Subgroups — #36523903

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029332	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019072	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₃₅₆	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic
6	#26018648	C₂	1	2	yes	maximal
7	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
10	C₈×Dic₁₇₈	C₂²	1	8	yes	—
11	#19151722	—	2	4	no	—
12	C₄×Dic₃₅₆	C₄	1	2	yes	—
13	C₄×Dic₃₅₆	C₄	1	2	yes	—
14	C₄×Dic₃₅₆	C₂²	1	2	yes	—
15	C₄×Dic₃₅₆	C₂²	1	2	yes	—
16	#19151651	C₂²	1	2	yes	—
17	#19151648	—	2	4	no	—
18	C₂×#15667946	C₄	1	1	yes	characteristic
19	C₂×#15667937	C₄	1	2	yes	—
20	#19151351	C₂²	1	2	yes	—
21	#19151349	C₂²	1	2	yes	—
22	#19151330	C₂²	1	8	yes	—
23	#431×C₈₉	C₂²	1	1	yes	characteristic
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#322×C₈₉	C₂²	1	2	yes	—
27	C₈²×C₈₉	C₂²	1	2	yes	—
28	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₅₆	C₈	1	8	yes	complemented
30	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
31	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₇₈	—	2	2	no	—
33	C₄×Dic₁₇₈	—	2	2	no	—
34	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₇₈	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
37	C₂×#13083553	—	2	2	no	—
38	C₂×C₄×Dic₈₉	—	2	2	no	—
39	#15667946	C₂×C₄	1	2	yes	—
40	#15667946	C₂×C₄	1	2	yes	—
41	C₄⋊#10623494	C₂³	1	4	yes	—
42	#15667941	—	2	8	no	—
43	#15667937	C₂×C₄	1	4	yes	—
44	#15667937	C₂×C₄	1	4	yes	—
45	C₂²⋊#10623494	—	2	2	no	—
46	C₄×#10623494	—	2	2	no	—
47	C₈×Dic₈₉	—	2	8	no	—
48	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	—	2	1	no	—
52	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
53	C₂²⋊C₈×C₈₉	—	2	1	no	—
54	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
61	C₂×Dic₁₇₈	—	2	2	no	—
62	C₂×Dic₁₇₈	—	2	4	no	—
63	C₂×Dic₁₇₈	C₂×C₈	1	8	yes	—
64	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
65	C₂²×Dic₈₉	—	2	2	no	—
66	C₂²×Dic₈₉	—	2	2	no	—
67	Dic₃₅₆	C₂×C₈	1	8	yes	—
68	Dic₃₅₆	C₂×C₈	1	8	yes	—
69	C₂×#10623494	—	2	4	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
72	#13083553	—	2	4	no	—
73	#13083553	—	2	4	no	—
74	C₄×Dic₈₉	—	2	4	no	—
75	C₄×Dic₈₉	—	2	4	no	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
87	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	#41	1	1	yes	characteristic
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	4	1	no	—
94	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₈₉	#131	1	2	yes	—
99	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₈₉	—	2	4	no	—
103	C₂×Dic₈₉	—	2	8	no	—
104	C₂×Dic₈₉	—	2	4	no	—
105	C₂×Dic₈₉	—	8	2	no	—
106	Dic₁₇₈	—	2	4	no	—
107	Dic₁₇₈	—	2	8	no	—
108	Dic₁₇₈	C₂²×C₈	1	8	yes	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	—	2	4	no	—
112	C₈×C₈₉	—	4	2	no	—
113	C₈×C₈₉	C₂²×C₈	1	2	yes	—
114	C₈×C₈₉	C₂²×C₈	1	2	yes	—
115	#10623494	—	4	4	no	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	C₈×D₄	1	2	yes	—
119	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	C₈×D₄	1	2	yes	—
124	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
125	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
126	C₄×C₈₉	#578	1	2	yes	—
127	Dic₈₉	—	2	8	no	—
128	Dic₈₉	—	4	4	no	—
129	#17856	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	2	no	—
131	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
132	C₂×C₈₉	#2950	1	1	yes	characteristic
133	C₂×C₈₉	#2957	1	1	yes	characteristic
134	#2982	—	89	2	no	—
135	C₂×C₄×Q₁₆	—	89	1	no	—
136	#1579	—	89	4	no	—
137	C₈×Q₁₆	—	89	4	no	—
138	#1564	—	89	1	no	—
139	#1479	—	89	2	no	—
140	#1450	D₈₉	1	1	yes	characteristic
141	C₈₉	#17856	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₁₆	—	89	1	no	—
143	C₂×C₄×Q₈	—	89	2	no	—
