Subgroups — #36445213

Order 22784.

502 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667937	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151330	C₂	1	1	yes	maximal, characteristic
5	#26018092	C₂	1	1	yes	maximal, characteristic
6	#26018087	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₇₈	—	2	2	no	—
10	C₈×Dic₁₇₈	C₄	1	4	yes	complemented
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×#15667941	—	2	1	no	—
14	C₂×#15667937	C₄	1	2	yes	—
15	C₂×C₈×Dic₈₉	—	2	1	no	—
16	C₂×C₈×Dic₈₉	—	2	1	no	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
18	#19151361	C₄	1	2	yes	—
19	#19151352	—	2	1	no	—
20	#19151330	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#322×C₈₉	—	2	2	no	—
25	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	—	2	1	no	—
27	C₂²×#10623494	—	2	1	no	—
28	C₄×Dic₁₇₈	—	2	1	no	—
29	C₄×Dic₁₇₈	—	2	1	no	—
30	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
34	C₂×#13083553	—	2	1	no	—
35	C₂×C₄×Dic₈₉	—	2	1	no	—
36	C₂×C₄×Dic₈₉	—	2	1	no	—
37	C₄⋊#10623494	—	2	1	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	C₂×C₄	1	2	yes	—
40	C₄⋊#10623494	C₂×C₄	1	2	yes	—
41	#15667941	—	2	4	no	—
42	#15667937	C₈	1	8	yes	complemented
43	C₈×Dic₈₉	—	2	4	no	—
44	C₈×Dic₈₉	—	4	2	no	—
45	#15667919	—	2	1	no	—
46	C₂²×C₈×C₈₉	—	2	1	no	—
47	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₈₉	—	2	1	no	—
52	C₄×C₈×C₈₉	—	2	1	no	—
53	C₄×C₈×C₈₉	—	2	1	no	—
54	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₈₉	D₄	1	2	yes	—
57	C₂×Dic₁₇₈	—	2	1	no	—
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×Dic₁₇₈	C₄²	1	2	yes	—
60	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
61	C₂²×Dic₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	—	2	2	no	—
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	—	4	1	no	—
67	C₂×#10623494	—	4	1	no	—
68	C₂×#10623494	—	4	1	no	—
69	C₂×#10623494	—	4	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	C₄²	1	2	yes	—
73	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
74	#13083553	—	2	2	no	—
75	#13083553	—	2	2	no	—
76	C₄×Dic₈₉	—	2	2	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	C₄×Dic₈₉	—	4	1	no	—
79	C₄×Dic₈₉	—	4	1	no	—
80	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	2	2	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	4	2	no	—
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
94	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	#41	1	1	yes	characteristic
98	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₄×D₄	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	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₈₉	#131	1	2	yes	—
108	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	2	4	no	—
114	C₂×Dic₈₉	—	2	2	no	—
115	C₂×Dic₈₉	—	2	4	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	Dic₁₇₈	—	2	2	no	—
120	Dic₁₇₈	—	2	4	no	—
121	Dic₁₇₈	C₄×C₈	1	4	yes	complemented
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	—	2	4	no	—
125	C₈×C₈₉	—	4	2	no	—
126	#10623494	—	4	2	no	—
127	#10623494	—	4	2	no	—
128	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₈₉	C₈×D₄	1	2	yes	—
130	C₂²×C₈₉	#440	1	1	yes	characteristic
131	C₂²×C₈₉	#441	1	1	yes	characteristic
132	C₂²×C₈₉	#444	1	2	yes	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
136	C₄×C₈₉	#405	1	1	yes	characteristic
137	C₄×C₈₉	#417	1	1	yes	characteristic
138	C₄×C₈₉	#419	1	1	yes	characteristic
139	C₄×C₈₉	C₈×D₄	1	2	yes	—
140	C₄×C₈₉	#440	1	1	yes	characteristic
141	C₄×C₈₉	#441	1	1	yes	characteristic
142	Dic₈₉	—	2	4	no	—
143	Dic₈₉	—	4	4	no	—
144	Dic₈₉	—	4	2	no	—
145	#8201	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	SD₁₆⋊C₈	1	2	yes	—
147	C₂×C₈₉	#1579	1	2	yes	—
148	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₈₉	#1763	1	1	yes	characteristic
150	C₂×C₈₉	#1765	1	1	yes	characteristic
151	C₂×C₈×Q₈	—	89	1	no	—
152	C₈×C₄⋊C₄	—	89	1	no	—
153	C₄×#103	—	89	1	no	—
154	C₂×Q₈⋊C₈	—	89	1	no	—
155	C₂×#322	D₈₉	1	1	yes	characteristic
156	#1279	—	89	1	no	—
157	#1277	—	89	1	no	—
