Subgroups — C₂×#26019099 #36523945

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151725	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#13083564	C₂	1	1	yes	maximal, characteristic
4	#26019099	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151350	C₂	1	1	yes	maximal, characteristic
6	C₂×#19151330	C₂	1	1	yes	maximal, characteristic
7	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×#13083564	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	#19151725	—	2	2	no	—
12	#19151725	C₂²	1	4	yes	—
13	Q₈×#10623494	—	2	2	no	—
14	Q₈×#10623494	C₂²	1	4	yes	—
15	C₄×#13083564	C₄	1	4	yes	—
16	C₄×#13083564	C₂²	1	4	yes	—
17	C₂×#15667945	C₄	1	1	yes	characteristic
18	C₂×C₄⋊#10623494	—	2	1	no	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667941	—	2	1	no	—
21	C₂×#15667936	C₄	1	1	yes	characteristic
22	C₂×#15667935	C₄	1	1	yes	characteristic
23	C₂×#15667922	—	2	1	no	—
24	C₂×C₄×#10623494	—	2	1	no	—
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
27	#19151350	C₂²	1	4	yes	—
28	#19151330	C₂²	1	4	yes	—
29	#19151320	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	Q₈⋊C₈×C₈₉	C₂²	1	4	yes	—
34	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#13083564	—	2	4	no	—
36	C₂×#13083564	C₂×C₄	1	4	yes	—
37	C₂²×#10623494	—	2	1	no	—
38	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
39	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
40	C₂²×#10623494	D₄	1	1	yes	characteristic
41	C₂²×#10623494	D₄	1	1	yes	characteristic
42	C₄×Dic₁₇₈	—	2	1	no	—
43	C₄×Dic₁₇₈	—	2	1	no	—
44	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₇₈	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083553	—	2	1	no	—
48	C₂×C₄×Dic₈₉	—	2	1	no	—
49	#15667945	C₂×C₄	1	4	yes	—
50	C₄⋊#10623494	—	2	4	no	—
51	C₄⋊#10623494	—	2	1	no	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	#15667941	—	2	4	no	—
54	#15667936	C₂×C₄	1	4	yes	—
55	#15667935	C₂×C₄	1	4	yes	—
56	#15667922	—	2	4	no	—
57	C₄×#10623494	—	2	4	no	—
58	C₄×#10623494	C₂×C₄	1	2	yes	—
59	C₄×#10623494	D₄	1	2	yes	—
60	C₄×#10623494	C₂³	1	2	yes	—
61	C₄×#10623494	C₂³	1	2	yes	—
62	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	—	2	1	no	—
64	C₂²×C₈×C₈₉	—	2	1	no	—
65	C₄×Q₈×C₈₉	—	2	1	no	—
66	C₄×Q₈×C₈₉	—	2	1	no	—
67	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
68	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	—	2	1	no	—
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	—	2	1	no	—
74	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
75	C₄×C₈×C₈₉	—	2	1	no	—
76	C₄×C₈×C₈₉	C₂³	1	2	yes	—
77	C₂×Dic₁₇₈	—	2	2	no	—
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₈₉	—	2	1	no	—
82	C₂²×Dic₈₉	—	2	1	no	—
83	#13083564	—	2	8	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₂×#10623494	—	2	2	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	—	4	2	no	—
89	C₂×#10623494	Q₁₆	1	4	yes	complemented
90	C₂×#10623494	C₂²×C₄	1	2	yes	—
91	C₂×#10623494	C₂²×C₄	1	2	yes	—
92	C₂×#10623494	C₂×D₄	1	2	yes	—
93	C₂×#10623494	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
96	#13083553	—	2	2	no	—
97	#13083553	—	2	2	no	—
98	C₄×Dic₈₉	—	2	2	no	—
99	C₄×Dic₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	—	2	2	no	—
102	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
103	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	4	2	no	—
112	C₂×C₈×C₈₉	—	4	2	no	—
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	2	1	no	—
