Subgroups — C₂×C₄×#15667957 #36707697

Order 22784.

742 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667957	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
4	C₄×#15667957	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151403	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151400	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₈	—	2	1	no	—
12	C₂×C₄×D₃₅₆	—	2	1	no	—
13	C₂×#15667957	C₄	1	8	yes	direct product factor
14	C₂×#15667957	C₂²	1	4	yes	—
15	C₄×Dic₃₅₆	—	2	2	no	—
16	C₄×Dic₃₅₆	C₂²	1	4	yes	complemented
17	C₄×D₇₁₂	—	2	2	no	—
18	C₄×D₇₁₂	C₂²	1	4	yes	—
19	C₂×#15667946	C₂²	1	1	yes	characteristic
20	C₂×#15667946	C₂²	1	1	yes	characteristic
21	C₂×#15667937	—	2	1	no	—
22	C₂×#15667934	—	2	1	no	—
23	#19151426	C₂²	1	4	yes	complemented
24	#19151403	C₂²	1	4	yes	complemented
25	#19151400	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
30	C₂²×Dic₁₇₈	—	2	1	no	—
31	C₂²×D₃₅₆	—	2	1	no	—
32	C₂²×C₄×D₈₉	—	4	1	no	—
33	C₂×Dic₃₅₆	—	2	1	no	—
34	C₂×Dic₃₅₆	—	2	2	no	—
35	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	complemented
36	C₂×Dic₃₅₆	C₂³	1	2	yes	—
37	C₂×D₇₁₂	—	2	1	no	—
38	C₂×D₇₁₂	—	2	2	no	—
39	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
40	C₂×D₇₁₂	C₂³	1	2	yes	—
41	C₄×Dic₁₇₈	—	2	4	no	—
42	C₄×Dic₁₇₈	—	4	2	no	—
43	C₄×D₃₅₆	—	2	4	no	—
44	C₄×D₃₅₆	—	4	2	no	—
45	C₂×C₄⋊Dic₈₉	—	2	1	no	—
46	C₂×C₄⋊Dic₈₉	—	2	1	no	—
47	C₂×#13083553	—	4	1	no	—
48	C₂×#13083551	—	4	1	no	—
49	C₂×C₄×Dic₈₉	—	4	1	no	—
50	#15667957	C₂×C₄	1	16	yes	direct product factor
51	#15667946	—	2	1	no	—
52	#15667946	—	2	1	no	—
53	#15667946	C₂³	1	2	yes	—
54	#15667946	C₂³	1	2	yes	—
55	#15667937	—	2	4	no	—
56	#15667934	—	2	4	no	—
57	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
60	C₂×C₁₆×C₈₉	—	2	2	no	—
61	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
62	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
63	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
64	C₄×C₈×C₈₉	D₄	1	2	yes	—
65	C₄×C₈×C₈₉	C₂³	1	2	yes	—
66	C₂³×D₈₉	—	4	1	no	—
67	C₂×Dic₁₇₈	—	2	4	no	—
68	C₂×Dic₁₇₈	—	2	2	no	—
69	C₂×Dic₁₇₈	—	4	1	no	—
70	C₂×Dic₁₇₈	—	4	2	no	—
71	C₂×D₃₅₆	—	2	4	no	—
72	C₂×D₃₅₆	—	2	2	no	—
73	C₂×D₃₅₆	—	4	1	no	—
74	C₂×D₃₅₆	—	4	2	no	—
75	C₂²×Dic₈₉	—	4	1	no	—
76	C₂²×Dic₈₉	—	4	1	no	—
77	C₂²×Dic₈₉	—	4	1	no	—
78	C₂×C₄×D₈₉	—	4	4	no	—
79	C₂×C₄×D₈₉	—	8	4	no	—
80	Dic₃₅₆	—	2	4	no	—
81	Dic₃₅₆	—	2	2	no	—
82	Dic₃₅₆	C₂²×C₄	1	4	yes	complemented
83	D₇₁₂	—	2	4	no	—
84	D₇₁₂	—	2	2	no	—
85	D₇₁₂	C₂²×C₄	1	4	yes	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	#13083553	—	4	4	no	—
91	#13083551	—	4	4	no	—
92	C₄×Dic₈₉	—	4	4	no	—
93	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	#41	1	2	yes	—
101	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₄²×C₈₉	D₈	1	2	yes	—
103	C₄²×C₈₉	C₂×D₄	1	2	yes	—
104	C₁₆×C₈₉	—	2	2	no	—
105	C₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
106	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₈₉	—	4	2	no	—
108	C₂²×D₈₉	—	4	4	no	—
109	C₂²×D₈₉	—	8	2	no	—
110	C₂²×D₈₉	—	8	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	SD₃₂	1	4	yes	complemented
113	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₈₉	#135	1	2	yes	—
118	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	—	4	4	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	8	2	no	—
124	C₂×Dic₈₉	—	8	2	no	—
125	D₃₅₆	—	2	4	no	—
126	D₃₅₆	—	4	2	no	—
127	D₃₅₆	—	4	4	no	—
128	C₄×D₈₉	—	8	8	no	—
