Subgroups — C₂×#26019020 #36523447

Order 22784.

758 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
3	#26019020	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	C₂²×#13083558	—	2	1	no	—
8	C₂²×#13083558	C₄	1	2	yes	—
9	#19151622	—	2	2	no	—
10	#19151622	C₂²	1	4	yes	—
11	C₂×C₄⋊#10623494	—	2	1	no	—
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₄×#13083558	C₄	1	4	yes	—
14	C₄×#13083558	C₂²	1	4	yes	—
15	C₂×#15667923	C₄	1	1	yes	characteristic
16	C₂×#15667923	C₄	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	2	yes	—
18	#19151350	C₂²	1	8	yes	complemented
19	#19151348	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	#432×C₈₉	—	2	2	no	—
22	#432×C₈₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
26	C₂×#97×C₈₉	—	2	1	no	—
27	C₂×#13083558	—	2	8	no	—
28	C₂×#13083558	—	2	4	no	—
29	C₂×#13083558	—	4	4	no	—
30	C₂×#13083558	C₂×C₄	1	8	yes	—
31	C₂²×#10623494	—	2	2	no	—
32	C₂²×#10623494	C₂×C₄	1	2	yes	—
33	C₂²×#10623494	C₂×C₄	1	2	yes	—
34	C₄⋊#10623494	—	2	4	no	—
35	C₄⋊#10623494	—	2	2	no	—
36	C₄⋊#10623494	C₂³	1	4	yes	—
37	#15667923	C₂×C₄	1	4	yes	—
38	#15667923	C₂×C₄	1	4	yes	—
39	C₄×#10623494	C₂×C₄	1	4	yes	—
40	C₄×#10623494	C₂³	1	4	yes	—
41	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	—	2	1	no	—
45	C₂²×C₈×C₈₉	—	2	1	no	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	2	yes	—
50	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
54	C₄⋊C₈×C₈₉	—	2	1	no	—
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
57	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
58	C₂²⋊C₈×C₈₉	—	2	4	no	—
59	#97×C₈₉	—	2	4	no	—
60	#13083558	—	2	16	no	—
61	#13083558	—	4	8	no	—
62	C₂×#10623494	—	2	4	no	—
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	—	4	4	no	—
65	C₂×#10623494	M₁₆	1	8	yes	complemented
66	C₂×#10623494	C₂²×C₄	1	4	yes	—
67	C₂×#10623494	C₂²×C₄	1	4	yes	—
68	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	4	2	no	—
73	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	4	2	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
90	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
91	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
92	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
94	C₄²×C₈₉	C₂×D₄	1	2	yes	—
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
97	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₂³×C₈₉	—	2	2	no	—
100	C₂³×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	2	no	—
102	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	4	no	—
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	4	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₈₉	#137	1	2	yes	—
124	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₈₉	—	4	2	no	—
128	C₈×C₈₉	—	4	2	no	—
129	#10623494	—	2	4	no	—
130	#10623494	—	4	4	no	—
131	#10623494	C₂×M₁₆	1	8	yes	complemented
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	4	4	no	—
136	C₂²×C₈₉	#423	1	2	yes	—
137	C₂²×C₈₉	#428	1	2	yes	—
138	C₂²×C₈₉	#430	1	2	yes	—
139	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	#423	1	2	yes	—
146	C₄×C₈₉	#428	1	2	yes	—
147	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
148	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
150	C₂×#1569	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	—	4	2	no	—
152	C₂×C₈₉	#1569	1	4	yes	complemented
153	C₂×C₈₉	C₂×#423	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#428	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
156	C₂×#432	D₈₉	1	1	yes	characteristic
157	C₂×#423	—	89	1	no	—
158	C₂×C₄×M₁₆	—	89	1	no	—
159	#1569	—	89	8	no	—
160	C₂×#335	—	89	2	no	—
161	C₂×#334	—	89	2	no	—
162	C₈₉	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	89	2	no	—
164	C₂²×M₁₆	—	178	1	no	—
165	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
166	#432	—	2	2	no	—
167	#432	C₂×D₈₉	1	4	yes	—
168	#423	—	89	4	no	—
169	#423	—	178	2	no	—
