Subgroups — #36562890

Order 22784.

1062 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151978	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151706	C₂	1	4	yes	maximal
3	C₄×#15668012	C₂	1	2	yes	maximal
4	#26021123	C₂	1	4	yes	maximal
5	#26021054	C₂	1	2	yes	maximal
6	#26020409	C₂	1	1	yes	maximal, characteristic
7	C₄×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₅₆	C₂²	1	4	yes	—
9	C₂×#15668012	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₇₈	C₂²	1	4	yes	—
11	#19151978	C₄	1	8	yes	complemented
12	#19151706	—	2	16	no	—
13	C₂×#15667944	C₂²	1	4	yes	—
14	C₂×#15667937	C₂²	1	8	yes	—
15	#19151504	C₂²	1	4	yes	—
16	#19151496	—	2	4	no	—
17	#19151468	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₈₉	—	2	4	no	—
19	C₂×#97×C₈₉	C₂²	1	2	yes	—
20	C₂×#97×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄³×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₇₈	C₂³	1	4	yes	—
25	C₂×Dic₃₅₆	—	2	8	no	—
26	C₂×Dic₃₅₆	—	2	8	no	—
27	C₂×Dic₃₅₆	C₂×C₄	1	16	yes	—
28	#15668012	—	2	4	no	—
29	#15668012	C₂×C₄	1	8	yes	—
30	C₄×Dic₁₇₈	—	2	16	no	—
31	C₂×C₄⋊Dic₈₉	—	2	4	no	—
32	C₂×C₄⋊Dic₈₉	—	2	4	no	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
34	C₂×#13083553	—	2	4	no	—
35	C₂×C₄×Dic₈₉	—	2	4	no	—
36	#15667944	—	2	8	no	—
37	#15667937	—	2	16	no	—
38	#15667919	—	2	8	no	—
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
41	C₂×C₄²×C₈₉	D₄	1	4	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	#97×C₈₉	—	2	4	no	—
46	#97×C₈₉	D₄	1	8	yes	—
47	C₄×C₈×C₈₉	—	2	2	no	—
48	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₂×Dic₁₇₈	—	2	8	no	—
50	C₂×Dic₁₇₈	—	2	8	no	—
51	C₂×Dic₁₇₈	—	2	4	no	—
52	C₂×Dic₁₇₈	C₂²×C₄	1	8	yes	—
53	C₂²×Dic₈₉	—	2	4	no	—
54	C₂²×Dic₈₉	—	2	4	no	—
55	Dic₃₅₆	—	2	32	no	—
56	C₄⋊Dic₈₉	—	2	16	no	—
57	C₄⋊Dic₈₉	—	2	8	no	—
58	C₄⋊Dic₈₉	—	4	8	no	—
59	#13083553	—	4	8	no	—
60	C₄×Dic₈₉	—	4	8	no	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	2	no	—
66	C₂×C₈×C₈₉	—	2	2	no	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₈₉	#41	1	4	yes	—
71	C₄²×C₈₉	—	2	8	no	—
72	C₄²×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
74	C₄²×C₈₉	C₂×D₄	1	4	yes	—
75	C₄²×C₈₉	#41	1	2	yes	—
76	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	4	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	4	no	—
80	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₈₉	#127	1	4	yes	—
82	C₂×C₄×C₈₉	#137	1	8	yes	—
83	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
87	C₂×Dic₈₉	—	2	8	no	—
88	C₂×Dic₈₉	—	4	4	no	—
89	C₂×Dic₈₉	—	4	4	no	—
90	C₂×Dic₈₉	—	4	4	no	—
91	C₂×Dic₈₉	—	4	8	no	—
92	Dic₁₇₈	—	2	16	no	—
93	Dic₁₇₈	—	4	8	no	—
94	C₈×C₈₉	—	2	4	no	—
95	C₈×C₈₉	C₄×D₄	1	8	yes	—
96	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
98	C₂²×C₈₉	#532	1	1	yes	characteristic
99	C₂²×C₈₉	C₂×#137	1	4	yes	—
100	C₄×C₈₉	—	2	4	no	—
101	C₄×C₈₉	—	4	2	no	—
102	C₄×C₈₉	#441	1	8	yes	complemented
103	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
104	C₄×C₈₉	#532	1	2	yes	—
105	Dic₈₉	—	4	8	no	—
106	Dic₈₉	—	8	4	no	—
107	#21691	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
109	C₂×C₈₉	C₂×#441	1	2	yes	—
110	C₂×C₈₉	#3152	1	2	yes	—
111	C₂×C₈₉	#3185	1	2	yes	—
112	C₂×#494	—	89	1	no	—
113	C₂×#441	—	89	4	no	—
114	C₄×C₄⋊Q₈	—	89	2	no	—
115	#1971	—	89	4	no	—
116	#1956	—	89	2	no	—
117	#1844	—	89	1	no	—
118	C₄×#97	D₈₉	1	1	yes	characteristic
119	C₈₉	#21691	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	89	4	no	—
