Subgroups — C₂×#26018086 #36443883

Order 22784.

830 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018086	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10623494	—	2	1	no	—
6	C₂³×#10623494	C₄	1	2	yes	—
7	C₂×C₄⋊#10623494	—	2	2	no	—
8	C₂×C₄⋊#10623494	—	2	1	no	—
9	C₂×C₄⋊#10623494	C₄	1	4	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄×#10623494	—	2	2	no	—
12	C₂×C₄×#10623494	—	2	2	no	—
13	C₂×C₄×#10623494	C₄	1	2	yes	—
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂³×C₈×C₈₉	—	2	1	no	—
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
18	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
19	C₂×C₄⋊C₈×C₈₉	C₄	1	4	yes	—
20	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
21	C₂²×#10623494	—	2	4	no	—
22	C₂²×#10623494	—	2	4	no	—
23	C₂²×#10623494	—	2	4	no	—
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	4	2	no	—
26	C₂²×#10623494	—	4	2	no	—
27	C₂²×#10623494	C₈	1	8	yes	complemented
28	C₂²×#10623494	C₂×C₄	1	4	yes	—
29	C₄⋊#10623494	—	2	4	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	C₄⋊#10623494	C₂×C₄	1	4	yes	—
32	C₄×#10623494	—	2	4	no	—
33	C₄×#10623494	—	2	2	no	—
34	C₄×#10623494	—	2	2	no	—
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	—	2	4	no	—
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂²×C₈×C₈₉	—	4	2	no	—
41	C₂²×C₈×C₈₉	—	4	2	no	—
42	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	D₄	1	2	yes	—
48	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₈₉	—	2	4	no	—
51	C₄⋊C₈×C₈₉	—	2	2	no	—
52	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
53	C₂×#10623494	—	2	16	no	—
54	C₂×#10623494	—	2	4	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	4	8	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₂×#10623494	—	4	2	no	—
60	C₂×#10623494	C₂×C₈	1	8	yes	complemented
61	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	—	2	4	no	—
63	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
74	C₂²×C₄×C₈₉	D₈	1	2	yes	—
75	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	4	no	—
82	C₂×C₈×C₈₉	—	4	8	no	—
83	C₂×C₈×C₈₉	—	4	2	no	—
84	C₂×C₈×C₈₉	—	4	2	no	—
85	C₄²×C₈₉	C₄²	1	2	yes	—
86	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
90	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
91	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₈₉	#95	1	2	yes	—
96	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
97	C₂³×C₈₉	#106	1	2	yes	—
98	C₂³×C₈₉	#107	1	2	yes	—
99	C₂³×C₈₉	#108	1	4	yes	—
100	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	—	2	4	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	#95	1	2	yes	—
106	C₂×C₄×C₈₉	#95	1	2	yes	—
107	C₂×C₄×C₈₉	#95	1	2	yes	—
108	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
109	C₂×C₄×C₈₉	#97	1	4	yes	—
110	C₂×C₄×C₈₉	C₂²⋊C₈	1	8	yes	—
111	C₂×C₄×C₈₉	#102	1	4	yes	—
112	C₂×C₄×C₈₉	#102	1	4	yes	—
113	C₂×C₄×C₈₉	#103	1	4	yes	—
114	C₂×C₄×C₈₉	#103	1	4	yes	—
115	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
116	C₂×C₄×C₈₉	#106	1	2	yes	—
117	C₂×C₄×C₈₉	#107	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
130	C₈×C₈₉	—	4	4	no	—
131	#10623494	—	2	8	no	—
132	#10623494	—	4	4	no	—
133	C₂²×C₈₉	#334	1	8	yes	—
