Subgroups — C₈×#15667930 #36444992

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667930	C₂	1	1	yes	maximal, characteristic
3	#26018575	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
5	#26018103	C₂	1	1	yes	maximal, characteristic
6	#26018101	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₇₈	—	2	1	no	—
10	C₈×Dic₁₇₈	—	2	1	no	—
11	C₈×Dic₁₇₈	—	2	1	no	—
12	C₈×Dic₁₇₈	—	2	1	no	—
13	#19151651	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×#15667941	C₄	1	2	yes	—
16	C₂×#15667930	C₄	1	2	yes	—
17	C₄×#13083558	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
19	#19151366	C₄	1	2	yes	—
20	#19151323	C₄	1	4	yes	—
21	#19151320	—	2	1	no	—
22	C₈×#10623494	—	2	1	no	—
23	#431×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#322×C₈₉	—	2	1	no	—
27	C₈²×C₈₉	—	2	1	no	—
28	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
29	C₂×#13083558	C₂×C₄	1	2	yes	—
30	C₂²×#10623494	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₇₈	—	2	1	no	—
32	C₄×Dic₁₇₈	—	2	1	no	—
33	C₄×Dic₁₇₈	—	2	1	no	—
34	C₄×Dic₁₇₈	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×#13083553	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	—	2	1	no	—
40	C₄⋊#10623494	—	2	1	no	—
41	#15667941	—	2	2	no	—
42	#15667941	—	2	2	no	—
43	#15667930	C₈	1	8	yes	direct product factor
44	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
45	#15667923	—	2	1	no	—
46	C₄×#10623494	—	2	1	no	—
47	C₄×#10623494	—	2	1	no	—
48	C₈×Dic₈₉	—	2	2	no	—
49	C₈×Dic₈₉	—	2	2	no	—
50	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	—	2	1	no	—
55	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
56	#97×C₈₉	—	2	1	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	—	2	2	no	—
66	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
67	C₂²×Dic₈₉	C₄²	1	2	yes	—
68	C₂²×Dic₈₉	C₄²	1	2	yes	—
69	#13083558	—	2	2	no	—
70	#13083558	C₂×C₈	1	4	yes	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	4	2	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	#13083553	—	2	2	no	—
78	#13083553	—	2	2	no	—
79	C₄×Dic₈₉	—	2	2	no	—
80	C₄×Dic₈₉	—	2	2	no	—
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	M₁₆×C₈₉	—	2	2	no	—
85	M₁₆×C₈₉	C₂×C₈	1	4	yes	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	1	no	—
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₈₉	#41	1	1	yes	characteristic
97	C₄²×C₈₉	#41	1	1	yes	characteristic
98	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	4	1	no	—
102	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₈₉	#131	1	2	yes	—
110	C₂×Dic₈₉	—	2	2	no	—
111	C₂×Dic₈₉	—	2	2	no	—
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	C₄×C₈	1	4	yes	—
115	Dic₁₇₈	—	2	4	no	—
116	Dic₁₇₈	—	2	4	no	—
117	C₈×C₈₉	—	2	1	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	—	2	4	no	—
121	C₈×C₈₉	—	4	2	no	—
122	C₈×C₈₉	#101	1	2	yes	complemented
123	#10623494	—	2	4	no	—
124	#10623494	—	4	2	no	—
125	#10623494	—	4	2	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
130	C₂²×C₈₉	#405	1	1	yes	characteristic
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
135	C₄×C₈₉	#413	1	1	yes	characteristic
136	C₄×C₈₉	C₈×D₄	1	2	yes	—
137	C₄×C₈₉	C₈×D₄	1	2	yes	—
138	Dic₈₉	—	2	4	no	—
139	Dic₈₉	—	4	2	no	—
140	C₈×#101	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₈₉	C₄×#101	1	1	yes	characteristic
145	C₂×C₈₉	#1757	1	1	yes	characteristic
146	C₂×C₈×Q₈	—	89	1	no	—
147	C₄×#101	—	89	1	no	—
148	#1461	—	89	1	no	—
149	C₈×M₁₆	—	89	1	no	—
150	C₈×M₁₆	D₈₉	1	1	yes	characteristic
151	#1283	—	89	1	no	—
152	#1283	—	89	1	no	—
153	C₈₉	C₈×#101	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄×Q₈	—	89	1	no	—
155	C₈×Q₈	—	178	1	no	—
156	C₈×Q₈	—	178	1	no	—
157	C₈×Q₈	—	178	1	no	—
158	C₈×Q₈	—	178	1	no	—
159	#431	—	89	1	no	—
160	#431	C₂×D₈₉	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	89	2	no	—
162	C₂×C₄⋊C₈	—	89	1	no	—
163	C₂×#101	—	89	2	no	—
164	C₄×M₁₆	—	89	1	no	—
