Subgroups — C₂×#26018296 #36465649

Order 22784.

434 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151982	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018296	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
6	C₂²×#13083563	—	2	1	no	—
7	C₂²×D₇₁₂	C₄	1	1	yes	characteristic
8	C₂×#15668000	—	2	1	no	—
9	#19151982	—	2	1	no	—
10	#19151982	—	2	1	no	—
11	#19151982	C₄	1	2	yes	—
12	#19151982	C₂²	1	2	yes	—
13	C₂×C₄⋊#10623494	—	2	1	no	—
14	#19151349	—	2	1	no	—
15	#19151349	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₉	—	2	1	no	—
19	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂²×D₃₅₆	—	2	1	no	—
22	C₂×Dic₃₅₆	—	2	1	no	—
23	C₂×Dic₃₅₆	—	4	2	no	—
24	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
25	C₂×#13083563	—	2	2	no	—
26	C₂×#13083563	—	2	2	no	—
27	C₂×#13083563	—	4	2	no	—
28	C₂×#13083563	—	4	2	no	—
29	C₂×D₇₁₂	—	2	1	no	—
30	C₂×D₇₁₂	—	4	2	no	—
31	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
32	C₂²×#10623494	—	4	1	no	—
33	#15668000	—	2	2	no	—
34	#15668000	—	2	2	no	—
35	#15668000	—	4	1	no	—
36	#15668000	—	4	1	no	—
37	C₂×#13083551	—	4	1	no	—
38	C₂×#13083551	—	4	1	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	—	4	1	no	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	—	2	2	no	—
45	C₂×C₁₆×C₈₉	—	4	2	no	—
46	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂³×D₈₉	—	4	1	no	—
51	C₂×Dic₁₇₈	—	2	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂×Dic₁₇₈	—	4	2	no	—
54	C₂×Dic₁₇₈	—	4	1	no	—
55	C₂×D₃₅₆	—	2	2	no	—
56	C₂×D₃₅₆	—	4	2	no	—
57	C₂×D₃₅₆	—	4	2	no	—
58	C₂×D₃₅₆	—	4	1	no	—
59	C₂²×Dic₈₉	—	4	1	no	—
60	Dic₃₅₆	—	4	2	no	—
61	Dic₃₅₆	—	4	2	no	—
62	#13083563	—	4	4	no	—
63	#13083563	—	4	2	no	—
64	#13083563	—	4	2	no	—
65	D₇₁₂	—	4	2	no	—
66	D₇₁₂	—	4	2	no	—
67	C₂×#10623494	—	4	2	no	—
68	C₂×#10623494	—	8	2	no	—
69	#13083551	—	4	2	no	—
70	#13083551	—	4	2	no	—
71	#13083551	—	4	2	no	—
72	#13083551	—	4	2	no	—
73	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	D₈	1	2	yes	—
80	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₈₉	D₈	1	1	yes	characteristic
85	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
86	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₈₉	—	4	2	no	—
88	C₂³×C₈₉	#102	1	1	yes	characteristic
89	C₂²×D₈₉	—	4	2	no	—
90	C₂²×D₈₉	—	8	2	no	—
91	C₂²×D₈₉	—	8	2	no	—
92	C₂²×D₈₉	—	8	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	#100	1	2	yes	—
96	C₂×C₄×C₈₉	#102	1	2	yes	—
97	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×Dic₈₉	—	4	2	no	—
104	C₂×Dic₈₉	—	8	2	no	—
105	D₃₅₆	—	4	2	no	—
106	D₃₅₆	—	4	2	no	—
107	D₃₅₆	—	8	2	no	—
108	Dic₁₇₈	—	4	2	no	—
109	Dic₁₇₈	—	4	2	no	—
110	Dic₁₇₈	—	8	2	no	—
111	C₈×C₈₉	—	2	1	no	—
112	C₈×C₈₉	—	2	1	no	—
113	C₈×C₈₉	#102	1	2	yes	—
114	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
116	#10623494	—	8	2	no	—
117	C₂²×C₈₉	#331	1	2	yes	—
118	C₂²×C₈₉	#359	1	2	yes	—
119	C₂²×C₈₉	#360	1	2	yes	—
120	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
121	C₂×D₈₉	—	8	2	no	—
122	C₂×D₈₉	—	8	2	no	—
123	C₂×D₈₉	—	8	2	no	—
124	C₂×D₈₉	—	16	4	no	—
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	#331	1	2	yes	—
130	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
131	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
132	Dic₈₉	—	8	2	no	—
133	C₂×#1360	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	#1360	1	4	yes	complemented
135	C₂×C₈₉	C₂×#331	1	1	yes	characteristic
136	C₂×C₈₉	C₂×#359	1	1	yes	characteristic
137	C₂×C₈₉	C₂×#360	1	1	yes	characteristic
138	D₈₉	—	16	2	no	—
139	C₂×#495	—	89	1	no	—
140	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
141	C₂×#334	—	89	1	no	—
142	#1360	—	89	4	no	—
143	C₈₉	C₂×#1360	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₁₆	—	89	1	no	—
145	C₂²×SD₁₆	—	178	1	no	—
146	C₂²×D₈	—	89	1	no	—
147	C₂×#124	—	178	1	no	—
148	C₂²×C₁₆	—	2	1	no	—
149	#495	—	89	2	no	—
150	#495	—	89	2	no	—
151	#495	—	178	1	no	—
152	#495	—	178	1	no	—
153	C₂×C₄⋊C₈	—	178	1	no	—
154	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
155	C₄⋊C₁₆	—	2	1	no	—