134	C₂²×C₈₉	#335	1	8	yes	—
135	C₂²×C₈₉	#336	1	4	yes	—
136	C₂²×C₈₉	#340	1	4	yes	—
137	C₂²×C₈₉	#341	1	4	yes	—
138	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
139	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
140	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
141	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
142	C₂²×C₈₉	C₂×#108	1	2	yes	—
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	#336	1	4	yes	—
147	C₄×C₈₉	#340	1	4	yes	—
148	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
149	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
150	C₄×C₈₉	C₂×C₂²⋊C₈	1	2	yes	—
151	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
155	C₂×#1277	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	#1277	1	8	yes	complemented
157	C₂×C₈₉	C₂×#334	1	2	yes	—
158	C₂×C₈₉	C₂×#335	1	2	yes	—
159	C₂×C₈₉	C₂×#336	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#341	1	1	yes	characteristic
162	C₂²×C₄⋊C₈	—	89	1	no	—
163	C₂²×C₄⋊C₈	D₈₉	1	1	yes	characteristic
164	C₂²×C₄×C₈	—	89	1	no	—
165	#1277	—	89	4	no	—
166	C₈₉	C₂×#1277	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₈	—	2	1	no	—
168	C₂³×C₈	—	89	2	no	—
169	C₂³×C₈	—	178	1	no	—
170	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	2	1	no	—
172	C₂×C₄⋊C₈	—	2	2	no	—
173	C₂×C₄⋊C₈	—	89	4	no	—
174	C₂×C₄⋊C₈	—	89	2	no	—
175	C₂×C₄⋊C₈	—	178	2	no	—
176	C₂×C₄⋊C₈	—	178	1	no	—
177	C₂×C₄⋊C₈	Dic₈₉	1	4	yes	—
178	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
179	C₂×C₄×C₈	—	89	2	no	—
180	C₂×C₄×C₈	—	89	2	no	—
181	C₂×C₄×C₈	—	178	2	no	—
182	C₂×C₄×C₈	—	178	2	no	—
183	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	4	2	no	—
189	C₂²×C₈	—	4	2	no	—
190	C₂²×C₈	—	89	8	no	—
191	C₂²×C₈	—	89	4	no	—
192	C₂²×C₈	—	356	2	no	—
193	C₂²×C₈	—	356	2	no	—
194	C₂²×C₈	—	178	4	no	—
195	C₂²×C₈	—	178	2	no	—
196	C₂²×C₈	—	178	4	no	—
197	C₂²×C₈	—	178	4	no	—
198	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₈₉	1	2	yes	—
201	C₂×C₄²	D₃₅₆	1	2	yes	—
202	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
203	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
205	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	89	4	no	—
209	C₄⋊C₈	—	178	4	no	—
210	C₄⋊C₈	—	178	2	no	—
211	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
212	C₄×C₈	—	178	4	no	—
213	C₄×C₈	—	178	2	no	—
214	C₄×C₈	—	178	2	no	—
215	C₂⁴	#13083553	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
218	C₂²×C₄	#13083551	1	2	yes	—
219	C₂²×C₄	#13083551	1	2	yes	—
220	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
221	C₂²×C₄	#13083553	1	1	yes	characteristic
222	C₂²×C₄	#13083553	1	2	yes	—
223	C₂²×C₄	#13083553	1	1	yes	characteristic
224	C₂²×C₄	#13083553	1	1	yes	characteristic
225	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
226	C₂²×C₄	C₈×D₈₉	1	4	yes	—
227	C₂²×C₄	#13083557	1	4	yes	—
228	C₂²×C₄	#13083559	1	2	yes	—
229	C₂²×C₄	#13083561	1	2	yes	—
230	C₂²×C₄	#13083562	1	2	yes	—
231	C₂²×C₄	#13083564	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	8	no	—
239	C₂×C₈	—	89	8	no	—
240	C₂×C₈	—	356	2	no	—
241	C₂×C₈	—	356	2	no	—
242	C₂×C₈	—	356	8	no	—
243	C₂×C₈	—	178	4	no	—
244	C₂×C₈	—	178	4	no	—