165	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
166	C₂×C₄×C₈	—	89	2	no	—
167	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
168	#343	—	89	2	no	—
169	#324	—	89	4	no	—
170	#322	—	2	1	no	—
171	#322	—	178	1	no	—
172	C₈²	—	2	1	no	—
173	C₈²	—	178	1	no	—
174	C₂²×Q₈	—	89	1	no	—
175	C₂×M₁₆	—	89	2	no	—
176	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
177	C₂²×C₈	—	89	2	no	—
178	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
180	C₄×Q₈	—	178	1	no	—
181	C₄×Q₈	—	178	1	no	—
182	C₄×Q₈	—	178	1	no	—
183	C₄×Q₈	—	178	1	no	—
184	C₂×C₄⋊C₄	—	89	2	no	—
185	C₂×C₄⋊C₄	—	89	1	no	—
186	C₂×C₄²	—	89	2	no	—
187	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	178	2	no	—
190	C₄⋊C₈	—	178	2	no	—
191	C₄⋊C₈	—	178	1	no	—
192	C₄⋊C₈	—	178	1	no	—
193	C₄⋊C₈	—	178	1	no	—
194	#101	—	89	8	no	—
195	C₂²⋊C₈	—	89	2	no	—
196	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
197	#97	—	2	1	no	—
198	#97	—	178	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	178	2	no	—
202	C₄×C₈	—	178	2	no	—
203	C₄×C₈	—	178	1	no	—
204	C₄×C₈	—	178	1	no	—
205	C₄×C₈	D₃₅₆	1	1	yes	characteristic
206	C₄×C₈	D₃₅₆	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂×Q₈	—	89	4	no	—
210	C₂×Q₈	—	178	1	no	—
211	C₂×Q₈	—	178	1	no	—
212	C₂×Q₈	—	178	2	no	—
213	C₂²×C₄	—	89	2	no	—
214	C₂²×C₄	—	89	2	no	—
215	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	89	4	no	—
220	M₁₆	—	178	2	no	—
221	M₁₆	C₈×D₈₉	1	4	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	356	2	no	—
227	C₂×C₈	—	178	2	no	—
228	C₂×C₈	—	178	2	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	#13083551	1	1	yes	characteristic
231	C₂×C₈	#13083551	1	1	yes	characteristic
232	C₂×C₈	#13083551	1	1	yes	characteristic
233	C₂×C₈	#13083551	1	1	yes	characteristic
234	C₄⋊C₄	—	178	2	no	—
235	C₄⋊C₄	—	178	1	no	—
236	C₄⋊C₄	—	178	2	no	—
237	C₄⋊C₄	—	178	1	no	—
238	C₄²	—	178	2	no	—
239	C₄²	—	178	2	no	—
240	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
241	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
242	C₄²	#13083574	1	1	yes	characteristic
243	C₄²	#13083574	1	1	yes	characteristic
244	C₂³	C₄²×D₈₉	1	1	yes	characteristic
245	Q₈	—	178	4	no	—
246	Q₈	—	178	4	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	89	4	no	—
251	C₂×C₄	—	356	2	no	—
252	C₂×C₄	—	178	2	no	—
253	C₂×C₄	—	178	2	no	—
254	C₂×C₄	—	178	2	no	—
255	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
256	C₂×C₄	#15667969	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
262	C₂×C₄	#15668018	1	2	yes	—
263	C₈	—	2	4	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	4	2	no	—
268	C₈	—	356	2	no	—
269	C₈	—	356	2	no	—
270	C₈	—	178	4	no	—
271	C₈	#15667930	1	2	yes	direct product factor
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₄×#13083551	1	1	yes	characteristic
275	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
276	C₂²	#19151552	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	356	2	no	—
281	C₄	—	178	4	no	—
282	C₄	C₂×#15667930	1	1	yes	characteristic
283	C₄	#19151580	1	1	yes	characteristic
284	C₄	C₈×C₂²⋊D₈₉	1	2	yes	—
285	C₄	C₈×D₃₅₆	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₈×#13083551	1	1	yes	characteristic
289	C₂	C₄×#15667930	1	1	yes	characteristic
290	C₂	#26020053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#13083551
commutator	C₂²×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₈₉	C₂×C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	C₈×#101
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
43. #15667930	C₈	direct (N × H)
122. C₈×C₈₉	#101	semidirect
153. C₈₉	C₈×#101	semidirect
271. C₈	#15667930	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×C₈×Dic₁₇₈	2
C₄×#15667930	2
#26018575	2
C₈×#13083558	2
#26018103	2
#26018101	2
C₈×M₁₆×C₈₉	2
C₈×#101	89

Subgroups — C8×#15667930 #36444992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#15667930 #36444992