156	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
157	#334	—	89	2	no	—
158	#334	—	178	1	no	—
159	C₂²×Q₈	—	178	1	no	—
160	C₂²×D₄	—	178	1	no	—
161	C₂×Q₁₆	—	89	2	no	—
162	C₂×Q₁₆	—	356	2	no	—
163	C₂×Q₁₆	—	178	1	no	—
164	C₂×SD₁₆	—	356	2	no	—
165	C₂×SD₁₆	—	356	2	no	—
166	C₂×SD₁₆	—	178	2	no	—
167	C₂×SD₁₆	—	178	2	no	—
168	C₂×D₈	—	89	2	no	—
169	C₂×D₈	—	356	2	no	—
170	C₂×D₈	—	178	1	no	—
171	C₂²×C₈	—	356	1	no	—
172	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
174	#124	—	356	1	no	—
175	#124	—	356	1	no	—
176	#124	—	178	2	no	—
177	#124	—	178	2	no	—
178	C₂×C₂²⋊C₄	—	356	1	no	—
179	C₂×C₂²⋊C₄	—	356	1	no	—
180	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	4	2	no	—
183	C₄⋊C₈	—	356	1	no	—
184	C₄⋊C₈	—	178	2	no	—
185	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₄×C₈	D₃₅₆	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
189	C₂⁴	—	356	1	no	—
190	C₂×Q₈	—	356	1	no	—
191	C₂×Q₈	—	356	2	no	—
192	C₂×Q₈	—	356	2	no	—
193	C₂×Q₈	—	178	2	no	—
194	C₂×D₄	—	356	2	no	—
195	C₂×D₄	—	356	2	no	—
196	C₂×D₄	—	356	1	no	—
197	C₂×D₄	—	178	2	no	—
198	C₂²×C₄	—	356	1	no	—
199	C₂²×C₄	#13083551	1	1	yes	characteristic
200	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
201	C₂²×C₄	#13083563	1	1	yes	characteristic
202	Q₁₆	—	356	2	no	—
203	Q₁₆	—	356	2	no	—
204	SD₁₆	—	356	2	no	—
205	SD₁₆	—	356	2	no	—
206	SD₁₆	—	356	4	no	—
207	D₈	—	356	2	no	—
208	D₈	—	356	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	356	2	no	—
211	C₂×C₈	—	712	2	no	—
212	C₂×C₈	#13083551	1	1	yes	characteristic
213	C₂×C₈	#13083551	1	1	yes	characteristic
214	C₂×C₈	#13083559	1	2	yes	—
215	C₂×C₈	#13083561	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
218	C₂²⋊C₄	—	356	2	no	—
219	C₂²⋊C₄	—	356	2	no	—
220	C₂²⋊C₄	—	356	2	no	—
221	C₂²⋊C₄	—	356	2	no	—
222	C₄²	#13083551	1	1	yes	characteristic
223	C₄²	D₇₁₂	1	1	yes	characteristic
224	C₄²	#13083563	1	1	yes	characteristic
225	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
226	C₁₆	—	4	2	no	—
227	C₂³	—	356	2	no	—
228	C₂³	—	712	2	no	—
229	C₂³	—	712	2	no	—
230	C₂³	—	712	2	no	—
231	C₂³	#15667934	1	1	yes	characteristic
232	Q₈	—	356	2	no	—
233	Q₈	—	356	2	no	—
234	Q₈	—	712	2	no	—
235	D₄	—	356	2	no	—
236	D₄	—	356	2	no	—
237	D₄	—	712	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	712	2	no	—
242	C₂×C₄	#15667928	1	2	yes	—
243	C₂×C₄	#15667932	1	2	yes	—
244	C₂×C₄	#15667934	1	1	yes	characteristic
245	C₂×C₄	#15667934	1	1	yes	characteristic
246	C₂×C₄	#15667934	1	1	yes	characteristic
247	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
248	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
249	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	712	2	no	—
253	C₈	#15667932	1	2	yes	—
254	C₈	C₂×#13083559	1	1	yes	characteristic
255	C₈	C₂×#13083561	1	1	yes	characteristic
256	C₂²	—	1424	4	no	—
257	C₂²	—	712	2	no	—
258	C₂²	—	712	2	no	—
259	C₂²	—	712	2	no	—
260	C₂²	#19151341	1	2	yes	—
261	C₂²	#19151406	1	2	yes	—
262	C₂²	#19151409	1	2	yes	—
263	C₂²	C₂×#15667934	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	712	2	no	—
269	C₄	#19151341	1	2	yes	—
270	C₄	C₂×#15667928	1	1	yes	characteristic
271	C₄	C₂×#15667932	1	1	yes	characteristic
272	C₂	—	1424	2	no	—
273	C₂	#26018296	1	4	yes	direct product factor
274	C₂	C₂×#19151341	1	1	yes	characteristic
275	C₂	C₂×#19151406	1	1	yes	characteristic
276	C₂	C₂×#19151409	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	#102
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	C₂×#1360
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151349	C₂	semidirect
3. #26018296	C₂	direct (N × H)
4. C₂×C₄⋊C₁₆×C₈₉	C₂	semidirect
15. #19151349	C₂²	semidirect
19. C₄⋊C₁₆×C₈₉	C₂²	semidirect
134. C₂×C₈₉	#1360	semidirect
140. C₂×C₄⋊C₁₆	D₈₉	semidirect
143. C₈₉	C₂×#1360	semidirect
156. C₄⋊C₁₆	C₂×D₈₉	semidirect
273. C₂	#26018296	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×#19151982	2
C₂×#19151349	2
#26018296	2
C₂×C₄⋊C₁₆×C₈₉	2
C₂×#1360	89

Subgroups — C2×#26018296 #36465649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018296 #36465649