Subgroups — C₂×#26018165 #36444841

Order 22784.

658 conjugacy classes of subgroups.

The subgroup lattice has 268 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	2	yes	maximal
2	#26018165	C₂	1	4	yes	maximal, direct product factor
3	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	C₄	1	4	yes	—
5	#19151622	—	2	2	no	—
6	#19151622	—	2	2	no	—
7	#19151622	C₄	1	4	yes	—
8	#19151622	C₂²	1	4	yes	—
9	C₂×C₄⋊#10623494	—	2	2	no	—
10	C₂×C₄⋊#10623494	—	2	2	no	—
11	C₂²×M₁₆×C₈₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	#423×C₈₉	—	2	2	no	—
14	#423×C₈₉	C₄	1	2	yes	—
15	#423×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
17	C₂×#13083558	—	2	4	no	—
18	C₂×#13083558	—	4	8	no	—
19	C₂×#13083558	C₂×C₄	1	8	yes	—
20	C₂²×#10623494	—	2	4	no	—
21	C₄⋊#10623494	—	2	4	no	—
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₄⋊#10623494	—	2	4	no	—
25	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₈₉	—	2	2	no	—
28	C₂×M₁₆×C₈₉	—	4	4	no	—
29	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	—	2	2	no	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	2	yes	—
36	C₂×C₄²×C₈₉	D₄	1	2	yes	—
37	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	—	2	4	no	—
41	C₄⋊C₈×C₈₉	—	2	4	no	—
42	#13083558	—	4	8	no	—
43	#13083558	—	4	8	no	—
44	C₂×#10623494	—	2	8	no	—
45	C₂×#10623494	—	4	8	no	—
46	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	—	2	4	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	4	no	—
50	C₂²×C₄×C₈₉	—	2	2	no	—
51	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	M₁₆×C₈₉	—	4	4	no	—
64	M₁₆×C₈₉	—	4	4	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	—	4	4	no	—
67	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
71	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
73	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₈₉	C₂×D₄	1	2	yes	—
75	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₈₉	—	2	4	no	—
78	C₂³×C₈₉	—	2	2	no	—
79	C₂³×C₈₉	#95	1	1	yes	characteristic
80	C₂³×C₈₉	#95	1	1	yes	characteristic
81	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₈₉	—	2	4	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	4	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	4	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#95	1	2	yes	—
98	C₂×C₄×C₈₉	#95	1	2	yes	—
99	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#100	1	4	yes	—
101	C₂×C₄×C₈₉	#100	1	2	yes	—
102	C₂×C₄×C₈₉	#101	1	4	yes	—
103	C₂×C₄×C₈₉	#101	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₈×C₈₉	—	4	4	no	—
109	#10623494	—	4	8	no	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	4	4	no	—
115	C₂²×C₈₉	#343	1	4	yes	—
116	C₂²×C₈₉	#343	1	2	yes	—
117	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	#343	1	4	yes	—
123	C₄×C₈₉	#343	1	2	yes	—
124	C₄×C₈₉	C₂×#100	1	2	yes	—
125	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
126	C₄×C₈₉	C₂×#101	1	2	yes	—
127	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
128	C₂×#1309	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	—	4	1	no	—
130	C₂×C₈₉	—	4	1	no	—
131	C₂×C₈₉	#1309	1	4	yes	complemented
132	C₂×C₈₉	C₂×#343	1	2	yes	—
133	C₂×C₈₉	C₂×#343	1	1	yes	characteristic
134	C₂×#423	—	89	2	no	—
135	C₂×#423	D₈₉	1	1	yes	characteristic
136	#1309	—	89	4	no	—
137	C₈₉	C₂×#1309	1	1	yes	characteristic, complemented, Sylow
138	C₂²×M₁₆	—	89	4	no	—
139	C₂²×M₁₆	Dic₈₉	1	2	yes	—
140	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
141	#423	—	2	2	no	—
142	#423	—	89	4	no	—
143	#423	—	89	4	no	—
144	#423	—	178	2	no	—
145	#423	—	178	2	no	—
146	#423	Dic₈₉	1	2	yes	—
147	#423	C₂×D₈₉	1	2	yes	—
148	C₂×C₄⋊C₈	—	2	2	no	—
149	C₂×C₄⋊C₈	—	178	2	no	—
150	C₂×C₄⋊C₈	—	178	2	no	—
151	C₂³×C₄	C₄×D₈₉	1	2	yes	—
152	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	4	4	no	—
155	C₂×M₁₆	—	89	8	no	—
156	C₂×M₁₆	—	356	8	no	—
157	C₂×M₁₆	—	178	4	no	—
158	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	178	4	no	—
161	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
162	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₈₉	1	2	yes	—
164	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
165	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
169	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	178	4	no	—
173	C₄⋊C₈	—	178	4	no	—
174	C₄⋊C₈	—	178	4	no	—
175	C₄⋊C₈	—	178	4	no	—
176	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
182	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
183	C₂²×C₄	#13083551	1	2	yes	—
184	C₂²×C₄	#13083551	1	2	yes	—
185	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
187	C₂²×C₄	#13083553	1	2	yes	—
188	C₂²×C₄	#13083553	1	2	yes	—
189	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
193	M₁₆	—	4	4	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	356	8	no	—
196	M₁₆	—	356	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	356	8	no	—
200	C₂×C₈	—	178	8	no	—
201	C₄²	#13083551	1	2	yes	—
202	C₄²	#13083551	1	2	yes	—
203	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
204	C₄²	#13083553	1	2	yes	—
205	C₄²	#13083553	1	2	yes	—
206	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
207	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
209	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
210	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
211	C₂³	—	2	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#15667919	1	1	yes	characteristic
214	C₂³	#15667919	1	1	yes	characteristic
215	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	#15667919	1	1	yes	characteristic
231	C₂×C₄	#15667919	1	2	yes	—
232	C₂×C₄	#15667919	1	2	yes	—
233	C₂×C₄	#15667919	1	1	yes	characteristic
234	C₂×C₄	#15667928	1	4	yes	—
235	C₂×C₄	#15667929	1	2	yes	—
236	C₂×C₄	#15667930	1	4	yes	—
237	C₂×C₄	#15667931	1	2	yes	—
238	C₂×C₄	C₂×#13083551	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂×#13083553	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
242	C₈	—	4	4	no	—
243	C₈	—	356	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	4	no	—
249	C₂²	#19151366	1	4	yes	—
250	C₂²	#19151367	1	2	yes	—
251	C₂²	C₂×#15667919	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	#19151366	1	4	yes	—
257	C₄	#19151367	1	2	yes	—
258	C₄	C₂×#15667928	1	2	yes	—
259	C₄	C₂×#15667929	1	1	yes	characteristic
260	C₄	C₂×#15667930	1	2	yes	—
261	C₄	C₂×#15667931	1	1	yes	characteristic
262	C₂	—	4	1	no	—
263	C₂	—	4	1	no	—
264	C₂	#26018165	1	4	yes	direct product factor
265	C₂	C₂×#19151366	1	2	yes	—
266	C₂	C₂×#19151367	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#423×C₈₉	C₂
socle	C₂³×C₈₉	#95
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#423	D₈₉
hypocenter	C₈₉	C₂×#1309
LCS	G ▷ C₂³×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #26018165	C₂	direct (N × H)
131. C₂×C₈₉	#1309	semidirect
137. C₈₉	C₂×#1309	semidirect
264. C₂	#26018165	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
#26018165	2
C₂×#423×C₈₉	2
C₂×#1309	89

Subgroups — C2×#26018165 #36444841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018165 #36444841