Subgroups — #36470510

Order 22784.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151415	C₂	1	2	yes	maximal
3	#26020424	C₂	1	1	yes	maximal, characteristic
4	#26018361	C₂	1	2	yes	maximal
5	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	—	2	1	no	—
8	C₄×Dic₃₅₆	—	2	1	no	—
9	C₄×Dic₃₅₆	—	2	1	no	—
10	C₄×Dic₃₅₆	C₄	1	2	yes	—
11	C₄×Dic₃₅₆	C₄	1	2	yes	—
12	C₂×#15667948	—	2	1	no	—
13	C₂×#15667948	C₂²	1	2	yes	—
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	#19151622	—	2	1	no	—
16	C₂×#15667937	—	2	1	no	—
17	#19151415	—	2	4	no	—
18	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₁₆×C₈₉	—	2	1	no	—
21	C₄⋊C₁₆×C₈₉	—	2	1	no	—
22	C₂²×Dic₁₇₈	—	2	1	no	—
23	C₂×Dic₃₅₆	—	2	1	no	—
24	C₂×Dic₃₅₆	—	2	1	no	—
25	C₂×Dic₃₅₆	—	2	1	no	—
26	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
27	C₂×Dic₃₅₆	D₄	1	2	yes	—
28	C₂×Dic₃₅₆	Q₈	1	2	yes	—
29	C₂×#13083558	—	2	2	no	—
30	C₄×Dic₁₇₈	—	2	2	no	—
31	C₄×Dic₁₇₈	—	2	2	no	—
32	C₄×Dic₁₇₈	—	4	1	no	—
33	C₄×Dic₁₇₈	—	4	1	no	—
34	C₂×C₄⋊Dic₈₉	—	2	1	no	—
35	C₂×#13083553	—	4	1	no	—
36	C₂×C₄×Dic₈₉	—	4	1	no	—
37	#15667948	—	2	2	no	—
38	#15667948	—	2	2	no	—
39	#15667948	—	4	2	no	—
40	#15667946	—	2	1	no	—
41	#15667946	C₂×C₄	1	2	yes	—
42	C₄⋊#10623494	—	4	1	no	—
43	C₄⋊#10623494	—	4	1	no	—
44	#15667937	—	2	2	no	—
45	#15667937	—	2	2	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	M₃₂×C₈₉	—	2	2	no	—
50	M₃₂×C₈₉	—	2	2	no	—
51	C₂×C₁₆×C₈₉	—	2	2	no	—
52	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂×Dic₁₇₈	—	2	2	no	—
57	C₂×Dic₁₇₈	—	2	2	no	—
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×Dic₁₇₈	—	4	1	no	—
60	C₂×Dic₁₇₈	—	4	1	no	—
61	C₂×Dic₁₇₈	—	4	1	no	—
62	C₂²×Dic₈₉	—	4	1	no	—
63	C₂²×Dic₈₉	—	4	1	no	—
64	Dic₃₅₆	—	2	2	no	—
65	Dic₃₅₆	—	2	2	no	—
66	Dic₃₅₆	—	2	2	no	—
67	Dic₃₅₆	C₄⋊C₄	1	4	yes	—
68	#13083558	—	4	2	no	—
69	#13083558	—	4	2	no	—
70	C₂×#10623494	—	4	2	no	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	—	4	1	no	—
73	#13083553	—	4	2	no	—
74	#13083553	—	4	2	no	—
75	C₄×Dic₈₉	—	4	2	no	—
76	C₄×Dic₈₉	—	4	2	no	—
77	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
90	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₈₉	D₈	1	1	yes	characteristic
93	C₄²×C₈₉	D₈	1	1	yes	characteristic
94	C₁₆×C₈₉	—	4	2	no	—
95	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
108	C₂×Dic₈₉	—	4	2	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	8	2	no	—
113	Dic₁₇₈	—	2	4	no	—
114	Dic₁₇₈	—	4	2	no	—
115	Dic₁₇₈	—	4	2	no	—
116	Dic₁₇₈	—	4	2	no	—
117	C₈×C₈₉	—	2	1	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₈₉	#117	1	1	yes	characteristic
121	C₈×C₈₉	#121	1	1	yes	characteristic
122	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
123	#10623494	—	8	2	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
127	C₂²×C₈₉	#461	1	1	yes	characteristic
128	C₂²×C₈₉	#476	1	1	yes	characteristic
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	#362	1	2	yes	complemented
133	C₄×C₈₉	#380	1	1	yes	characteristic
134	C₄×C₈₉	#417	1	1	yes	characteristic
135	C₄×C₈₉	#459	1	1	yes	characteristic
136	C₄×C₈₉	#474	1	1	yes	characteristic
137	Dic₈₉	—	4	4	no	—
138	Dic₈₉	—	8	2	no	—
139	#12819	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	#1802	1	1	yes	characteristic
143	C₂×C₈₉	C₂×#362	1	1	yes	characteristic
144	C₂×C₈₉	#2148	1	1	yes	characteristic
145	C₂×C₄×Q₁₆	—	89	1	no	—
146	#2150	D₈₉	1	1	yes	characteristic
147	C₂×#362	—	89	2	no	—
148	#1848	—	89	1	no	—
149	#1387	—	89	2	no	—
150	C₈₉	#12819	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	89	1	no	—
152	C₂×C₄×Q₈	—	178	1	no	—
153	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
154	C₄×Q₁₆	—	89	2	no	—
155	C₄×Q₁₆	—	89	2	no	—
156	C₄×Q₁₆	—	178	1	no	—
157	C₄×Q₁₆	—	178	1	no	—
158	C₂×#108	—	89	2	no	—
159	C₂×#108	—	178	1	no	—
160	C₂×#107	—	89	1	no	—
161	#423	—	178	1	no	—
162	C₂×#103	—	178	1	no	—
