Subgroups — #36447349

Order 22784.

398 conjugacy classes of subgroups.

The subgroup lattice has 262 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	#26020245	C₂	1	2	yes	maximal
3	#26019008	C₂	1	1	yes	maximal, characteristic
4	#26018126	C₂	1	2	yes	maximal
5	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₇₈	—	2	1	no	—
8	C₈×Dic₁₇₈	—	2	1	no	—
9	C₈×Dic₁₇₈	—	2	1	no	—
10	C₈×Dic₁₇₈	—	2	1	no	—
11	#19151622	C₂²	1	2	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×#15667941	C₄	1	2	yes	—
14	C₂×#15667930	—	2	2	no	—
15	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
16	#19151366	—	2	2	no	—
17	#19151349	—	2	1	no	—
18	#19151349	—	2	1	no	—
19	#423×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#334×C₈₉	—	2	1	no	—
22	#334×C₈₉	—	2	1	no	—
23	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
24	C₂×#13083558	—	2	2	no	—
25	C₂²×#10623494	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₇₈	—	2	1	no	—
27	C₄×Dic₁₇₈	—	2	1	no	—
28	C₄×Dic₁₇₈	—	2	1	no	—
29	C₄×Dic₁₇₈	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
31	C₂×#13083553	Q₈	1	2	yes	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
33	C₄⋊#10623494	—	2	1	no	—
34	C₄⋊#10623494	—	2	1	no	—
35	C₄⋊#10623494	—	2	2	no	—
36	C₄⋊#10623494	—	2	2	no	—
37	#15667941	—	2	2	no	—
38	#15667941	—	2	2	no	—
39	#15667930	—	4	4	no	—
40	C₈×Dic₈₉	—	2	2	no	—
41	C₈×Dic₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₈₉	—	2	2	no	—
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂×Dic₁₇₈	—	2	1	no	—
53	C₂×Dic₁₇₈	—	2	1	no	—
54	C₂×Dic₁₇₈	—	2	2	no	—
55	C₂×Dic₁₇₈	SD₁₆	1	4	yes	—
56	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
57	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
58	#13083558	—	4	2	no	—
59	#13083558	—	4	2	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₄⋊Dic₈₉	—	2	1	no	—
64	C₄⋊Dic₈₉	—	2	1	no	—
65	#13083553	—	2	2	no	—
66	#13083553	—	2	2	no	—
67	C₄×Dic₈₉	—	2	2	no	—
68	C₄×Dic₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
72	M₁₆×C₈₉	—	4	2	no	—
73	M₁₆×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	—	2	2	no	—
75	C₂×C₈×C₈₉	—	4	1	no	—
76	C₂×C₈×C₈₉	—	4	2	no	—
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	#41	1	1	yes	characteristic
84	C₄²×C₈₉	#41	1	1	yes	characteristic
85	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	4	1	no	—
89	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₈₉	#135	1	2	yes	—
97	C₂×Dic₈₉	—	2	2	no	—
98	C₂×Dic₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	2	2	no	—
100	C₂×Dic₈₉	—	4	2	no	—
101	C₂×Dic₈₉	#106	1	4	yes	—
102	Dic₁₇₈	—	2	4	no	—
103	Dic₁₇₈	—	2	4	no	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	—	2	1	no	—
106	C₈×C₈₉	—	2	1	no	—
107	C₈×C₈₉	—	8	2	no	—
108	C₈×C₈₉	#101	1	2	yes	complemented
109	#10623494	—	4	2	no	—
110	#10623494	—	8	2	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	#380	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
115	C₂²×C₈₉	#427	1	1	yes	characteristic
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
120	C₄×C₈₉	#413	1	1	yes	characteristic
121	C₄×C₈₉	#465	1	2	yes	—
122	C₄×C₈₉	#465	1	2	yes	—
123	Dic₈₉	—	2	4	no	—
124	Dic₈₉	—	4	2	no	—
125	#8615	—	89	1	no	maximal, Sylow
126	C₂×C₈₉	—	2	1	no	—
127	C₂×C₈₉	—	2	1	no	—
128	C₂×C₈₉	#1801	1	1	yes	characteristic
129	C₂×C₈₉	#1807	1	1	yes	characteristic
130	C₂×C₈₉	#1813	1	1	yes	characteristic
131	C₂×C₈×Q₈	—	89	1	no	—
132	#1801	—	89	2	no	—
133	#1566	—	89	1	no	—
134	#1566	D₈₉	1	1	yes	characteristic
135	#1292	—	89	2	no	—
136	C₈₉	#8615	1	1	yes	characteristic, complemented, Sylow
137	C₂×C₄×Q₈	—	89	1	no	—
