Subgroups — C₂×#26019013 #36523433

Order 22784.

766 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	C₂×#19151655	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	#26019013	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151350	C₂	1	2	yes	maximal, complemented
5	C₂×#19151348	C₂	1	2	yes	maximal, complemented
6	C₂×#432×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10623494	C₄	1	1	yes	characteristic
8	C₂³×#10623494	C₄	1	1	yes	characteristic
9	#19151655	—	2	2	no	—
10	#19151655	C₂²	1	4	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₂²⋊#10623494	—	2	1	no	—
13	C₂×#15667923	—	2	1	no	—
14	C₂×C₄×#10623494	C₄	1	4	yes	—
15	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
16	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10623494	C₂²	1	4	yes	—
20	#19151350	C₂²	1	8	yes	complemented
21	#19151348	C₂²	1	8	yes	complemented
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	#432×C₈₉	—	2	2	no	—
24	#432×C₈₉	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
27	C₂×#97×C₈₉	—	2	1	no	—
28	C₂²×#10623494	—	2	4	no	—
29	C₂²×#10623494	—	2	4	no	—
30	C₂²×#10623494	—	2	2	no	—
31	C₂²×#10623494	C₂×C₄	1	4	yes	—
32	C₂²×#10623494	C₂×C₄	1	4	yes	—
33	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
34	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
35	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
36	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
37	C₄⋊#10623494	—	2	2	no	—
38	C₄⋊#10623494	C₂³	1	4	yes	—
39	C₂²⋊#10623494	—	2	4	no	—
40	#15667923	—	2	4	no	—
41	C₄×#10623494	C₂×C₄	1	4	yes	—
42	C₄×#10623494	C₂×C₄	1	2	yes	—
43	C₄×#10623494	C₂×C₄	1	2	yes	—
44	C₄×#10623494	C₂×C₄	1	4	yes	—
45	C₄×#10623494	C₂³	1	2	yes	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	—	2	2	no	—
51	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	2	yes	—
55	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
59	C₄⋊C₈×C₈₉	—	2	2	no	—
60	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
61	C₂²⋊C₈×C₈₉	—	2	4	no	—
62	#97×C₈₉	—	2	4	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	2	8	no	—
66	C₂×#10623494	—	2	8	no	—
67	C₂×#10623494	—	2	4	no	—
68	C₂×#10623494	—	4	4	no	—
69	C₂×#10623494	M₁₆	1	4	yes	—
70	C₂×#10623494	M₁₆	1	4	yes	—
71	C₂×#10623494	C₂²×C₄	1	2	yes	—
72	C₂×#10623494	C₂²×C₄	1	2	yes	—
73	C₂×#10623494	C₂²×C₄	1	2	yes	—
74	C₂×#10623494	C₂²×C₄	1	2	yes	—
75	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	—	2	4	no	—
77	C₂²×C₄×C₈₉	—	2	2	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	—	2	4	no	—
80	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
84	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₈₉	—	2	4	no	—
93	C₂×C₈×C₈₉	—	4	4	no	—
94	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
96	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
97	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	2	yes	—
100	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
102	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
103	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₂³×C₈₉	—	2	2	no	—
105	C₂³×C₈₉	—	2	2	no	—
106	C₂³×C₈₉	#108	1	4	yes	—
107	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	4	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	4	no	—
118	C₂×C₄×C₈₉	—	2	4	no	—
119	C₂×C₄×C₈₉	—	2	4	no	—
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₈₉	#135	1	4	yes	—
129	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₈₉	—	4	4	no	—
133	#10623494	—	2	2	no	—
134	#10623494	—	2	2	no	—
135	#10623494	—	4	4	no	—
136	#10623494	C₂×M₁₆	1	4	yes	—
137	#10623494	C₂×M₁₆	1	4	yes	—
138	C₂²×C₈₉	—	2	8	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	#423	1	2	yes	—
142	C₂²×C₈₉	#427	1	2	yes	—
143	C₂²×C₈₉	C₂×#108	1	4	yes	—
144	C₂²×C₈₉	C₂×#108	1	2	yes	—
145	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	#423	1	2	yes	—
152	C₄×C₈₉	#427	1	2	yes	—
153	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
155	C₄×C₈₉	C₂×#135	1	2	yes	—
156	C₂×#1568	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	4	no	—
158	C₂×C₈₉	#1568	1	4	yes	complemented
159	C₂×C₈₉	C₂×#423	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
161	C₂×C₈₉	C₂²×#108	1	1	yes	characteristic
162	C₂×#432	—	89	1	no	—
163	C₂×#432	D₈₉	1	1	yes	characteristic
164	C₂²×C₄×C₈	—	89	1	no	—
165	#1568	—	89	8	no	—
166	C₂×#335	—	89	2	no	—
167	C₂×#334	—	89	2	no	—
168	C₈₉	C₂×#1568	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	89	1	no	—
170	C₂³×C₈	—	89	1	no	—
171	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
