	2	4	no	—
117	#10623494	—	4	4	no	—
118	#10623494	—	4	4	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	4	4	no	—
123	C₂²×C₈₉	#424	1	2	yes	—
124	C₂²×C₈₉	#427	1	2	yes	—
125	C₂²×C₈₉	#430	1	2	yes	—
126	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	2	2	no	—
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	#424	1	2	yes	—
133	C₄×C₈₉	#427	1	2	yes	—
134	C₄×C₈₉	C₂×#131	1	2	yes	—
135	C₄×C₈₉	C₂×#135	1	2	yes	—
136	C₂×#1572	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	4	2	no	—
138	C₂×C₈₉	#1572	1	4	yes	complemented
139	C₂×C₈₉	C₂×#424	1	1	yes	characteristic
140	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
141	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
142	C₂×#432	—	89	1	no	—
143	C₂×#431	—	89	1	no	—
144	C₂×#431	D₈₉	1	1	yes	characteristic
145	#1572	—	89	8	no	—
146	C₂×#335	—	89	2	no	—
147	C₂×#334	—	89	2	no	—
148	C₈₉	C₂×#1572	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
150	#432	—	89	4	no	—
151	#432	—	178	2	no	—
152	#431	—	89	4	no	—
153	#431	—	178	2	no	—
154	#431	Dic₈₉	1	4	yes	—
155	#431	C₂×D₈₉	1	4	yes	—
156	C₂×C₄⋊C₈	—	89	2	no	—
157	C₂×C₄⋊C₈	—	89	2	no	—
158	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
159	C₂×C₂²⋊C₈	—	178	1	no	—
160	C₂×C₂²⋊C₈	—	178	1	no	—
161	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
162	C₂×#97	—	178	1	no	—
163	C₂×C₄×C₈	—	178	1	no	—
164	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
166	#335	—	89	8	no	—
167	#334	—	89	8	no	—
168	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
169	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
171	C₂²×C₈	—	178	2	no	—
172	C₂²×C₈	—	178	2	no	—
173	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
174	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
175	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
176	C₂×C₄²	Dic₁₇₈	1	2	yes	—
177	C₂×C₄²	D₃₅₆	1	2	yes	—
178	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
182	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	C₄⋊C₈	—	89	4	no	—
184	C₄⋊C₈	—	89	4	no	—
185	C₄⋊C₈	—	178	2	no	—
186	C₄⋊C₈	—	178	2	no	—
187	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
188	C₂²⋊C₈	—	178	4	no	—
189	C₂²⋊C₈	—	178	4	no	—
190	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
191	#97	—	178	4	no	—
192	C₄×C₈	—	178	4	no	—
193	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
194	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
195	C₄×C₈	C₂²×D₈₉	1	2	yes	—
196	C₄×C₈	C₂²×D₈₉	1	2	yes	—
197	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
203	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
210	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	356	4	no	—
216	C₂×C₈	—	356	4	no	—
217	C₂×C₈	—	178	4	no	—
218	C₂×C₈	—	178	4	no	—
219	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
220	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
221	C₄²	C₄⋊Dic₈₉	1	2	yes	—
222	C₄²	C₄⋊Dic₈₉	1	2	yes	—
223	C₄²	C₄⋊Dic₈₉	1	2	yes	—
224	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
225	C₄²	C₂²×Dic₈₉	1	2	yes	—
226	C₄²	C₂×D₃₅₆	1	2	yes	—
227	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
228	C₄²	C₂×Dic₁₇₈	1	2	yes	—
229	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
230	C₄²	C₂³×D₈₉	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
235	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
250	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
252	C₂×C₄	#15668019	1	4	yes	—
253	C₂×C₄	#15668034	1	4	yes	—
254	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
256	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	—	2	4	no	—
259	C₈	—	356	4	no	—
260	C₈	—	356	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	4	no	—
265	C₂²	#19151625	1	2	yes	—
266	C₂²	#19151636	1	2	yes	—
267	C₂²	#19151647	1	2	yes	—
268	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	#19151625	1	2	yes	—
275	C₄	#19151636	1	2	yes	—
276	C₄	C₂×#15668019	1	2	yes	—
277	C₄	C₂×#15668034	1	2	yes	—
278	C₂	—	4	2	no	—
279	C₂	#26019032	1	4	yes	direct product factor
280	C₂	C₂×#19151625	1	1	yes	characteristic
281	C₂	C₂×#19151636	1	1	yes	characteristic
282	C₂	C₂×#19151647	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#431×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#431	D₈₉
hypocenter	C₈₉	C₂×#1572
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #26019032	C₂	direct (N × H)
4. C₂×#19151351	C₂	semidirect
5. C₂×#19151349	C₂	semidirect
17. #19151351	C₂²	semidirect
18. #19151349	C₂²	semidirect
138. C₂×C₈₉	#1572	semidirect
148. C₈₉	C₂×#1572	semidirect
279. C₂	#26019032	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×#19151655	2
C₂×#19151651	2
#26019032	2
C₂×#19151351	2
C₂×#19151349	2
C₂×#431×C₈₉	2
C₂×#1572	89

Subgroups — C2×#26019032 #36523495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019032 #36523495