144	C₈×Q₈	—	89	8	no	—
145	C₈×Q₈	—	178	4	no	—
146	C₄×Q₁₆	—	89	2	no	—
147	C₄×Q₁₆	—	89	2	no	—
148	C₄×Q₁₆	—	89	2	no	—
149	C₄×Q₁₆	—	89	2	no	—
150	#431	—	89	2	no	—
151	#431	—	178	4	no	—
152	#431	C₂×D₈₉	1	1	yes	characteristic
153	#431	C₂×D₈₉	1	1	yes	characteristic
154	C₂×#107	—	89	1	no	—
155	C₂×#103	—	89	2	no	—
156	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
157	#335	—	89	2	no	—
158	#334	—	89	2	no	—
159	Q₈⋊C₈	—	89	8	no	—
160	#322	C₂×D₈₉	1	2	yes	—
161	C₈²	C₂×D₈₉	1	2	yes	—
162	C₂²×Q₈	—	89	2	no	—
163	C₂×Q₁₆	—	89	8	no	—
164	C₂×Q₁₆	—	89	2	no	—
165	C₂×Q₁₆	—	89	2	no	—
166	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
168	C₄×Q₈	—	89	4	no	—
169	C₄×Q₈	—	89	4	no	—
170	C₄×Q₈	—	178	2	no	—
171	C₄×Q₈	—	178	2	no	—
172	C₂×C₄⋊C₄	—	89	2	no	—
173	C₂×C₄⋊C₄	—	178	2	no	—
174	C₂×C₄²	—	178	2	no	—
175	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
176	#107	—	89	2	no	—
177	#107	—	89	2	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	89	4	no	—
180	C₄⋊C₈	—	178	8	no	—
181	C₄⋊C₈	D₃₅₆	1	2	yes	—
182	#103	—	89	4	no	—
183	#103	—	89	4	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	178	2	no	—
186	C₂²⋊C₈	D₃₅₆	1	2	yes	—
187	C₄×C₈	—	178	8	no	—
188	C₄×C₈	—	178	2	no	—
189	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
190	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₈₉	1	2	yes	—
192	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₈₉	1	2	yes	—
195	C₂×Q₈	—	89	8	no	—
196	C₂×Q₈	—	89	4	no	—
197	C₂×Q₈	—	178	2	no	—
198	C₂×Q₈	—	178	4	no	—
199	C₂²×C₄	—	178	2	no	—
200	C₂²×C₄	—	178	2	no	—
201	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
203	C₂²×C₄	#13083574	1	1	yes	characteristic
204	Q₁₆	—	89	8	no	—
205	Q₁₆	—	89	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	178	4	no	—
209	C₂×C₈	C₈×D₈₉	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
215	C₄⋊C₄	—	89	4	no	—
216	C₄⋊C₄	—	178	2	no	—
217	C₄⋊C₄	—	178	4	no	—
218	C₄⋊C₄	—	178	4	no	—
219	C₄²	—	178	4	no	—
220	C₄²	—	178	4	no	—
221	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
222	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
223	C₄²	#13083574	1	1	yes	characteristic
224	C₄²	C₂³×D₈₉	1	1	yes	characteristic
225	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
226	Q₈	—	89	8	no	—
227	Q₈	—	178	8	no	—
228	Q₈	—	178	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	178	4	no	—
232	C₂×C₄	—	178	8	no	—
233	C₂×C₄	—	178	4	no	—
234	C₂×C₄	—	712	2	no	—
235	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
236	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
237	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
238	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
239	C₂×C₄	#15668018	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	356	4	no	—
248	C₈	C₂×C₈×D₈₉	1	2	yes	—
249	C₈	C₂×C₈×D₈₉	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₈×D₃₅₆	1	2	yes	—
253	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	356	4	no	—
258	C₄	—	178	8	no	—
259	C₄	C₈×D₃₅₆	1	2	yes	—
260	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
261	C₄	C₂×#15668018	1	1	yes	characteristic
262	C₄	#19152415	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₈×D₃₅₆	1	1	yes	characteristic
265	C₂	#26029121	1	1	yes	characteristic
266	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1450	D₈₉
hypocenter	C₈₉	#17856
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #26019072	C₂	semidirect
4. C₈×Dic₃₅₆	C₂	semidirect
29. C₂×Dic₃₅₆	C₈	semidirect
141. C₈₉	#17856	semidirect

Maximal subgroups

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

subgroup	index
#26029332	2
C₂×C₄×Dic₃₅₆	2
#26019072	2
C₈×Dic₃₅₆	2
#26019000	2
#26018648	2
C₈₉×#1450	2
#17856	89