172	#432	—	2	2	no	—
173	#432	—	89	4	no	—
174	#432	—	178	2	no	—
175	#432	C₂×D₈₉	1	4	yes	—
176	C₂×C₄⋊C₈	—	89	2	no	—
177	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
178	C₂×C₂²⋊C₈	—	2	1	no	—
179	C₂×C₂²⋊C₈	—	178	1	no	—
180	C₂×#97	—	2	1	no	—
181	C₂×#97	—	178	1	no	—
182	C₂×C₄×C₈	—	89	1	no	—
183	C₂×C₄×C₈	—	89	4	no	—
184	C₂×C₄×C₈	—	89	1	no	—
185	C₂×C₄×C₈	—	89	1	no	—
186	C₂×C₄×C₈	—	89	1	no	—
187	C₂×C₄×C₈	—	89	4	no	—
188	#335	—	89	8	no	—
189	#334	—	89	8	no	—
190	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
191	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
192	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	89	1	no	—
195	C₂²×C₈	—	89	1	no	—
196	C₂²×C₈	—	89	4	no	—
197	C₂²×C₈	—	89	4	no	—
198	C₂²×C₈	—	89	1	no	—
199	C₂²×C₈	—	89	1	no	—
200	C₂²×C₈	—	178	4	no	—
201	C₂²×C₈	—	178	4	no	—
202	C₂²×C₈	—	178	2	no	—
203	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
204	C₂×C₄²	Dic₁₇₈	1	2	yes	—
205	C₂×C₄²	C₄×D₈₉	1	2	yes	—
206	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
210	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	89	4	no	—
213	C₄⋊C₈	—	178	2	no	—
214	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₂²⋊C₈	—	178	4	no	—
217	#97	—	2	4	no	—
218	#97	—	178	4	no	—
219	C₄×C₈	—	89	4	no	—
220	C₄×C₈	—	89	2	no	—
221	C₄×C₈	—	89	2	no	—
222	C₄×C₈	—	89	4	no	—
223	C₄×C₈	—	89	2	no	—
224	C₄×C₈	—	89	2	no	—
225	C₂⁴	#13083553	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	#13083553	1	1	yes	characteristic
231	C₂²×C₄	#13083553	1	2	yes	—
232	C₂²×C₄	#13083553	1	1	yes	characteristic
233	C₂²×C₄	#13083553	1	2	yes	—
234	C₂²×C₄	#13083553	1	2	yes	—
235	C₂²×C₄	#13083553	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
241	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	89	4	no	—
245	C₂×C₈	—	89	4	no	—
246	C₂×C₈	—	89	2	no	—
247	C₂×C₈	—	89	2	no	—
248	C₂×C₈	—	89	2	no	—
249	C₂×C₈	—	89	2	no	—
250	C₂×C₈	—	356	4	no	—
251	C₂×C₈	—	178	8	no	—
252	C₂×C₈	—	178	8	no	—
253	C₂×C₈	—	178	4	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	—	178	2	no	—
256	C₄²	#13083553	1	2	yes	—
257	C₄²	#13083553	1	2	yes	—
258	C₄²	#13083553	1	2	yes	—
259	C₄²	C₂×C₄×D₈₉	1	2	yes	—
260	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
262	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
263	C₄²	C₂×Dic₁₇₈	1	2	yes	—
264	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
265	C₄²	C₂³×D₈₉	1	1	yes	characteristic
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	#15667947	1	4	yes	—
269	C₂³	C₂×#13083553	1	2	yes	—
270	C₂³	C₂×#13083553	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	C₂×#13083553	1	2	yes	—
283	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
284	C₂×C₄	C₂×#13083553	1	2	yes	—
285	C₂×C₄	C₂×#13083553	1	2	yes	—
286	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
287	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
289	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
290	C₂×C₄	#15668032	1	4	yes	—
291	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
294	C₈	—	4	4	no	—
295	C₈	—	89	4	no	—
296	C₈	—	89	4	no	—
297	C₈	—	356	4	no	—
298	C₈	—	178	2	no	—
299	C₈	—	178	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	8	no	—
303	C₂²	#19151621	1	2	yes	—
304	C₂²	#19151634	1	2	yes	—
305	C₂²	C₂×#15667947	1	4	yes	—
306	C₂²	C₂×#15667947	1	2	yes	—
307	C₂²	C₂²×#13083553	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	#19151621	1	2	yes	—
314	C₄	#19151634	1	2	yes	—
315	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
316	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
317	C₄	C₂×#15668032	1	2	yes	—
318	C₂	—	2	4	no	—
319	C₂	#26019013	1	4	yes	direct product factor
320	C₂	C₂×#19151621	1	1	yes	characteristic
321	C₂	C₂×#19151634	1	1	yes	characteristic
322	C₂	C₂²×#15667947	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#432×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#432	D₈₉
hypocenter	C₈₉	C₂×#1568
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #26019013	C₂	direct (N × H)
4. C₂×#19151350	C₂	semidirect
5. C₂×#19151348	C₂	semidirect
20. #19151350	C₂²	semidirect
21. #19151348	C₂²	semidirect
158. C₂×C₈₉	#1568	semidirect
168. C₈₉	C₂×#1568	semidirect
319. C₂	#26019013	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×#19151655	2
C₂²×C₄×#10623494	2
#26019013	2
C₂×#19151350	2
C₂×#19151348	2
C₂×#432×C₈₉	2
C₂×#1568	89

Subgroups — C2×#26019013 #36523433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019013 #36523433