Subgroups — C₂×#26020423 #36554883

Order 22784.

1046 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667947	C₂	1	4	yes	maximal
2	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
4	#26020423	C₂	1	8	yes	maximal, direct product factor
5	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂³×#10623494	C₂²	1	2	yes	—
8	C₂×#15667947	—	2	16	no	—
9	C₂×#15667947	C₂²	1	16	yes	—
10	#19151622	—	2	2	no	—
11	#19151622	C₂²	1	4	yes	—
12	C₂×C₄⋊#10623494	—	2	2	no	—
13	C₂×C₄×#10623494	C₄	1	4	yes	—
14	C₂×C₄×#10623494	C₄	1	2	yes	—
15	C₂×C₄×#10623494	C₄	1	2	yes	—
16	C₂×C₄×#10623494	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#423×C₈₉	—	2	2	no	—
20	#423×C₈₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	8	no	—
24	C₂×#13083558	C₂³	1	4	yes	—
25	C₂²×#10623494	—	2	2	no	—
26	C₂²×#10623494	C₂×C₄	1	4	yes	—
27	C₂²×#10623494	C₂×C₄	1	4	yes	—
28	C₂²×#10623494	D₄	1	8	yes	—
29	C₂²×#10623494	Q₈	1	8	yes	—
30	C₂²×#10623494	C₂³	1	4	yes	—
31	#15667947	—	2	32	no	—
32	C₄⋊#10623494	—	2	8	no	—
33	C₄×#10623494	C₂×C₄	1	8	yes	—
34	C₄×#10623494	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₈₉	—	2	2	no	—
38	C₂×M₁₆×C₈₉	—	2	8	no	—
39	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	2	yes	—
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
48	C₄⋊C₈×C₈₉	—	2	8	no	—
49	#13083558	—	2	8	no	—
50	#13083558	—	4	4	no	—
51	C₂×#10623494	—	2	4	no	—
52	C₂×#10623494	—	4	4	no	—
53	C₂×#10623494	C₄⋊C₄	1	16	yes	—
54	C₂×#10623494	C₄⋊C₄	1	16	yes	—
55	C₂×#10623494	C₂²×C₄	1	8	yes	—
56	C₂×#10623494	C₂×D₄	1	8	yes	—
57	C₂×#10623494	C₂×Q₈	1	8	yes	—
58	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	—	2	8	no	—
60	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₈₉	—	2	8	no	—
73	M₁₆×C₈₉	—	4	4	no	—
74	C₂×C₈×C₈₉	—	2	4	no	—
75	C₂×C₈×C₈₉	—	4	4	no	—
76	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
78	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
80	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
81	C₄²×C₈₉	C₂×D₄	1	4	yes	—
82	C₂³×C₈₉	—	2	4	no	—
83	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₈₉	C₄⋊Q₈	1	4	yes	—
86	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	—	2	4	no	—
88	C₂×C₄×C₈₉	—	2	8	no	—
89	C₂×C₄×C₈₉	—	2	8	no	—
90	C₂×C₄×C₈₉	#108	1	8	yes	complemented
91	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₈₉	#127	1	2	yes	—
98	C₂×C₄×C₈₉	#127	1	2	yes	—
99	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
101	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
106	C₈×C₈₉	—	4	4	no	—
107	#10623494	—	4	4	no	—
108	#10623494	C₂×C₄⋊C₄	1	16	yes	—
109	C₂²×C₈₉	—	2	8	no	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	—	2	2	no	—
112	C₂²×C₈₉	#384	1	2	yes	—
113	C₂²×C₈₉	#384	1	2	yes	—
114	C₂²×C₈₉	C₂×#108	1	4	yes	—
115	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
117	C₄×C₈₉	—	2	4	no	—
118	C₄×C₈₉	#384	1	2	yes	—
119	C₄×C₈₉	#384	1	2	yes	—
120	C₄×C₈₉	C₂×#108	1	8	yes	complemented
121	C₄×C₈₉	C₂²×C₄⋊C₄	1	2	yes	—
122	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
123	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₂×#1848	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	—	2	2	no	—
127	C₂×C₈₉	#1848	1	4	yes	complemented
128	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
129	C₂×C₈₉	C₂²×#108	1	2	yes	—
130	C₂²×#108	—	89	4	no	—
131	C₂×#423	—	89	1	no	—
132	C₂×#423	D₈₉	1	1	yes	characteristic
133	C₂²×C₄×C₈	—	89	1	no	—
134	#1848	—	89	8	no	—
135	C₈₉	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
136	C₂²×M₁₆	—	89	2	no	—
137	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
138	C₂³×C₈	—	89	2	no	—
139	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
140	C₂×#108	—	89	16	no	—
141	C₂×#108	—	178	16	no	—
142	#423	—	2	2	no	—
143	#423	—	89	4	no	—
144	#423	—	178	2	no	—
145	#423	C₂×D₈₉	1	4	yes	—
146	C₂×C₄⋊C₈	—	2	2	no	—
147	C₂×C₄⋊C₈	—	178	2	no	—
