Subgroups — C₂×#26019101 #36523947

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×#19151722	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#26019101	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151328	C₂	1	1	yes	maximal, characteristic
6	C₂×#19151328	C₂	1	1	yes	maximal, characteristic
7	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
8	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	#19151725	—	2	2	no	—
13	#19151725	C₂²	1	4	yes	—
14	#19151722	—	2	2	no	—
15	#19151722	C₂²	1	4	yes	—
16	C₄×Dic₃₅₆	C₄	1	4	yes	—
17	C₄×Dic₃₅₆	C₂²	1	4	yes	—
18	C₂×#15667946	C₄	1	1	yes	characteristic
19	C₂×#15667941	—	2	1	no	—
20	C₂×#15667941	—	2	1	no	—
21	C₂×#15667937	C₄	1	1	yes	characteristic
22	C₂×#15667937	C₄	1	1	yes	characteristic
23	C₂×#15667922	—	2	1	no	—
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₈₉	—	2	1	no	—
27	#19151328	C₂²	1	4	yes	—
28	#19151328	C₂²	1	4	yes	—
29	#19151320	C₂²	1	4	yes	—
30	C₂×C₄⋊C₈×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	#335×C₈₉	C₂²	1	4	yes	—
34	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₃₅₆	—	2	4	no	—
37	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
38	C₂²×#10623494	—	2	1	no	—
39	C₂²×#10623494	D₄	1	1	yes	characteristic
40	C₂²×#10623494	D₄	1	1	yes	characteristic
41	C₄×Dic₁₇₈	—	2	1	no	—
42	C₄×Dic₁₇₈	—	2	1	no	—
43	C₄×Dic₁₇₈	—	2	1	no	—
44	C₄×Dic₁₇₈	—	2	1	no	—
45	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
46	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₇₈	C₂³	1	2	yes	—
48	C₄×Dic₁₇₈	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×#13083553	—	2	1	no	—
52	C₂×#13083553	—	2	1	no	—
53	C₂×C₄×Dic₈₉	—	2	1	no	—
54	C₂×C₄×Dic₈₉	—	2	1	no	—
55	#15667946	C₂×C₄	1	4	yes	—
56	#15667941	—	2	4	no	—
57	#15667941	—	2	4	no	—
58	#15667937	C₂×C₄	1	4	yes	—
59	#15667937	C₂×C₄	1	4	yes	—
60	#15667922	—	2	4	no	—
61	C₄×#10623494	—	2	1	no	—
62	C₄×#10623494	D₄	1	2	yes	—
63	C₄×#10623494	C₂³	1	2	yes	—
64	C₄×#10623494	C₂³	1	2	yes	—
65	C₈×Dic₈₉	—	2	4	no	—
66	C₂²×C₈×C₈₉	—	2	1	no	—
67	C₂²×C₈×C₈₉	—	2	1	no	—
68	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₈₉	—	2	1	no	—
72	C₄⋊C₈×C₈₉	—	2	1	no	—
73	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
74	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
75	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₈₉	C₂³	1	2	yes	—
77	C₂×Dic₁₇₈	—	2	1	no	—
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	2	2	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×Dic₁₇₈	—	2	1	no	—
82	C₂×Dic₁₇₈	M₁₆	1	4	yes	—
83	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	C₂²×Dic₈₉	—	2	1	no	—
87	C₂²×Dic₈₉	—	2	1	no	—
88	C₂²×Dic₈₉	—	2	1	no	—
89	Dic₃₅₆	—	2	8	no	—
90	C₂×#10623494	—	2	2	no	—
91	C₂×#10623494	—	2	2	no	—
92	C₂×#10623494	—	4	2	no	—
93	C₂×#10623494	Q₁₆	1	4	yes	complemented
94	C₂×#10623494	C₂×D₄	1	2	yes	—
95	C₂×#10623494	C₂×D₄	1	2	yes	—
96	C₄⋊Dic₈₉	—	2	1	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
99	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
100	#13083553	—	2	2	no	—
101	#13083553	—	2	2	no	—
102	#13083553	—	2	2	no	—
103	#13083553	—	2	2	no	—
104	C₄×Dic₈₉	—	2	2	no	—
105	C₄×Dic₈₉	—	2	2	no	—
106	C₄×Dic₈₉	—	2	2	no	—
107	C₄×Dic₈₉	—	2	2	no	—
108	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	4	2	no	—
116	C₂×C₈×C₈₉	—	4	2	no	—
117	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
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	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	#131	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
131	C₂×C₄×C₈₉	#135	1	2	yes	—
132	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	2	4	no	—
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	2	2	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	Dic₁₇₈	—	2	2	no	—
144	Dic₁₇₈	—	2	4	no	—
145	Dic₁₇₈	—	2	2	no	—
146	Dic₁₇₈	—	2	2	no	—
147	Dic₁₇₈	—	4	2	no	—
148	Dic₁₇₈	C₂×M₁₆	1	4	yes	—
149	C₈×C₈₉	—	2	2	no	—
150	C₈×C₈₉	—	2	2	no	—
151	C₈×C₈₉	—	4	2	no	—
152	C₈×C₈₉	—	4	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	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	#435	1	2	yes	—
163	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
164	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
165	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
166	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
