Subgroups — C₂×C₄×#15667956 #36707693

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×#15667956	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#15667956	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151425	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151401	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151399	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	—	2	1	no	—
11	C₂×#15667956	C₄	1	8	yes	direct product factor
12	C₂×#15667956	C₂²	1	4	yes	—
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	C₄×Dic₃₅₆	—	2	2	no	—
16	C₄×Dic₃₅₆	C₂²	1	4	yes	complemented
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667937	—	2	1	no	—
19	#19151425	C₂²	1	4	yes	complemented
20	#19151401	C₂²	1	4	yes	complemented
21	#19151399	C₂²	1	4	yes	—
22	C₄×#13083549	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₈₉	—	2	1	no	—
25	C₄×D₈×C₈₉	—	2	2	no	—
26	C₄×D₈×C₈₉	C₂²	1	4	yes	—
27	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₈₉	—	2	1	no	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₇₈	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	2	no	—
33	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	complemented
34	C₂×Dic₃₅₆	C₂³	1	2	yes	—
35	C₄×Dic₁₇₈	—	2	4	no	—
36	C₄×Dic₁₇₈	—	4	2	no	—
37	C₂×C₄⋊Dic₈₉	—	2	1	no	—
38	C₂×#13083553	—	4	1	no	—
39	C₂×C₄×Dic₈₉	—	4	1	no	—
40	#15667956	C₂×C₄	1	16	yes	direct product factor
41	C₂×#13083549	—	2	2	no	—
42	C₂×#13083549	C₂×C₄	1	4	yes	complemented
43	C₂×#13083549	C₂³	1	2	yes	—
44	C₂×#13083549	C₂³	1	2	yes	—
45	#15667946	—	2	1	no	—
46	#15667946	C₂³	1	2	yes	—
47	#15667937	—	2	4	no	—
48	C₂²×D₄×C₈₉	—	2	1	no	—
49	C₂³×C₄×C₈₉	—	4	1	no	—
50	C₂×D₈×C₈₉	—	2	1	no	—
51	C₂×D₈×C₈₉	—	2	2	no	—
52	C₂×D₈×C₈₉	C₂×C₄	1	4	yes	—
53	C₂×D₈×C₈₉	C₂³	1	2	yes	—
54	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₈₉	—	2	4	no	—
57	C₄×D₄×C₈₉	—	4	2	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
60	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
61	#107×C₈₉	—	2	1	no	—
62	#107×C₈₉	C₂³	1	2	yes	—
63	#102×C₈₉	—	2	4	no	—
64	C₄×C₈×C₈₉	D₄	1	2	yes	—
65	C₄×C₈×C₈₉	C₂³	1	2	yes	—
66	C₂×Dic₁₇₈	—	2	4	no	—
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×Dic₁₇₈	—	4	2	no	—
70	C₂²×Dic₈₉	—	4	1	no	—
71	C₂²×Dic₈₉	—	4	1	no	—
72	Dic₃₅₆	—	2	4	no	—
73	Dic₃₅₆	—	2	2	no	—
74	Dic₃₅₆	C₂²×C₄	1	4	yes	complemented
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	4	1	no	—
77	#13083553	—	4	4	no	—
78	C₄×Dic₈₉	—	4	4	no	—
79	#13083549	—	2	2	no	—
80	#13083549	C₂²×C₄	1	4	yes	complemented
81	C₂⁴×C₈₉	—	4	1	no	—
82	C₂×D₄×C₈₉	—	2	4	no	—
83	C₂×D₄×C₈₉	—	2	2	no	—
84	C₂×D₄×C₈₉	—	4	1	no	—
85	C₂×D₄×C₈₉	—	4	2	no	—
86	C₂²×C₄×C₈₉	—	4	1	no	—
87	C₂²×C₄×C₈₉	—	4	4	no	—
88	C₂²×C₄×C₈₉	—	8	4	no	—
89	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	D₈×C₈₉	—	2	4	no	—
93	D₈×C₈₉	—	2	2	no	—
94	D₈×C₈₉	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	#41	1	2	yes	—
100	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₂²⋊C₄×C₈₉	—	4	4	no	—
104	C₄²×C₈₉	D₈	1	2	yes	—
105	C₄²×C₈₉	C₂×D₄	1	2	yes	—
106	C₂³×C₈₉	—	4	2	no	—
107	C₂³×C₈₉	—	4	4	no	—
108	C₂³×C₈₉	—	8	2	no	—
109	C₂³×C₈₉	—	8	2	no	—
110	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
