Subgroups — #36443873

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	2	yes	maximal
2	#26018085	C₂	1	4	yes	maximal, complemented
3	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151655	—	2	2	no	—
5	#19151655	C₄	1	4	yes	complemented
6	#19151651	—	2	2	no	—
7	#19151651	C₂²	1	4	yes	—
8	#19151622	C₄	1	4	yes	complemented
9	#19151562	—	2	2	no	—
10	C₄×#13083558	—	2	2	no	—
11	C₂×#117×C₈₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₈₉	—	2	2	no	—
15	#376×C₈₉	C₄	1	2	yes	—
16	C₄³×C₈₉	C₂²	1	2	yes	—
17	C₂×#13083558	—	2	4	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄⋊#10623494	C₂×C₄	1	8	yes	—
20	C₂²⋊#10623494	—	2	4	no	—
21	C₂²⋊#10623494	—	2	4	no	—
22	#15667923	—	2	4	no	—
23	C₄×#10623494	—	2	4	no	—
24	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₈₉	D₄	1	2	yes	—
26	#117×C₈₉	—	2	8	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄²×C₈₉	—	2	4	no	—
30	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	D₄	1	2	yes	—
37	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	#95×C₈₉	—	2	4	no	—
41	#13083558	—	4	8	no	—
42	C₂×#10623494	—	2	8	no	—
43	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	—	2	4	no	—
45	C₂²×C₄×C₈₉	—	2	4	no	—
46	C₂²×C₄×C₈₉	—	4	4	no	—
47	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	2	8	no	—
60	C₂²⋊C₄×C₈₉	—	2	8	no	—
61	C₄²×C₈₉	—	2	4	no	—
62	C₄²×C₈₉	—	4	8	no	—
63	C₄²×C₈₉	C₄²	1	2	yes	—
64	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₈₉	M₁₆	1	8	yes	complemented
71	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
72	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₈₉	—	2	2	no	—
75	C₂³×C₈₉	—	4	2	no	—
76	C₂³×C₈₉	#95	1	1	yes	characteristic
77	C₂³×C₈₉	#95	1	1	yes	characteristic
78	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	4	no	—
83	C₂×C₄×C₈₉	—	2	4	no	—
84	C₂×C₄×C₈₉	—	2	4	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	8	no	—
88	C₂×C₄×C₈₉	—	4	2	no	—
89	C₂×C₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	—	4	2	no	—
91	C₂×C₄×C₈₉	—	4	8	no	—
92	C₂×C₄×C₈₉	#95	1	2	yes	—
93	C₂×C₄×C₈₉	#95	1	2	yes	—
94	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
101	#10623494	—	4	8	no	—
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	—	2	1	no	—
104	C₂²×C₈₉	—	2	1	no	—
105	C₂²×C₈₉	—	2	2	no	—
106	C₂²×C₈₉	—	2	2	no	—
107	C₂²×C₈₉	—	4	4	no	—
108	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
109	C₂²×C₈₉	#407	1	2	yes	—
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	—	2	1	no	—
112	C₄×C₈₉	—	2	2	no	—
113	C₄×C₈₉	—	2	2	no	—
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	—	4	8	no	—
116	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
117	C₄×C₈₉	#421	1	4	yes	—
118	C₄×C₈₉	#423	1	2	yes	—
119	#7868	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	—	2	2	no	—
121	C₂×C₈₉	—	4	1	no	—
122	C₂×C₈₉	—	4	1	no	—
123	C₂×C₈₉	#1728	1	1	yes	characteristic
124	C₂×C₈₉	#1733	1	2	yes	—
125	#2920	—	89	2	no	—
126	C₄×#117	D₈₉	1	1	yes	characteristic
127	#1276	—	89	4	no	—
128	C₈₉	#7868	1	1	yes	characteristic, complemented, Sylow
129	C₂×#117	Dic₈₉	1	2	yes	—
130	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
131	#432	—	89	4	no	—
132	#432	—	178	2	no	—
133	#431	—	89	4	no	—
134	#431	—	178	2	no	—
135	#423	—	89	4	no	—
136	#407	—	178	2	no	—
137	C₄×M₁₆	—	178	2	no	—
138	C₄×C₄⋊C₄	—	2	2	no	—
139	C₄×C₂²⋊C₄	—	2	2	no	—
140	#376	Dic₈₉	1	2	yes	—
141	C₄³	C₂×D₈₉	1	2	yes	—
142	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
143	C₂³×C₄	C₂²⋊D₈₉	1	2	yes	—
144	C₂×M₁₆	—	178	4	no	—
145	#117	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
150	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
151	C₂×C₄²	C₄×D₈₉	1	2	yes	—
152	C₂×C₄²	C₄×D₈₉	1	2	yes	—
153	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
154	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
156	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
158	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
159	C₄⋊C₈	—	89	8	no	—
160	C₄⋊C₈	—	178	4	no	—
161	C₂²⋊C₈	—	178	4	no	—
162	C₂²⋊C₈	—	178	4	no	—
163	#97	—	178	4	no	—
164	C₄×C₈	—	178	4	no	—
165	#95	—	2	4	no	—
166	C₂⁴	C₂²⋊Dic₈₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
171	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
172	C₂²×C₄	#13083551	1	2	yes	—
173	C₂²×C₄	#13083551	1	2	yes	—
174	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
176	C₂²×C₄	#13083553	1	2	yes	—
177	C₂²×C₄	#13083553	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
182	M₁₆	—	356	8	no	—
183	C₂×C₈	—	178	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	C₄×Dic₈₉	1	2	yes	—
189	C₄²	#13083551	1	2	yes	—
190	C₄²	#13083551	1	2	yes	—
191	C₄²	#13083553	1	2	yes	—
192	C₄²	#13083553	1	2	yes	—
193	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
194	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
195	C₄²	#13083558	1	8	yes	complemented
196	C₄²	C₂×C₄×D₈₉	1	2	yes	—
197	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
198	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	#15667919	1	1	yes	characteristic
202	C₂³	#15667919	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	#15667919	1	2	yes	—
218	C₂×C₄	#15667919	1	2	yes	—
219	C₂×C₄	#15667919	1	1	yes	characteristic
220	C₂×C₄	#15667919	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
222	C₂×C₄	C₂×#13083551	1	2	yes	—
223	C₂×C₄	C₂×#13083553	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
225	C₂×C₄	C₂×#13083558	1	4	yes	—
226	C₈	—	356	8	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×#15667919	1	1	yes	characteristic
234	C₂²	#19151562	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	C₄×#13083558	1	2	yes	—
242	C₄	#19151616	1	4	yes	—
243	C₄	#19151622	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	#26019946	1	1	yes	characteristic
248	C₂	#26019962	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#117×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#117	D₈₉
hypocenter	C₈₉	#7868
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #26018085	C₂	semidirect
5. #19151655	C₄	semidirect
8. #19151622	C₄	semidirect
70. C₄²×C₈₉	M₁₆	semidirect
128. C₈₉	#7868	semidirect
195. C₄²	#13083558	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26018085	2
C₄×#117×C₈₉	2
#7868	89