129	Dic₁₇₈	—	2	4	no	—
130	Dic₁₇₈	—	4	2	no	—
131	Dic₁₇₈	—	4	4	no	—
132	C₈×C₈₉	—	2	2	no	—
133	C₈×C₈₉	C₄×D₄	1	2	yes	—
134	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
135	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
136	C₂²×C₈₉	C₄×D₈	1	2	yes	—
137	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
138	C₂²×C₈₉	#508	1	2	yes	—
139	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₈₉	—	4	4	no	—
141	C₂×D₈₉	—	8	2	no	—
142	C₂×D₈₉	—	8	2	no	—
143	C₂×D₈₉	—	8	8	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	C₄×D₈	1	2	yes	—
146	C₄×C₈₉	C₂×SD₃₂	1	4	yes	complemented
147	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
149	Dic₈₉	—	4	4	no	—
150	Dic₈₉	—	8	2	no	—
151	Dic₈₉	—	8	2	no	—
152	C₂×C₄×SD₃₂	—	89	1	no	maximal, Sylow
153	C₂×C₈₉	C₄×SD₃₂	1	4	yes	complemented
154	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
156	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
157	D₈₉	—	8	4	no	—
158	C₂²×SD₃₂	—	89	1	no	—
159	C₂×C₄×Q₁₆	—	89	1	no	—
160	C₂×C₄×D₈	—	89	1	no	—
161	C₄×SD₃₂	—	89	8	no	—
162	C₂×#367	—	89	1	no	—
163	C₂×#358	—	89	1	no	—
164	C₂×#357	—	89	1	no	—
165	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
166	C₈₉	C₂×C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	—	89	1	no	—
168	C₂²×D₈	—	89	1	no	—
169	C₂×C₄×Q₈	—	178	1	no	—
170	C₂×C₄×D₄	—	178	1	no	—
171	C₂×SD₃₂	—	89	8	no	—
172	C₂×SD₃₂	—	89	4	no	—
173	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
175	C₄×Q₁₆	—	89	4	no	—
176	C₄×Q₁₆	—	178	2	no	—
177	C₄×D₈	—	89	4	no	—
178	C₄×D₈	—	178	2	no	—
179	C₂×#107	—	89	1	no	—
180	C₂×#107	—	89	1	no	—
181	C₂×#103	—	178	1	no	—
182	C₂×#102	—	178	1	no	—
183	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
184	#367	—	89	4	no	—
185	#358	—	89	4	no	—
186	#357	—	89	4	no	—
187	C₄×C₁₆	C₂×D₈₉	1	4	yes	complemented
188	C₂²×Q₈	—	178	1	no	—
189	C₂²×D₄	—	178	1	no	—
190	C₂³×C₄	—	356	1	no	—
191	C₂×Q₁₆	—	89	4	no	—
192	C₂×Q₁₆	—	89	2	no	—
193	C₂×Q₁₆	—	178	1	no	—
194	C₂×Q₁₆	—	178	2	no	—
195	C₂×D₈	—	89	4	no	—
196	C₂×D₈	—	89	2	no	—
197	C₂×D₈	—	178	1	no	—
198	C₂×D₈	—	178	2	no	—
199	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
201	C₄×Q₈	—	356	2	no	—
202	C₄×Q₈	—	178	4	no	—
203	C₄×D₄	—	356	2	no	—
204	C₄×D₄	—	178	4	no	—
205	C₂×C₄⋊C₄	—	356	1	no	—
206	C₂×C₄⋊C₄	—	178	1	no	—
207	C₂×C₄⋊C₄	—	178	1	no	—
208	C₂×C₂²⋊C₄	—	356	1	no	—
209	C₂×C₄²	—	356	1	no	—
210	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
211	SD₃₂	—	89	16	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
214	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
215	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
216	#107	—	89	2	no	—
217	#107	—	89	2	no	—
218	#107	—	178	1	no	—
219	#107	—	178	1	no	—
220	#103	—	178	4	no	—
221	#102	—	178	4	no	—
222	C₄×C₈	D₃₅₆	1	2	yes	—
223	C₄×C₈	C₂²×D₈₉	1	2	yes	—
224	C₂⁴	—	356	1	no	—
225	C₂×Q₈	—	356	1	no	—
226	C₂×Q₈	—	356	2	no	—
227	C₂×Q₈	—	178	4	no	—
228	C₂×Q₈	—	178	2	no	—
229	C₂×D₄	—	356	1	no	—
230	C₂×D₄	—	356	2	no	—
231	C₂×D₄	—	178	4	no	—
232	C₂×D₄	—	178	2	no	—
233	C₂²×C₄	—	356	1	no	—
234	C₂²×C₄	—	356	1	no	—
235	C₂²×C₄	—	356	1	no	—
236	C₂²×C₄	—	356	4	no	—
237	C₂²×C₄	—	712	4	no	—
238	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
239	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
241	Q₁₆	—	89	4	no	—
242	Q₁₆	—	178	4	no	—
243	Q₁₆	—	178	2	no	—
244	D₈	—	89	4	no	—
245	D₈	—	178	4	no	—
246	D₈	—	178	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