158	C₈₉	#8201	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	—	89	1	no	—
160	C₈×Q₈	—	89	4	no	—
161	C₈×Q₈	—	178	2	no	—
162	C₂×C₄⋊C₈	—	89	1	no	—
163	C₂×C₄⋊C₈	—	89	1	no	—
164	C₂×C₄⋊C₈	—	178	1	no	—
165	C₂×#103	—	89	2	no	—
166	C₂×C₄×C₈	—	178	1	no	—
167	C₂×C₄×C₈	—	178	1	no	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
171	C₄×C₄⋊C₄	—	89	1	no	—
172	#340	—	89	2	no	—
173	#336	—	178	1	no	—
174	Q₈⋊C₈	—	89	4	no	—
175	#322	—	2	2	no	—
176	C₂²×Q₈	—	89	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	178	1	no	—
179	C₂²×C₈	—	178	1	no	—
180	C₂²×C₈	C₄×D₈₉	1	2	yes	—
181	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
182	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
183	C₄×Q₈	—	89	2	no	—
184	C₄×Q₈	—	89	2	no	—
185	C₄×Q₈	—	178	1	no	—
186	C₄×Q₈	—	178	1	no	—
187	C₂×C₄⋊C₄	—	89	1	no	—
188	C₂×C₄⋊C₄	—	89	2	no	—
189	C₂×C₄⋊C₄	—	178	1	no	—
190	C₂×C₄²	—	178	1	no	—
191	C₂×C₄²	—	178	1	no	—
192	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
193	C₄⋊C₈	—	89	2	no	—
194	C₄⋊C₈	—	89	2	no	—
195	C₄⋊C₈	—	178	4	no	—
196	C₄⋊C₈	—	178	1	no	—
197	C₄⋊C₈	—	178	1	no	—
198	#103	—	89	8	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	356	2	no	—
203	C₄×C₈	—	178	4	no	—
204	C₄×C₈	C₄×D₈₉	1	2	yes	—
205	C₄×C₈	C₄×D₈₉	1	2	yes	—
206	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
207	#95	—	178	1	no	—
208	C₂×Q₈	—	89	4	no	—
209	C₂×Q₈	—	89	2	no	—
210	C₂×Q₈	—	178	1	no	—
211	C₂×Q₈	—	178	2	no	—
212	C₂²×C₄	—	178	2	no	—
213	C₂²×C₄	—	178	1	no	—
214	C₂²×C₄	—	178	1	no	—
215	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
217	C₂²×C₄	#13083574	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	356	1	no	—
227	C₂×C₈	—	356	1	no	—
228	C₂×C₈	—	356	1	no	—
229	C₂×C₈	—	356	1	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	#13083551	1	2	yes	—
233	C₂×C₈	#13083551	1	2	yes	—
234	C₂×C₈	C₈×D₈₉	1	4	yes	—
235	C₄⋊C₄	—	89	4	no	—
236	C₄⋊C₄	—	89	2	no	—
237	C₄⋊C₄	—	178	2	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	2	no	—
240	C₄⋊C₄	—	178	2	no	—
241	C₄²	—	356	1	no	—
242	C₄²	—	356	1	no	—
243	C₄²	—	178	2	no	—
244	C₄²	—	178	2	no	—
245	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
247	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
248	C₄²	#13083574	1	1	yes	characteristic
249	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
250	Q₈	—	89	4	no	—
251	Q₈	—	178	4	no	—
252	Q₈	—	178	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	356	2	no	—
257	C₂×C₄	—	178	4	no	—
258	C₂×C₄	—	178	2	no	—
259	C₂×C₄	—	178	4	no	—
260	C₂×C₄	—	178	2	no	—
261	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
263	C₂×C₄	#15667969	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
268	C₂×C₄	#15668018	1	2	yes	—
269	C₂×C₄	#15668040	1	1	yes	characteristic
270	C₂×C₄	#15668045	1	1	yes	characteristic
271	C₂×C₄	#15668045	1	1	yes	characteristic
272	C₂×C₄	#15668045	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	356	2	no	—
278	C₈	—	356	2	no	—
279	C₂²	C₄×#13083551	1	1	yes	characteristic
280	C₂²	C₈×D₃₅₆	1	2	yes	—
281	C₂²	#19151700	1	1	yes	characteristic
282	C₂²	#19151706	1	1	yes	characteristic
283	C₂²	#19151721	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	356	2	no	—
287	C₄	—	356	4	no	—
288	C₄	—	178	4	no	—
289	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
290	C₄	#19151552	1	1	yes	characteristic
291	C₄	#19151600	1	1	yes	characteristic
292	C₄	#19151608	1	1	yes	characteristic
293	C₄	C₈×C₂²⋊D₈₉	1	2	yes	—
294	C₄	#19151700	1	1	yes	characteristic
295	C₄	#19151706	1	1	yes	characteristic
296	C₂	#26019066	1	2	yes	—
297	C₂	#26019072	1	2	yes	—
298	C₂	C₈×#13083551	1	1	yes	characteristic
299	C₂	#26020083	1	1	yes	characteristic
300	C₂	#26020096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8201
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₇₈	C₄	semidirect
20. #19151330	C₄	semidirect
42. #15667937	C₈	semidirect
121. Dic₁₇₈	C₄×C₈	semidirect
158. C₈₉	#8201	semidirect

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
C₈×C₄⋊Dic₈₉	2
C₄×#15667937	2
C₂×#19151330	2
#26018092	2
#26018087	2
C₂×#322×C₈₉	2
#8201	89