121	C₂×C₄⋊Q₈	—	89	2	no	—
122	C₂×C₄×Q₈	—	89	4	no	—
123	#494	—	89	8	no	—
124	#441	—	178	16	no	—
125	C₂×#106	—	89	4	no	—
126	C₂×#103	—	89	8	no	—
127	C₂×#97	C₂×D₈₉	1	2	yes	—
128	C₂×#97	C₂×D₈₉	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
131	#391	—	89	4	no	—
132	#389	—	178	4	no	—
133	#384	—	89	2	no	—
134	C₄×C₄⋊C₄	—	178	4	no	—
135	C₄³	C₂×D₈₉	1	1	yes	characteristic
136	C₂²×Q₈	—	89	4	no	—
137	C₂×Q₁₆	—	89	16	no	—
138	C₂×Q₁₆	—	178	8	no	—
139	C₂×Q₁₆	—	178	8	no	—
140	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
141	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
142	C₄⋊Q₈	—	89	8	no	—
143	C₄⋊Q₈	—	178	4	no	—
144	C₄×Q₈	—	178	16	no	—
145	C₂×C₄⋊C₄	—	89	4	no	—
146	C₂×C₄⋊C₄	—	178	4	no	—
147	C₂×C₄⋊C₄	—	178	4	no	—
148	C₂×C₄⋊C₄	—	178	4	no	—
149	C₂×C₄²	—	178	4	no	—
150	C₂×C₄²	D₃₅₆	1	4	yes	—
151	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
154	#106	—	178	8	no	—
155	#103	—	178	16	no	—
156	#97	—	2	4	no	—
157	#97	D₃₅₆	1	8	yes	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	C₄×D₈₉	1	4	yes	—
160	#95	—	178	8	no	—
161	C₂×Q₈	—	89	8	no	—
162	C₂×Q₈	—	178	8	no	—
163	C₂×Q₈	—	178	8	no	—
164	C₂×Q₈	—	178	4	no	—
165	C₂²×C₄	—	178	4	no	—
166	C₂²×C₄	—	178	4	no	—
167	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
168	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
169	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
170	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
171	Q₁₆	—	178	32	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
175	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
176	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
177	C₂×C₈	#13083574	1	4	yes	—
178	C₄⋊C₄	—	356	8	no	—
179	C₄⋊C₄	—	356	8	no	—
180	C₄⋊C₄	—	178	8	no	—
181	C₄⋊C₄	—	178	16	no	—
182	C₄²	—	2	8	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	356	8	no	—
185	C₄²	C₂×C₄×D₈₉	1	2	yes	—
186	C₄²	C₂×D₃₅₆	1	4	yes	—
187	C₄²	#13083574	1	2	yes	—
188	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
189	Q₈	—	356	8	no	—
190	Q₈	—	178	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	356	4	no	—
195	C₂×C₄	—	356	4	no	—
196	C₂×C₄	—	356	8	no	—
197	C₂×C₄	—	356	4	no	—
198	C₂×C₄	—	178	8	no	—
199	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
200	C₂×C₄	#15668009	1	4	yes	—
201	C₂×C₄	#15668045	1	8	yes	—
202	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
204	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
205	C₂×C₄	C₂×#13083574	1	2	yes	—
206	C₈	—	2	4	no	—
207	C₈	C₄×D₃₅₆	1	8	yes	—
208	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
209	C₂²	C₂×#15668009	1	1	yes	characteristic
210	C₂²	#19152148	1	1	yes	characteristic
211	C₂²	C₂×#15668045	1	4	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	4	2	no	—
214	C₄	—	356	8	no	—
215	C₄	—	712	4	no	—
216	C₄	#19151706	1	8	yes	complemented
217	C₄	C₂×C₄×D₃₅₆	1	2	yes	—
218	C₄	#19152148	1	2	yes	—
219	C₂	C₄×#15668009	1	1	yes	characteristic
220	C₂	C₂×#19151706	1	2	yes	—
221	C₂	#26030711	1	2	yes	—
222	C₂	#26030997	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#97	D₈₉
hypocenter	C₈₉	#21691
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #19151978	C₄	semidirect
102. C₄×C₈₉	#441	semidirect
119. C₈₉	#21691	semidirect
216. C₄	#19151706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151978	2
C₂×#19151706	2
C₄×#15668012	2
#26021123	2
#26021054	2
#26020409	2
C₄×#97×C₈₉	2
#21691	89