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
117	C₄²×C₈₉	—	2	2	no	—
118	C₄²×C₈₉	—	2	2	no	—
119	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₈₉	#41	1	1	yes	characteristic
122	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
124	Q₈×C₈₉	—	2	2	no	—
125	Q₈×C₈₉	—	2	4	no	—
126	Q₈×C₈₉	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	4	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₈₉	#131	1	2	yes	—
137	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₈₉	#135	1	2	yes	—
139	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	Dic₁₇₈	—	2	2	no	—
147	Dic₁₇₈	—	2	2	no	—
148	Dic₁₇₈	—	4	2	no	—
149	C₈×C₈₉	—	4	2	no	—
150	C₈×C₈₉	—	4	2	no	—
151	#10623494	—	2	2	no	—
152	#10623494	—	2	2	no	—
153	#10623494	—	2	2	no	—
154	#10623494	—	4	2	no	—
155	#10623494	C₂×Q₁₆	1	4	yes	complemented
156	C₂²×C₈₉	#435	1	2	yes	—
157	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
158	C₂²×C₈₉	#444	1	2	yes	—
159	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₈₉	—	2	4	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	4	2	no	—
164	C₄×C₈₉	#435	1	2	yes	—
165	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
166	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
167	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
168	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
169	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	Dic₈₉	—	4	2	no	—
172	C₂×#1584	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	#1584	1	4	yes	complemented
174	C₂×C₈₉	C₂×#435	1	1	yes	characteristic
175	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
176	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
177	C₂×#446	—	89	1	no	—
178	C₂×C₈×Q₈	—	89	1	no	—
179	C₂×C₄×Q₁₆	—	89	1	no	—
180	#1584	—	89	8	no	—
181	C₂×#335	—	89	1	no	—
182	C₂×Q₈⋊C₈	—	89	1	no	—
183	C₂×Q₈⋊C₈	D₈₉	1	1	yes	characteristic
184	C₂×#322	—	89	1	no	—
185	C₈₉	C₂×#1584	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₁₆	—	89	1	no	—
187	C₂×C₄×Q₈	—	89	1	no	—
188	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
189	#446	—	89	4	no	—
190	#446	—	178	2	no	—
191	C₈×Q₈	—	89	4	no	—
192	C₈×Q₈	—	178	2	no	—
193	C₄×Q₁₆	—	89	4	no	—
194	C₄×Q₁₆	—	89	4	no	—
195	C₂×#107	—	89	1	no	—
196	C₂×C₄⋊C₈	—	89	1	no	—
197	C₂×C₄⋊C₈	—	178	1	no	—
198	C₂×C₄⋊C₈	—	178	1	no	—
199	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
200	C₂×#103	—	89	1	no	—
201	C₂×#103	—	89	1	no	—
202	C₂×#97	—	178	1	no	—
203	C₂×C₄×C₈	—	89	1	no	—
204	C₂×C₄×C₈	—	89	1	no	—
205	C₂×C₄×C₈	—	178	1	no	—
206	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
207	#335	—	89	4	no	—
208	Q₈⋊C₈	—	89	4	no	—
209	Q₈⋊C₈	C₂×D₈₉	1	4	yes	—
210	#322	—	89	4	no	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
213	C₂×Q₁₆	—	89	4	no	—
214	C₂×Q₁₆	—	178	4	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₂²×C₈	—	89	1	no	—
221	C₂²×C₈	—	178	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	89	2	no	—
225	C₄×Q₈	—	89	2	no	—
226	C₄×Q₈	—	178	1	no	—
227	C₄×Q₈	—	178	1	no	—
228	C₄×Q₈	C₄×D₈₉	1	2	yes	—
229	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	89	1	no	—
232	C₂×C₄⋊C₄	—	178	1	no	—
233	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	—	178	1	no	—
236	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
237	#107	—	89	4	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	89	2	no	—
240	C₄⋊C₈	—	178	4	no	—
241	C₄⋊C₈	—	178	4	no	—
242	C₄⋊C₈	—	178	1	no	—
243	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
