Subgroups — C₂×#26019023 #36523484

Order 22784.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#26019023	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151349	C₂	1	4	yes	maximal, complemented
5	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	—	2	1	no	—
7	C₂³×#10623494	—	2	1	no	—
8	#19151622	—	2	2	no	—
9	#19151622	C₂²	1	4	yes	—
10	C₂×C₄⋊#10623494	—	2	1	no	—
11	C₂×C₄⋊#10623494	—	2	2	no	—
12	C₂×C₄⋊#10623494	—	2	1	no	—
13	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
14	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	#19151349	C₂²	1	16	yes	complemented
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#431×C₈₉	C₄	1	4	yes	—
19	#431×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
21	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
23	C₂×#13083558	—	2	4	no	—
24	C₂×#13083558	—	4	4	no	—
25	C₂²×#10623494	—	2	4	no	—
26	C₂²×#10623494	—	2	2	no	—
27	C₂²×#10623494	—	2	2	no	—
28	C₂²×#10623494	—	4	4	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	—	2	4	no	—
31	C₄⋊#10623494	—	2	4	no	—
32	C₄⋊#10623494	—	2	2	no	—
33	C₄⋊#10623494	C₂³	1	4	yes	—
34	C₄⋊#10623494	C₂³	1	4	yes	—
35	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	2	yes	—
43	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
47	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
48	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₈₉	C₂³	1	4	yes	—
51	#13083558	—	4	8	no	—
52	C₂×#10623494	—	2	4	no	—
53	C₂×#10623494	—	2	4	no	—
54	C₂×#10623494	—	4	8	no	—
55	C₂×#10623494	—	4	4	no	—
56	C₂×#10623494	—	4	4	no	—
57	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	—	2	4	no	—
59	C₂²×C₄×C₈₉	—	4	2	no	—
60	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	SD₁₆	1	4	yes	—
67	C₂²×C₄×C₈₉	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₈₉	—	2	8	no	—
75	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
76	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
81	C₄²×C₈₉	C₂×D₄	1	2	yes	—
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
84	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₂³×C₈₉	—	2	2	no	—
87	C₂³×C₈₉	—	2	2	no	—
88	C₂³×C₈₉	#106	1	4	yes	—
89	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	4	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	#106	1	8	yes	—
99	C₂×C₄×C₈₉	#106	1	4	yes	—
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#131	1	4	yes	—
107	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
108	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
114	C₈×C₈₉	—	2	8	no	—
115	#10623494	—	4	4	no	—
116	#10623494	—	4	4	no	—
117	C₂²×C₈₉	—	2	8	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	#424	1	2	yes	—
121	C₂²×C₈₉	C₂×#106	1	4	yes	—
122	C₂²×C₈₉	C₂×#106	1	2	yes	—
123	C₂²×C₈₉	#430	1	2	yes	—
124	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	#424	1	2	yes	—
130	C₄×C₈₉	C₂×#106	1	2	yes	—
131	C₄×C₈₉	C₂×#106	1	4	yes	—
132	C₄×C₈₉	C₂×#131	1	2	yes	—
133	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
134	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
135	C₂×#1570	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	2	4	no	—
137	C₂×C₈₉	#1570	1	4	yes	complemented
138	C₂×C₈₉	C₂×#424	1	1	yes	characteristic
139	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
140	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
141	C₂×#431	D₈₉	1	1	yes	characteristic
142	C₂×#423	—	89	1	no	—
143	C₂²×C₄⋊C₈	—	89	1	no	—
144	#1570	—	89	8	no	—
145	C₂×#334	—	89	4	no	—
146	C₈₉	C₂×#1570	1	1	yes	characteristic, complemented, Sylow
147	C₂²×M₁₆	—	178	1	no	—
148	C₂³×C₈	—	178	1	no	—
149	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
150	#431	Dic₈₉	1	4	yes	—
151	#431	C₂×D₈₉	1	4	yes	—
152	#423	—	89	4	no	—
153	#423	—	178	2	no	—
154	C₂×C₄⋊C₈	—	89	2	no	—
155	C₂×C₄⋊C₈	—	89	2	no	—
156	C₂×C₄⋊C₈	—	89	4	no	—
157	C₂×C₄⋊C₈	—	178	2	no	—
