Subgroups — C₂×#26019000 #36523402

Order 22784.

830 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	2	yes	maximal
2	#26019000	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151351	C₂	1	2	yes	maximal, complemented
4	C₂×#19151349	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	—	2	4	no	—
7	#19151651	C₂²	1	8	yes	—
8	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
9	C₂×C₂²⋊#10623494	—	2	2	no	—
10	C₂×C₄×#10623494	—	2	2	no	—
11	#19151351	C₂²	1	8	yes	complemented
12	#19151349	C₂²	1	8	yes	complemented
13	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	4	yes	—
22	C₂²×#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₄⋊#10623494	C₂³	1	8	yes	—
25	C₂²⋊#10623494	—	2	8	no	—
26	C₄×#10623494	—	2	8	no	—
27	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	—	2	4	no	—
31	C₂²×C₈×C₈₉	C₈	1	8	yes	complemented
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	2	yes	—
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
46	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₈₉	C₂³	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂×#10623494	—	2	8	no	—
53	C₂×#10623494	—	4	8	no	—
54	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	—	2	4	no	—
56	C₂²×C₄×C₈₉	—	2	4	no	—
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₈₉	—	2	8	no	—
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	—	2	2	no	—
74	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
75	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₈₉	C₂×C₈	1	16	yes	complemented
77	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
82	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
83	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₈₉	C₂×D₄	1	2	yes	—
87	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
89	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₉	—	2	2	no	—
92	C₂³×C₈₉	—	2	2	no	—
93	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
94	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₄⋊C₈	1	8	yes	—
103	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
104	C₂×C₄×C₈₉	#108	1	4	yes	—
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
114	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₈₉	#135	1	4	yes	—
116	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₈₉	—	2	2	no	—
120	C₈×C₈₉	—	2	2	no	—
121	C₈×C₈₉	C₂²×C₈	1	4	yes	—
122	C₈×C₈₉	C₂²×C₈	1	4	yes	—
123	#10623494	—	4	8	no	—
124	C₂²×C₈₉	—	2	8	no	—
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	—	2	2	no	—
127	C₂²×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
128	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
129	C₂²×C₈₉	#427	1	2	yes	—
130	C₂²×C₈₉	C₂×#108	1	2	yes	—
131	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
136	C₄×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
137	C₄×C₈₉	#427	1	2	yes	—
138	C₄×C₈₉	C₂×#108	1	4	yes	—
139	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#135	1	2	yes	—
142	C₂×#1564	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	4	no	—
144	C₂×C₈₉	#1564	1	4	yes	complemented
145	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
147	C₂×C₈₉	C₂²×#108	1	1	yes	characteristic
148	C₂×#431	—	89	2	no	—
149	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
150	#1564	—	89	8	no	—
151	C₂×#335	—	89	2	no	—
152	C₂×#334	—	89	2	no	—
153	C₈₉	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
155	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
156	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
157	#431	—	89	8	no	—
158	#431	—	178	4	no	—
159	C₂×C₄⋊C₈	—	89	4	no	—
160	C₂×C₂²⋊C₈	—	178	2	no	—
161	C₂×C₄×C₈	—	178	2	no	—
162	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
163	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
164	C₂×C₄×C₈	Dic₈₉	1	4	yes	—
165	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₈₉	1	4	yes	—
168	#335	—	89	8	no	—
169	#334	—	89	8	no	—
170	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
171	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	178	4	no	—
175	C₂²×C₈	#10623494	1	8	yes	complemented
176	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
178	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
179	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
182	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
183	C₂×C₄²	Dic₁₇₈	1	2	yes	—
184	C₂×C₄²	D₃₅₆	1	2	yes	—
185	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
188	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
190	C₄⋊C₈	—	89	8	no	—
191	C₄⋊C₈	—	178	4	no	—
192	C₂²⋊C₈	—	178	8	no	—
193	C₄×C₈	—	178	8	no	—
194	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
195	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
196	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
197	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
198	C₄×C₈	C₂²×D₈₉	1	2	yes	—
199	C₄×C₈	C₂²×D₈₉	1	2	yes	—
200	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
206	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
207	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×#10623494	1	4	yes	—
210	C₂²×C₄	#13083558	1	4	yes	—
211	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
213	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
214	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	356	8	no	—
221	C₂×C₈	—	178	8	no	—
222	C₂×C₈	C₂×#10623494	1	4	yes	—
223	C₂×C₈	C₂×#10623494	1	4	yes	—
224	C₂×C₈	C₂×#10623494	1	16	yes	complemented
225	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
226	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
227	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
228	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
229	C₄²	C₄⋊Dic₈₉	1	2	yes	—
230	C₄²	C₄⋊Dic₈₉	1	2	yes	—
231	C₄²	C₄⋊Dic₈₉	1	2	yes	—
232	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
233	C₄²	C₂²×Dic₈₉	1	2	yes	—
234	C₄²	C₂×D₃₅₆	1	2	yes	—
235	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
236	C₄²	C₂×Dic₁₇₈	1	2	yes	—
237	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
238	C₄²	C₂³×D₈₉	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₄⋊#10623494	1	4	yes	—
242	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
243	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₄⋊#10623494	1	8	yes	—
251	C₂×C₄	C₄⋊#10623494	1	4	yes	—
252	C₂×C₄	#15667948	1	4	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
259	C₂×C₄	C₂²×#10623494	1	2	yes	—
260	C₂×C₄	C₂²×#10623494	1	4	yes	—
261	C₂×C₄	C₂×#13083558	1	4	yes	—
262	C₂×C₄	C₂×#13083558	1	2	yes	—
263	C₂×C₄	#15668034	1	4	yes	—
264	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	356	8	no	—
270	C₈	C₂²×#10623494	1	4	yes	—
271	C₈	C₂²×#10623494	1	4	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	8	no	—
275	C₂²	C₂×C₄⋊#10623494	1	4	yes	—
276	C₂²	C₂×C₄⋊#10623494	1	2	yes	—
277	C₂²	#19151636	1	2	yes	—
278	C₂²	C₂×#15667948	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	C₂×C₄⋊#10623494	1	2	yes	—
284	C₄	C₂×C₄⋊#10623494	1	4	yes	—
285	C₄	#19151636	1	2	yes	—
286	C₄	C₂×#15667948	1	4	yes	—
287	C₄	C₂³×#10623494	1	1	yes	characteristic
288	C₄	C₂²×#13083558	1	1	yes	characteristic
289	C₄	C₂×#15668034	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	#26019000	1	4	yes	direct product factor
292	C₂	C₂²×C₄⋊#10623494	1	1	yes	characteristic
293	C₂	C₂×#19151636	1	1	yes	characteristic
294	C₂	C₂²×#15667948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₃₅₆
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	C₂×#1564
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #26019000	C₂	direct (N × H)
3. C₂×#19151351	C₂	semidirect
4. C₂×#19151349	C₂	semidirect
11. #19151351	C₂²	semidirect
12. #19151349	C₂²	semidirect
31. C₂²×C₈×C₈₉	C₈	semidirect
76. C₂×C₈×C₈₉	C₂×C₈	semidirect
144. C₂×C₈₉	#1564	semidirect
153. C₈₉	C₂×#1564	semidirect
175. C₂²×C₈	#10623494	semidirect
224. C₂×C₈	C₂×#10623494	semidirect
291. C₂	#26019000	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×#19151651	2
#26019000	2
C₂×#19151351	2
C₂×#19151349	2
C₂²×C₄×C₈×C₈₉	2
C₂×#1564	89

Subgroups — C2×#26019000 #36523402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019000 #36523402