148	C₂×C₄×C₈	—	89	2	no	—
149	C₂×C₄×C₈	—	89	4	no	—
150	C₂×C₄×C₈	—	89	2	no	—
151	C₂×C₄×C₈	—	89	4	no	—
152	C₂³×C₄	Dic₁₇₈	1	2	yes	—
153	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	8	no	—
156	C₂×M₁₆	—	89	4	no	—
157	C₂×M₁₆	—	178	8	no	—
158	C₂×M₁₆	—	178	2	no	—
159	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	89	4	no	—
162	C₂²×C₈	—	89	4	no	—
163	C₂²×C₈	—	89	8	no	—
164	C₂²×C₈	—	89	8	no	—
165	C₂²×C₈	—	89	4	no	—
166	C₂²×C₈	—	178	2	no	—
167	C₂×C₄²	Dic₁₇₈	1	2	yes	—
168	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₈₉	1	2	yes	—
170	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
173	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
174	#108	—	178	32	no	—
175	C₄⋊C₈	—	2	8	no	—
176	C₄⋊C₈	—	178	8	no	—
177	C₄×C₈	—	89	8	no	—
178	C₄×C₈	—	89	8	no	—
179	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	#13083553	1	2	yes	—
182	C₂²×C₄	#13083553	1	4	yes	—
183	C₂²×C₄	#13083553	1	2	yes	—
184	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
186	C₂²×C₄	D₄×D₈₉	1	4	yes	—
187	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
189	C₂²×C₄	Q₈×D₈₉	1	4	yes	—
190	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
191	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
192	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
193	M₁₆	—	2	8	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	356	4	no	—
196	M₁₆	—	178	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	89	8	no	—
200	C₂×C₈	—	89	8	no	—
201	C₂×C₈	—	89	8	no	—
202	C₂×C₈	—	89	16	no	—
203	C₂×C₈	—	89	16	no	—
204	C₂×C₈	—	356	4	no	—
205	C₂×C₈	—	178	4	no	—
206	C₄²	#13083553	1	2	yes	—
207	C₄²	#13083553	1	4	yes	—
208	C₄²	#13083553	1	2	yes	—
209	C₄²	C₂×C₄×D₈₉	1	2	yes	—
210	C₄²	C₂×C₄×D₈₉	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₈₉	1	4	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂×#13083553	1	1	yes	characteristic
214	C₂³	C₂×#13083553	1	1	yes	characteristic
215	C₂³	#15668011	1	4	yes	—
216	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	#15667947	1	8	yes	complemented
221	C₂×C₄	C₄⋊C₄×D₈₉	1	4	yes	—
222	C₂×C₄	C₄⋊C₄×D₈₉	1	4	yes	—
223	C₂×C₄	C₂×#13083553	1	4	yes	—
224	C₂×C₄	C₂×#13083553	1	4	yes	—
225	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
226	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
227	C₂×C₄	#15668008	1	2	yes	—
228	C₂×C₄	#15668008	1	2	yes	—
229	C₂×C₄	#15668010	1	2	yes	—
230	C₂×C₄	#15668010	1	2	yes	—
231	C₂×C₄	#15668011	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₈₉	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	89	16	no	—
238	C₈	—	356	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	#19151467	1	2	yes	—
243	C₂²	#19151467	1	2	yes	—
244	C₂²	C₂×#15667947	1	4	yes	—
245	C₂²	C₂²×#13083553	1	1	yes	characteristic
246	C₂²	C₂×#15668011	1	2	yes	—
247	C₄	—	2	4	no	—
248	C₄	#19151467	1	2	yes	—
249	C₄	#19151467	1	2	yes	—
250	C₄	C₂×#15667947	1	8	yes	complemented
251	C₄	C₂×C₄⋊C₄×D₈₉	1	2	yes	—
252	C₄	C₂×#15668008	1	1	yes	characteristic
253	C₄	C₂×#15668010	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	#26020423	1	4	yes	direct product factor
257	C₂	C₂×#19151467	1	1	yes	characteristic
258	C₂	C₂²×#15667947	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#423×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#423	D₈₉
hypocenter	C₈₉	C₂×#1848
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #26020423	C₂	direct (N × H)
90. C₂×C₄×C₈₉	#108	semidirect
120. C₄×C₈₉	C₂×#108	semidirect
127. C₂×C₈₉	#1848	semidirect
135. C₈₉	C₂×#1848	semidirect
220. C₂×C₄	#15667947	semidirect
250. C₄	C₂×#15667947	semidirect
256. C₂	#26020423	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#15667947	2
C₂×#19151622	2
C₂²×C₄×#10623494	2
#26020423	2
C₂×#423×C₈₉	2
C₂×#1848	89

Subgroups — C2×#26020423 #36554883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26020423 #36554883