111	D₄×C₈₉	—	2	4	no	—
112	D₄×C₈₉	—	4	2	no	—
113	D₄×C₈₉	—	4	4	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	—	8	2	no	—
117	C₂×C₄×C₈₉	—	8	8	no	—
118	C₂×C₄×C₈₉	SD₃₂	1	4	yes	complemented
119	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	#135	1	2	yes	—
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	4	4	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	8	2	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	—	4	4	no	—
137	C₂²×C₈₉	—	8	2	no	—
138	C₂²×C₈₉	—	8	2	no	—
139	C₂²×C₈₉	—	8	8	no	—
140	C₂²×C₈₉	C₄×D₈	1	2	yes	—
141	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
142	C₂²×C₈₉	#508	1	2	yes	—
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	8	2	no	—
146	C₄×C₈₉	C₄×D₈	1	2	yes	—
147	C₄×C₈₉	C₂×SD₃₂	1	4	yes	complemented
148	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
150	Dic₈₉	—	4	4	no	—
151	Dic₈₉	—	8	2	no	—
152	C₂×C₄×SD₃₂	—	89	1	no	maximal, Sylow
153	C₂×C₈₉	—	8	4	no	—
154	C₂×C₈₉	C₄×SD₃₂	1	4	yes	complemented
155	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
156	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
158	C₂²×SD₃₂	—	89	1	no	—
159	C₂×C₄×Q₁₆	—	89	1	no	—
160	C₂×C₄×D₈	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
166	C₈₉	C₂×C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	—	89	1	no	—
168	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
169	C₂×C₄×Q₈	—	178	1	no	—
170	C₂×C₄×D₄	—	2	1	no	—
171	C₂×SD₃₂	—	89	8	no	—
172	C₂×SD₃₂	—	89	4	no	—
173	C₂²×C₁₆	—	89	1	no	—
174	C₂²×C₁₆	—	89	1	no	—
175	C₄×Q₁₆	—	89	4	no	—
176	C₄×Q₁₆	—	178	2	no	—
177	C₄×D₈	—	2	2	no	—
178	C₄×D₈	C₂×D₈₉	1	4	yes	—
179	C₂×#107	—	89	1	no	—
180	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
181	C₂×#103	—	178	1	no	—
182	C₂×#102	—	2	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	—	89	4	no	—
188	C₂²×Q₈	—	178	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂³×C₄	—	4	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	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₄×D₈₉	1	4	yes	—
198	C₂×D₈	C₂²×D₈₉	1	2	yes	—
199	C₂²×C₈	C₂²⋊D₈₉	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	—	2	4	no	—
204	C₄×D₄	—	4	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	356	1	no	—
207	C₂×C₄⋊C₄	—	178	1	no	—
208	C₂×C₂²⋊C₄	—	4	1	no	—
209	C₂×C₄²	—	356	1	no	—
210	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
211	SD₃₂	—	89	16	no	—
212	C₂×C₁₆	—	89	4	no	—
213	C₂×C₁₆	—	89	2	no	—
214	C₂×C₁₆	—	89	2	no	—
215	C₂×C₁₆	—	178	2	no	—
216	#107	—	2	1	no	—
217	#107	—	89	2	no	—
218	#107	—	178	1	no	—
219	#107	C₂²×D₈₉	1	2	yes	—
220	#103	—	178	4	no	—
221	#102	—	2	4	no	—
222	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
223	C₄×C₈	C₂²×D₈₉	1	2	yes	—
224	C₂⁴	—	4	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	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	8	4	no	—
236	C₂²×C₄	—	356	1	no	—
237	C₂²×C₄	—	356	1	no	—
238	C₂²×C₄	#13083559	1	1	yes	characteristic
239	C₂²×C₄	#13083559	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
241	Q₁₆	—	89	4	no	—
242	Q₁₆	—	178	4	no	—
243	Q₁₆	—	178	2	no	—
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₂×C₄×D₈₉	1	4	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