138	C₈×Q₈	—	178	1	no	—
139	C₈×Q₈	—	178	1	no	—
140	C₈×Q₈	—	178	1	no	—
141	C₈×Q₈	—	178	1	no	—
142	#423	—	89	2	no	—
143	#423	C₂×D₈₉	1	2	yes	—
144	C₂×C₄⋊C₈	—	89	2	no	—
145	C₂×C₄⋊C₈	—	89	1	no	—
146	C₂×#101	—	178	2	no	—
147	C₂×C₄×C₈	—	89	2	no	—
148	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
149	#343	—	178	2	no	—
150	#334	—	2	1	no	—
151	#334	—	2	1	no	—
152	#334	—	178	1	no	—
153	#334	—	178	1	no	—
154	C₂²×Q₈	—	89	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	178	2	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
160	C₄×Q₈	—	178	1	no	—
161	C₄×Q₈	—	178	1	no	—
162	C₄×Q₈	—	178	1	no	—
163	C₄×Q₈	—	178	1	no	—
164	C₂×C₄⋊C₄	—	89	1	no	—
165	C₂×C₄⋊C₄	—	89	2	no	—
166	C₂×C₄²	—	89	2	no	—
167	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	178	2	no	—
171	C₄⋊C₈	—	178	2	no	—
172	C₄⋊C₈	—	178	1	no	—
173	C₄⋊C₈	—	178	2	no	—
174	C₄⋊C₈	—	178	1	no	—
175	C₄⋊C₈	—	178	2	no	—
176	#101	—	356	4	no	—
177	C₄×C₈	—	178	2	no	—
178	C₄×C₈	—	178	2	no	—
179	C₄×C₈	D₃₅₆	1	1	yes	characteristic
180	C₄×C₈	D₃₅₆	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×Q₈	—	89	4	no	—
184	C₂×Q₈	—	178	1	no	—
185	C₂×Q₈	—	178	1	no	—
186	C₂×Q₈	—	178	2	no	—
187	C₂²×C₄	—	89	2	no	—
188	C₂²×C₄	—	89	2	no	—
189	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	356	2	no	—
195	M₁₆	—	356	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	356	2	no	—
200	C₂×C₈	—	356	2	no	—
201	C₂×C₈	—	178	2	no	—
202	C₂×C₈	#13083551	1	1	yes	characteristic
203	C₂×C₈	#13083551	1	1	yes	characteristic
204	C₂×C₈	#13083551	1	1	yes	characteristic
205	C₂×C₈	#13083551	1	1	yes	characteristic
206	C₄⋊C₄	—	178	1	no	—
207	C₄⋊C₄	—	178	1	no	—
208	C₄⋊C₄	—	178	2	no	—
209	C₄⋊C₄	—	178	2	no	—
210	C₄²	—	178	2	no	—
211	C₄²	—	178	2	no	—
212	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
213	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
214	C₄²	#13083573	1	1	yes	characteristic
215	C₄²	#13083575	1	1	yes	characteristic
216	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
217	Q₈	—	178	4	no	—
218	Q₈	—	178	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	89	4	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	—	178	2	no	—
225	C₂×C₄	—	178	2	no	—
226	C₂×C₄	—	178	2	no	—
227	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
228	C₂×C₄	#15667970	1	1	yes	characteristic
229	C₂×C₄	#15667983	1	1	yes	characteristic
230	C₂×C₄	#15667987	1	1	yes	characteristic
231	C₂×C₄	#15667988	1	1	yes	characteristic
232	C₂×C₄	#15667992	1	1	yes	characteristic
233	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
234	C₂×C₄	#15668033	1	2	yes	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	356	2	no	—
240	C₈	—	712	2	no	—
241	C₈	#15667930	1	2	yes	complemented
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#19151455	1	1	yes	characteristic
245	C₂²	#106×D₈₉	1	1	yes	characteristic
246	C₂²	#19151633	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	356	2	no	—
251	C₄	—	178	4	no	—
252	C₄	C₂×#15667930	1	1	yes	characteristic
253	C₄	#19151580	1	1	yes	characteristic
254	C₄	#19151824	1	2	yes	—
255	C₄	#19151828	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#26020245	1	1	yes	characteristic
259	C₂	#26020274	1	1	yes	characteristic
260	C₂	#26020299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151455
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#8615
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
108. C₈×C₈₉	#101	semidirect
136. C₈₉	#8615	semidirect
241. C₈	#15667930	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
#26020245	2
#26019008	2
#26018126	2
C₈₉×#1566	2
#8615	89