170	C₂×C₄⋊C₈	—	89	2	no	—
171	C₂×C₄⋊C₈	—	178	1	no	—
172	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
174	C₂×C₂²⋊C₈	—	2	1	no	—
175	C₄×M₁₆	—	89	4	no	—
176	C₄×M₁₆	—	89	4	no	—
177	C₂×#97	—	2	1	no	—
178	C₂×#97	—	89	1	no	—
179	C₂×#97	—	89	1	no	—
180	C₂×C₄×C₈	—	89	2	no	—
181	#335	—	89	8	no	—
182	#334	—	89	8	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	C₂×M₁₆	—	89	8	no	—
187	C₂×M₁₆	—	356	4	no	—
188	C₂×M₁₆	—	178	8	no	—
189	C₂×M₁₆	—	178	4	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	89	2	no	—
193	C₂²×C₈	—	89	2	no	—
194	C₂²×C₈	—	178	2	no	—
195	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
196	C₂×C₄²	Dic₁₇₈	1	2	yes	—
197	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₈₉	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
202	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	89	4	no	—
206	C₄⋊C₈	—	178	2	no	—
207	C₄⋊C₈	—	178	4	no	—
208	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
209	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
210	C₂²⋊C₈	—	2	4	no	—
211	#97	—	2	4	no	—
212	#97	—	89	4	no	—
213	#97	—	89	4	no	—
214	C₄×C₈	—	89	4	no	—
215	C₄×C₈	—	89	4	no	—
216	C₂⁴	#13083553	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
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	2	yes	—
225	C₂²×C₄	#13083553	1	1	yes	characteristic
226	C₂²×C₄	#13083553	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
233	M₁₆	—	356	8	no	—
234	M₁₆	—	178	16	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	89	8	no	—
240	C₂×C₈	—	89	4	no	—
241	C₂×C₈	—	89	4	no	—
242	C₂×C₈	—	356	4	no	—
243	C₂×C₈	—	178	4	no	—
244	C₂×C₈	—	178	4	no	—
245	C₄²	#13083553	1	2	yes	—
246	C₄²	#13083553	1	2	yes	—
247	C₄²	#13083553	1	2	yes	—
248	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₈₉	1	2	yes	—
250	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
251	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₄²	C₂×Dic₁₇₈	1	2	yes	—
253	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
254	C₄²	C₂³×D₈₉	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×#13083553	1	2	yes	—
259	C₂³	C₂×#13083553	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
272	C₂×C₄	C₂×#13083553	1	2	yes	—
273	C₂×C₄	C₂×#13083553	1	2	yes	—
274	C₂×C₄	C₂×#13083553	1	2	yes	—
275	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
276	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
277	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
278	C₂×C₄	#15668037	1	2	yes	—
279	C₂×C₄	#15668042	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
282	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	89	8	no	—
286	C₈	—	356	4	no	—
287	C₈	—	178	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#19151621	1	2	yes	—
293	C₂²	#19151641	1	2	yes	—
294	C₂²	#19151646	1	2	yes	—
295	C₂²	C₂²×#13083553	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	#19151621	1	2	yes	—
302	C₄	#19151641	1	2	yes	—
303	C₄	C₂×M₁₆×D₈₉	1	2	yes	—
304	C₄	C₂×#15668037	1	1	yes	characteristic
305	C₄	C₂×#15668042	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	#26019020	1	4	yes	direct product factor
308	C₂	C₂×#19151621	1	1	yes	characteristic
309	C₂	C₂×#19151641	1	1	yes	characteristic
310	C₂	C₂×#19151646	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×#1569
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. #26019020	C₂	direct (N × H)
4. C₂×#19151350	C₂	semidirect
5. C₂×#19151348	C₂	semidirect
18. #19151350	C₂²	semidirect
19. #19151348	C₂²	semidirect
65. C₂×#10623494	M₁₆	semidirect
131. #10623494	C₂×M₁₆	semidirect
152. C₂×C₈₉	#1569	semidirect
162. C₈₉	C₂×#1569	semidirect
307. C₂	#26019020	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×#19151622	2
C₂×C₄×#13083558	2
#26019020	2
C₂×#19151350	2
C₂×#19151348	2
C₂×#432×C₈₉	2
C₂×#1569	89

Subgroups — C2×#26019020 #36523447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019020 #36523447