250	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
251	C₂×C₈	#13083574	1	2	yes	—
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	356	4	no	—
256	C₄⋊C₄	—	356	1	no	—
257	C₄⋊C₄	—	178	2	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	356	4	no	—
260	C₄²	#13083559	1	2	yes	—
261	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
262	C₁₆	—	89	4	no	—
263	C₁₆	—	178	2	no	—
264	C₂³	—	4	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	8	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	C₂×#13083559	1	1	yes	characteristic
269	Q₈	—	356	4	no	—
270	Q₈	—	356	2	no	—
271	Q₈	—	178	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	8	no	—
279	C₂×C₄	—	356	4	no	—
280	C₂×C₄	—	356	2	no	—
281	C₂×C₄	—	356	2	no	—
282	C₂×C₄	—	712	2	no	—
283	C₂×C₄	#15667956	1	4	yes	direct product factor
284	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
285	C₂×C₄	C₂×#13083559	1	2	yes	—
286	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
287	C₂×C₄	C₂×#13083559	1	2	yes	—
288	C₂×C₄	#15668032	1	2	yes	—
289	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
292	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₈	C₂×#13083574	1	1	yes	characteristic
294	C₂²	—	4	4	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	8	no	—
297	C₂²	—	8	2	no	—
298	C₂²	C₄×#13083559	1	2	yes	—
299	C₂²	C₂×#15667956	1	2	yes	—
300	C₂²	#19152028	1	2	yes	—
301	C₂²	C₂²×#13083559	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	356	4	no	—
305	C₄	—	712	2	no	—
306	C₄	C₄×#13083559	1	2	yes	—
307	C₄	C₂×#15667956	1	4	yes	direct product factor
308	C₄	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₄	C₂×#15668032	1	1	yes	characteristic
310	C₂	—	8	4	no	—
311	C₂	C₄×#15667956	1	4	yes	direct product factor
312	C₂	C₂×C₄×#13083559	1	1	yes	characteristic
313	C₂	C₂²×#15667956	1	1	yes	characteristic
314	C₂	C₂×#19152028	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#13083559
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×D₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₈×C₈₉	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×D₈	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×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₅₆	C₂	semidirect
3. C₄×#15667956	C₂	direct (N × H)
4. C₂×#19151425	C₂	semidirect
5. C₂×#19151401	C₂	semidirect
7. C₂×C₄×#13083549	C₂	semidirect
11. C₂×#15667956	C₄	direct (N × H)
16. C₄×Dic₃₅₆	C₂²	semidirect
19. #19151425	C₂²	semidirect
20. #19151401	C₂²	semidirect
22. C₄×#13083549	C₂²	semidirect
33. C₂×Dic₃₅₆	C₂×C₄	semidirect
40. #15667956	C₂×C₄	direct (N × H)
42. C₂×#13083549	C₂×C₄	semidirect
74. Dic₃₅₆	C₂²×C₄	semidirect
80. #13083549	C₂²×C₄	semidirect
118. C₂×C₄×C₈₉	SD₃₂	semidirect
147. C₄×C₈₉	C₂×SD₃₂	semidirect
154. C₂×C₈₉	C₄×SD₃₂	semidirect
166. C₈₉	C₂×C₄×SD₃₂	semidirect
283. C₂×C₄	#15667956	direct (N × H)
307. C₄	C₂×#15667956	direct (N × H)
311. C₂	C₄×#15667956	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×#15667956	2
C₂×C₄×Dic₃₅₆	2
C₄×#15667956	2
C₂×#19151425	2
C₂×#19151401	2
C₂×#19151399	2
C₂×C₄×#13083549	2
C₂×C₄×D₈×C₈₉	2
C₂×C₄×SD₃₂	89

Subgroups — C2×C4×#15667956 #36707693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15667956 #36707693