249	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
250	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
251	C₂×C₈	#13083574	1	2	yes	—
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	356	4	no	—
254	C₄⋊C₄	—	356	1	no	—
255	C₄⋊C₄	—	356	1	no	—
256	C₄⋊C₄	—	178	2	no	—
257	C₄⋊C₄	—	178	2	no	—
258	C₂²⋊C₄	—	356	4	no	—
259	C₄²	—	356	4	no	—
260	C₄²	D₇₁₂	1	2	yes	—
261	C₄²	C₂×D₃₅₆	1	2	yes	—
262	C₁₆	—	2	2	no	—
263	C₁₆	C₂×C₄×D₈₉	1	4	yes	complemented
264	C₂³	—	356	4	no	—
265	C₂³	—	356	2	no	—
266	C₂³	—	712	2	no	—
267	C₂³	—	712	2	no	—
268	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
269	Q₈	—	356	4	no	—
270	Q₈	—	356	2	no	—
271	Q₈	—	178	4	no	—
272	D₄	—	356	4	no	—
273	D₄	—	356	2	no	—
274	D₄	—	178	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	356	4	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	356	2	no	—
280	C₂×C₄	—	712	2	no	—
281	C₂×C₄	—	712	2	no	—
282	C₂×C₄	—	712	8	no	—
283	C₂×C₄	#15667957	1	4	yes	direct product factor
284	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
285	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
286	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
287	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
288	C₂×C₄	#15668034	1	2	yes	—
289	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×D₃₅₆	1	2	yes	—
292	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₈	C₂×#13083574	1	1	yes	characteristic
294	C₂²	—	356	4	no	—
295	C₂²	—	712	2	no	—
296	C₂²	—	712	8	no	—
297	C₂²	—	712	2	no	—
298	C₂²	C₄×D₇₁₂	1	2	yes	—
299	C₂²	C₂×#15667957	1	2	yes	—
300	C₂²	#19152030	1	2	yes	—
301	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	356	4	no	—
304	C₄	—	712	2	no	—
305	C₄	—	712	2	no	—
306	C₄	C₄×D₇₁₂	1	2	yes	—
307	C₄	C₂×#15667957	1	4	yes	direct product factor
308	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
309	C₄	C₂×#15668034	1	1	yes	characteristic
310	C₂	—	712	4	no	—
311	C₂	C₄×#15667957	1	4	yes	direct product factor
312	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
313	C₂	C₂²×#15667957	1	1	yes	characteristic
314	C₂	C₂×#19152030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×C₄×SD₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₅₆	C₂	semidirect
4. C₄×#15667957	C₂	direct (N × H)
5. C₂×#19151426	C₂	semidirect
6. C₂×#19151403	C₂	semidirect
8. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
13. C₂×#15667957	C₄	direct (N × H)
16. C₄×Dic₃₅₆	C₂²	semidirect
23. #19151426	C₂²	semidirect
24. #19151403	C₂²	semidirect
29. C₄×C₁₆×C₈₉	C₂²	semidirect
35. C₂×Dic₃₅₆	C₂×C₄	semidirect
50. #15667957	C₂×C₄	direct (N × H)
61. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
82. Dic₃₅₆	C₂²×C₄	semidirect
105. C₁₆×C₈₉	C₂²×C₄	semidirect
112. C₂×C₄×C₈₉	SD₃₂	semidirect
146. C₄×C₈₉	C₂×SD₃₂	semidirect
153. C₂×C₈₉	C₄×SD₃₂	semidirect
165. C₂×C₄×C₁₆	D₈₉	semidirect
166. C₈₉	C₂×C₄×SD₃₂	semidirect
187. C₄×C₁₆	C₂×D₈₉	semidirect
213. C₂×C₁₆	C₄×D₈₉	semidirect
263. C₁₆	C₂×C₄×D₈₉	semidirect
283. C₂×C₄	#15667957	direct (N × H)
307. C₄	C₂×#15667957	direct (N × H)
311. C₂	C₄×#15667957	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×#15667957	2
C₂×C₄×Dic₃₅₆	2
C₂×C₄×D₇₁₂	2
C₄×#15667957	2
C₂×#19151426	2
C₂×#19151403	2
C₂×#19151400	2
C₂×C₄×C₁₆×C₈₉	2
C₂×C₄×SD₃₂	89

Subgroups — C2×C4×#15667957 #36707697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15667957 #36707697