244	#103	—	89	4	no	—
245	#103	—	89	4	no	—
246	#97	—	178	4	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	89	2	no	—
249	C₄×C₈	—	89	2	no	—
250	C₄×C₈	—	89	2	no	—
251	C₄×C₈	—	89	2	no	—
252	C₄×C₈	—	178	4	no	—
253	C₄×C₈	C₂²×D₈₉	1	2	yes	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	89	2	no	—
257	C₂×Q₈	—	178	2	no	—
258	C₂×Q₈	—	178	2	no	—
259	C₂×Q₈	—	178	1	no	—
260	C₂×Q₈	#13083557	1	4	yes	—
261	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	178	1	no	—
265	C₂²×C₄	—	178	1	no	—
266	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
268	C₂²×C₄	#13083573	1	1	yes	characteristic
269	Q₁₆	—	178	8	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	89	4	no	—
275	C₂×C₈	—	89	2	no	—
276	C₂×C₈	—	89	2	no	—
277	C₂×C₈	—	89	2	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	356	2	no	—
280	C₂×C₈	—	178	2	no	—
281	C₂×C₈	—	178	2	no	—
282	C₂×C₈	—	178	2	no	—
283	C₂×C₈	—	178	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	89	2	no	—
288	C₄⋊C₄	—	178	1	no	—
289	C₄⋊C₄	—	178	2	no	—
290	C₄⋊C₄	—	178	2	no	—
291	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	178	2	no	—
295	C₄²	—	178	2	no	—
296	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
297	C₄²	D₄×D₈₉	1	1	yes	characteristic
298	C₄²	#13083573	1	1	yes	characteristic
299	C₄²	C₂³×D₈₉	1	1	yes	characteristic
300	C₂³	#15667972	1	1	yes	characteristic
301	Q₈	—	2	4	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	356	2	no	—
304	Q₈	—	178	2	no	—
305	Q₈	—	178	2	no	—
306	Q₈	C₂×#13083557	1	4	yes	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	178	2	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	#15667972	1	1	yes	characteristic
317	C₂×C₄	#15667972	1	1	yes	characteristic
318	C₂×C₄	#15667972	1	1	yes	characteristic
319	C₂×C₄	C₂×#13083557	1	2	yes	—
320	C₂×C₄	#15668020	1	2	yes	—
321	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
322	C₂×C₄	#15668033	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
325	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	89	4	no	—
329	C₈	—	356	2	no	—
330	C₈	—	178	2	no	—
331	C₈	—	178	2	no	—
332	C₈	—	178	2	no	—
333	C₂²	#19151666	1	2	yes	—
334	C₂²	#19151690	1	2	yes	—
335	C₂²	#19151717	1	2	yes	—
336	C₂²	C₂×#15667972	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	4	no	—
340	C₄	—	4	2	no	—
341	C₄	—	356	2	no	—
342	C₄	—	356	2	no	—
343	C₄	#19151666	1	2	yes	—
344	C₄	#19151690	1	2	yes	—
345	C₄	C₂²×#13083557	1	1	yes	characteristic
346	C₄	C₂×#15668020	1	1	yes	characteristic
347	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
348	C₄	C₂×#15668033	1	1	yes	characteristic
349	C₂	#26019099	1	4	yes	direct product factor
350	C₂	C₂×#19151666	1	1	yes	characteristic
351	C₂	C₂×#19151690	1	1	yes	characteristic
352	C₂	C₂×#19151717	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×Q₈⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×Q₈⋊C₈	D₈₉
hypocenter	C₈₉	C₂×#1584
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
4. #26019099	C₂	direct (N × H)
89. C₂×#10623494	Q₁₆	semidirect
155. #10623494	C₂×Q₁₆	semidirect
173. C₂×C₈₉	#1584	semidirect
185. C₈₉	C₂×#1584	semidirect
349. C₂	#26019099	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×#19151725	2
C₂×Q₈×#10623494	2
C₂×C₄×#13083564	2
#26019099	2
C₂×#19151350	2
C₂×#19151330	2
C₂×#19151320	2
C₂×Q₈⋊C₈×C₈₉	2
C₂×#1584	89

Subgroups — C2×#26019099 #36523945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019099 #36523945