167	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
168	Dic₈₉	—	2	4	no	—
169	Dic₈₉	—	4	2	no	—
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	D₈₉	1	1	yes	characteristic
182	C₂×Q₈⋊C₈	—	89	1	no	—
183	C₂×Q₈⋊C₈	—	89	1	no	—
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	—	89	1	no	—
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	—	178	1	no	—
197	C₂×C₄⋊C₈	—	178	1	no	—
198	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
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	C₂×D₈₉	1	4	yes	—
208	Q₈⋊C₈	—	89	4	no	—
209	Q₈⋊C₈	—	89	4	no	—
210	#322	—	89	4	no	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂²×Q₈	—	89	1	no	—
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	—	178	1	no	—
220	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
221	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
222	C₄×Q₈	—	89	2	no	—
223	C₄×Q₈	—	89	2	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	—	178	1	no	—
229	C₄×Q₈	—	178	1	no	—
230	C₂×C₄⋊C₄	—	89	1	no	—
231	C₂×C₄⋊C₄	—	89	1	no	—
232	C₂×C₄⋊C₄	—	178	1	no	—
233	C₂×C₄⋊C₄	—	178	1	no	—
234	C₂×C₄²	—	178	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	—	2	1	no	—
240	C₄⋊C₈	—	178	4	no	—
241	C₄⋊C₈	—	178	4	no	—
242	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
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	—	89	2	no	—
248	C₄×C₈	—	89	2	no	—
249	C₄×C₈	—	89	2	no	—
250	C₄×C₈	—	178	4	no	—
251	C₄×C₈	—	178	1	no	—
252	C₄×C₈	C₄×D₈₉	1	2	yes	—
253	C₄×C₈	C₂²×D₈₉	1	2	yes	—
254	C₂×Q₈	—	89	4	no	—
255	C₂×Q₈	—	89	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	—	178	1	no	—
261	C₂×Q₈	—	178	2	no	—
262	C₂²×C₄	—	178	1	no	—
263	C₂²×C₄	—	178	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	#13083575	1	1	yes	characteristic
269	Q₁₆	—	178	8	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	89	4	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	C₂×C₄×D₈₉	1	2	yes	—
283	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
284	C₄⋊C₄	—	89	2	no	—
285	C₄⋊C₄	—	89	2	no	—
286	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₄⋊C₄	—	178	2	no	—
289	C₄⋊C₄	—	178	2	no	—
290	C₄⋊C₄	—	178	2	no	—
291	C₄⋊C₄	—	178	2	no	—
292	C₄²	—	178	2	no	—
293	C₄²	—	178	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	#13083575	1	1	yes	characteristic
299	C₄²	C₂³×D₈₉	1	1	yes	characteristic
300	C₂³	#15667974	1	1	yes	characteristic
301	Q₈	—	89	4	no	—
302	Q₈	—	356	2	no	—
303	Q₈	—	178	2	no	—
304	Q₈	—	178	2	no	—
305	Q₈	—	178	4	no	—
306	Q₈	—	178	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	—	178	2	no	—
313	C₂×C₄	—	178	2	no	—
314	C₂×C₄	—	178	4	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	#15667974	1	1	yes	characteristic
317	C₂×C₄	#15667974	1	1	yes	characteristic
318	C₂×C₄	#15667974	1	1	yes	characteristic
319	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
320	C₂×C₄	#15668018	1	2	yes	—
321	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
322	C₂×C₄	#15668031	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×#13083575	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	4	2	no	—
330	C₈	—	89	4	no	—
331	C₈	—	356	2	no	—
332	C₈	—	178	2	no	—
333	C₂²	#19151668	1	2	yes	—
334	C₂²	#19151692	1	2	yes	—
335	C₂²	#19151719	1	2	yes	—
336	C₂²	C₂×#15667974	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	356	2	no	—
340	C₄	—	356	2	no	—
341	C₄	—	356	2	no	—
342	C₄	—	178	4	no	—
343	C₄	#19151668	1	2	yes	—
344	C₄	#19151692	1	2	yes	—
345	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
346	C₄	C₂×#15668018	1	1	yes	characteristic
347	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
348	C₄	C₂×#15668031	1	1	yes	characteristic
349	C₂	#26019101	1	4	yes	direct product factor
350	C₂	C₂×#19151668	1	1	yes	characteristic
351	C₂	C₂×#19151692	1	1	yes	characteristic
352	C₂	C₂×#19151719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	C₂×#1584
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #26019101	C₂	direct (N × H)
93. C₂×#10623494	Q₁₆	semidirect
155. #10623494	C₂×Q₁₆	semidirect
173. C₂×C₈₉	#1584	semidirect
185. C₈₉	C₂×#1584	semidirect
349. C₂	#26019101	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×#19151722	2
C₂×C₄×Dic₃₅₆	2
#26019101	2
C₂×#19151328	2
C₂×#19151320	2
C₂×#335×C₈₉	2
C₂×#1584	89

Subgroups — C2×#26019101 #36523947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019101 #36523947