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₄⋊C₁₆	—	2	1	no	—
165	C₄⋊C₁₆	—	2	1	no	—
166	#362	—	178	4	no	—
167	C₂²×Q₈	—	178	1	no	—
168	C₂×Q₁₆	—	89	2	no	—
169	C₂×Q₁₆	—	89	2	no	—
170	C₂×Q₁₆	—	89	2	no	—
171	C₂×Q₁₆	—	178	1	no	—
172	C₂×Q₁₆	—	178	1	no	—
173	C₂×Q₁₆	—	178	1	no	—
174	C₂×M₁₆	—	178	2	no	—
175	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
177	C₄×Q₈	—	356	1	no	—
178	C₄×Q₈	—	356	1	no	—
179	C₄×Q₈	—	178	2	no	—
180	C₄×Q₈	—	178	2	no	—
181	C₂×C₄⋊C₄	—	356	1	no	—
182	C₂×C₄⋊C₄	—	178	1	no	—
183	C₂×C₄²	—	356	1	no	—
184	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	#108	—	356	2	no	—
189	#108	—	178	2	no	—
190	#108	—	178	2	no	—
191	#107	—	89	2	no	—
192	#107	—	178	1	no	—
193	C₄⋊C₈	—	356	1	no	—
194	C₄⋊C₈	—	356	1	no	—
195	#103	—	178	2	no	—
196	#103	—	178	2	no	—
197	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
198	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×Q₈	—	356	1	no	—
202	C₂×Q₈	—	356	1	no	—
203	C₂×Q₈	—	356	1	no	—
204	C₂×Q₈	—	178	2	no	—
205	C₂×Q₈	—	178	2	no	—
206	C₂×Q₈	—	178	2	no	—
207	C₂²×C₄	—	356	1	no	—
208	C₂²×C₄	—	356	1	no	—
209	C₂²×C₄	#13083563	1	1	yes	characteristic
210	C₂²×C₄	#13083563	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
212	Q₁₆	—	89	4	no	—
213	Q₁₆	—	178	2	no	—
214	Q₁₆	—	178	2	no	—
215	Q₁₆	—	178	2	no	—
216	M₁₆	—	356	2	no	—
217	M₁₆	—	356	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	356	2	no	—
221	C₂×C₈	#13083551	1	1	yes	characteristic
222	C₂×C₈	#13083551	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
224	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
227	C₂×C₈	#13083573	1	1	yes	characteristic
228	C₂×C₈	#13083575	1	1	yes	characteristic
229	C₄⋊C₄	—	356	1	no	—
230	C₄⋊C₄	—	356	2	no	—
231	C₄⋊C₄	—	356	2	no	—
232	C₄⋊C₄	—	178	2	no	—
233	C₄²	—	356	2	no	—
234	C₄²	—	356	2	no	—
235	C₄²	#13083551	1	1	yes	characteristic
236	C₄²	#13083551	1	1	yes	characteristic
237	C₄²	D₇₁₂	1	1	yes	characteristic
238	C₄²	D₇₁₂	1	1	yes	characteristic
239	C₁₆	—	4	2	no	—
240	C₂³	C₂×#13083563	1	1	yes	characteristic
241	Q₈	—	356	2	no	—
242	Q₈	—	356	2	no	—
243	Q₈	—	356	2	no	—
244	Q₈	—	178	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	356	2	no	—
248	C₂×C₄	—	356	2	no	—
249	C₂×C₄	—	356	2	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	712	2	no	—
252	C₂×C₄	#15667934	1	1	yes	characteristic
253	C₂×C₄	#15667934	1	1	yes	characteristic
254	C₂×C₄	#15667934	1	1	yes	characteristic
255	C₂×C₄	#15667934	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
257	C₂×C₄	#15667987	1	1	yes	characteristic
258	C₂×C₄	#15667992	1	1	yes	characteristic
259	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
260	C₂×C₄	#15668040	1	1	yes	characteristic
261	C₂×C₄	#15668045	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	712	2	no	—
265	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
266	C₈	#15667970	1	1	yes	characteristic
267	C₈	#15667983	1	1	yes	characteristic
268	C₈	#15667988	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#15667934	1	1	yes	characteristic
272	C₂²	#19151804	1	1	yes	characteristic
273	C₂²	#19151889	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	356	4	no	—
278	C₄	—	712	2	no	—
279	C₄	#19151415	1	2	yes	complemented
280	C₄	#19151455	1	1	yes	characteristic
281	C₄	#19151600	1	1	yes	characteristic
282	C₄	#19151794	1	1	yes	characteristic
283	C₄	#19151879	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#26020253	1	1	yes	characteristic
287	C₂	C₂×#19151415	1	1	yes	characteristic
288	C₂	#26022002	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#12819
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
132. C₄×C₈₉	#362	semidirect
150. C₈₉	#12819	semidirect
279. C₄	#19151415	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×Dic₃₅₆	2
C₂×#19151415	2
#26020424	2
#26018361	2
C₈₉×#2150	2
#12819	89