158	C₂×C₄⋊C₈	—	178	1	no	—
159	C₂×C₄⋊C₈	—	178	1	no	—
160	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
161	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
162	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
163	#334	—	89	16	no	—
164	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
165	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂×M₁₆	—	356	4	no	—
168	C₂×M₁₆	—	178	4	no	—
169	C₂²×C₈	—	356	4	no	—
170	C₂²×C₈	—	178	4	no	—
171	C₂²×C₈	—	178	2	no	—
172	C₂²×C₈	—	178	2	no	—
173	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
174	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
175	C₂×C₄²	Dic₁₇₈	1	2	yes	—
176	C₂×C₄²	D₃₅₆	1	2	yes	—
177	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
180	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	C₄⋊C₈	—	89	4	no	—
183	C₄⋊C₈	—	89	4	no	—
184	C₄⋊C₈	—	178	2	no	—
185	C₄⋊C₈	—	178	4	no	—
186	C₄⋊C₈	—	178	2	no	—
187	C₄⋊C₈	—	178	4	no	—
188	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
189	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
190	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
191	C₄×C₈	C₂²×D₈₉	1	4	yes	—
192	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
197	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
199	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
201	C₂²×C₄	#13083563	1	4	yes	—
202	C₂²×C₄	#13083563	1	4	yes	—
203	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
206	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
209	M₁₆	—	356	8	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	356	8	no	—
212	C₂×C₈	—	356	4	no	—
213	C₂×C₈	—	356	4	no	—
214	C₂×C₈	—	178	4	no	—
215	C₂×C₈	—	178	4	no	—
216	C₂×C₈	C₂²×Dic₈₉	1	8	yes	—
217	C₄²	C₄⋊Dic₈₉	1	2	yes	—
218	C₄²	C₄⋊Dic₈₉	1	2	yes	—
219	C₄²	C₄⋊Dic₈₉	1	2	yes	—
220	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
221	C₄²	C₂²×Dic₈₉	1	2	yes	—
222	C₄²	C₂×D₃₅₆	1	2	yes	—
223	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
224	C₄²	C₂×Dic₁₇₈	1	2	yes	—
225	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
226	C₄²	C₂³×D₈₉	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#15667944	1	4	yes	—
230	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
231	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#15667944	1	8	yes	—
240	C₂×C₄	#15667944	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
247	C₂×C₄	#15668019	1	4	yes	—
248	C₂×C₄	C₂×#13083563	1	4	yes	—
249	C₂×C₄	C₂×#13083563	1	2	yes	—
250	C₂×C₄	C₂×#13083563	1	2	yes	—
251	C₂×C₄	C₂×#13083563	1	4	yes	—
252	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
254	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	356	4	no	—
257	C₈	—	356	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	8	no	—
261	C₂²	#19151625	1	2	yes	—
262	C₂²	C₂×#15667944	1	4	yes	—
263	C₂²	C₂×#15667944	1	2	yes	—
264	C₂²	#19151647	1	2	yes	—
265	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	#19151625	1	2	yes	—
271	C₄	C₂×#15667944	1	4	yes	—
272	C₄	C₂×#15667944	1	2	yes	—
273	C₄	C₂×#15668019	1	2	yes	—
274	C₄	C₂²×#13083563	1	1	yes	characteristic
275	C₄	C₂²×#13083563	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	#26019023	1	4	yes	direct product factor
278	C₂	C₂×#19151625	1	1	yes	characteristic
279	C₂	C₂²×#15667944	1	1	yes	characteristic
280	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×#1570
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. #26019023	C₂	direct (N × H)
4. C₂×#19151349	C₂	semidirect
16. #19151349	C₂²	semidirect
137. C₂×C₈₉	#1570	semidirect
146. C₈₉	C₂×#1570	semidirect
277. C₂	#26019023	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×#19151622	2
C₂²×C₄⋊#10623494	2
#26019023	2
C₂×#19151349	2
C₂×#431×C₈₉	2
C₂×#1570	89

Subgroups — C2×#26019023 #36523484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019023 #36523484