245	C₂×C₈	—	178	16	no	—
246	C₂×C₈	—	178	4	no	—
247	C₄²	C₄×Dic₈₉	1	2	yes	—
248	C₄²	#13083551	1	2	yes	—
249	C₄²	#13083551	1	2	yes	—
250	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
251	C₄²	#13083553	1	2	yes	—
252	C₄²	C₄⋊Dic₈₉	1	2	yes	—
253	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
256	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
257	C₂³	#15667919	1	2	yes	—
258	C₂³	#15667941	1	4	yes	—
259	C₂³	#15667943	1	2	yes	—
260	C₂³	#15667945	1	2	yes	—
261	C₂³	#15667947	1	4	yes	—
262	C₂³	C₂×#13083553	1	1	yes	characteristic
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	#15667919	1	2	yes	—
268	C₂×C₄	#15667919	1	2	yes	—
269	C₂×C₄	#15667919	1	2	yes	—
270	C₂×C₄	C₈×Dic₈₉	1	4	yes	—
271	C₂×C₄	#15667922	1	4	yes	—
272	C₂×C₄	#15667924	1	8	yes	—
273	C₂×C₄	#15667932	1	4	yes	—
274	C₂×C₄	#15667933	1	4	yes	—
275	C₂×C₄	#15667935	1	4	yes	—
276	C₂×C₄	#15667936	1	4	yes	—
277	C₂×C₄	#15667941	1	4	yes	—
278	C₂×C₄	#15667943	1	2	yes	—
279	C₂×C₄	#15667945	1	2	yes	—
280	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
281	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
282	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
284	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
287	C₂×C₄	C₂×#13083557	1	2	yes	—
288	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
289	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
290	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
291	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
292	C₈	—	4	4	no	—
293	C₈	—	356	4	no	—
294	C₈	—	178	8	no	—
295	C₂²	#19151348	1	8	yes	—
296	C₂²	#19151350	1	8	yes	—
297	C₂²	#19151352	1	4	yes	—
298	C₂²	#19151360	1	4	yes	—
299	C₂²	#19151362	1	4	yes	—
300	C₂²	C₂×#15667919	1	1	yes	characteristic
301	C₂²	C₂×#15667941	1	2	yes	—
302	C₂²	C₂×#15667943	1	1	yes	characteristic
303	C₂²	C₂×#15667945	1	1	yes	characteristic
304	C₂²	C₂×#15667947	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	#19151352	1	4	yes	—
309	C₄	#19151360	1	4	yes	—
310	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
311	C₄	C₂×#15667922	1	1	yes	characteristic
312	C₄	C₂×#15667924	1	2	yes	—
313	C₄	C₂×#15667932	1	1	yes	characteristic
314	C₄	C₂×#15667933	1	1	yes	characteristic
315	C₄	C₂×#15667935	1	1	yes	characteristic
316	C₄	C₂×#15667936	1	1	yes	characteristic
317	C₂	#26018086	1	8	yes	direct product factor
318	C₂	C₂×#19151348	1	2	yes	—
319	C₂	C₂×#19151350	1	2	yes	—
320	C₂	C₂×#19151352	1	1	yes	characteristic
321	C₂	C₂×#19151360	1	1	yes	characteristic
322	C₂	C₂×#19151362	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂²×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂²×C₄⋊C₈	D₈₉
hypocenter	C₈₉	C₂×#1277
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #26018086	C₂	direct (N × H)
27. C₂²×#10623494	C₈	semidirect
60. C₂×#10623494	C₂×C₈	semidirect
156. C₂×C₈₉	#1277	semidirect
166. C₈₉	C₂×#1277	semidirect
317. C₂	#26018086	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×C₄⋊#10623494	2
C₂²×C₄×#10623494	2
#26018086	2
C₂²×C₄⋊C₈×C₈₉	2
C₂×#1277	89

Subgroups — C2×#26018086 #36443883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018086 #36443883