Subgroups — #36760023

Order 22784.

2162 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152327	C₂	1	2	yes	maximal
2	#26028650	C₂	1	3	yes	maximal
3	#26025283	C₂	1	12	yes	maximal
4	#26025240	C₂	1	6	yes	maximal
5	Q₈×C₄⋊Dic₈₉	C₂	1	4	yes	maximal
6	C₄×#15668012	C₂	1	3	yes	maximal
7	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19152327	—	2	16	no	—
9	C₂×#15668012	C₂²	1	6	yes	—
10	C₂×#15668012	C₂²	1	3	yes	—
11	C₂×#15668012	C₂²	1	6	yes	—
12	C₂×#15668003	C₂²	1	12	yes	—
13	C₂×Q₈×Dic₈₉	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₇₈	C₂²	1	6	yes	—
15	#19151530	C₂²	1	24	yes	—
16	#19151504	C₂²	1	6	yes	—
17	#19151497	C₂²	1	24	yes	—
18	#19151496	C₂²	1	12	yes	—
19	#19151468	C₂²	1	12	yes	—
20	#19151468	C₂²	1	12	yes	—
21	#19151468	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊Dic₈₉	C₂²	1	12	yes	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	6	yes	—
27	C₄³×C₈₉	C₂²	1	3	yes	—
28	C₂²×Dic₁₇₈	C₂³	1	6	yes	—
29	#15668012	—	2	24	no	—
30	#15668012	—	2	12	no	—
31	#15668012	—	4	12	no	—
32	#15668003	—	2	48	no	—
33	Q₈×Dic₈₉	—	2	16	no	—
34	C₄×Dic₁₇₈	—	2	24	no	—
35	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
36	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
37	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
38	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
41	C₂×#13083553	C₂³	1	12	yes	—
42	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
43	#15667919	C₂³	1	24	yes	—
44	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₈₉	D₄	1	8	yes	—
46	C₄×Q₈×C₈₉	Q₈	1	16	yes	complemented
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
48	C₂×C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
49	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
50	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	#95×C₈₉	C₂³	1	6	yes	—
55	C₂×Dic₁₇₈	—	2	12	no	—
56	C₂×Dic₁₇₈	—	2	12	no	—
57	C₂×Dic₁₇₈	—	2	12	no	—
58	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
59	C₂²×Dic₈₉	C₂⁴	1	4	yes	—
60	C₄⋊Dic₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	—	2	24	no	—
62	C₄⋊Dic₈₉	—	2	24	no	—
63	C₄⋊Dic₈₉	—	2	24	no	—
64	C₄⋊Dic₈₉	—	2	8	no	—
65	C₄⋊Dic₈₉	—	4	24	no	—
66	#13083553	—	2	24	no	—
67	C₄×Dic₈₉	—	2	24	no	—
68	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₈₉	C₂×Q₈	1	8	yes	—
70	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
71	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
72	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
73	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
75	C₄⋊C₄×C₈₉	C₂×D₄	1	12	yes	—
76	C₄⋊C₄×C₈₉	C₂×Q₈	1	24	yes	—
77	C₄²×C₈₉	—	2	6	no	—
78	C₄²×C₈₉	—	2	12	no	—
79	C₄²×C₈₉	C₂×D₄	1	12	yes	—
80	C₄²×C₈₉	C₂×Q₈	1	24	yes	—
81	C₄²×C₈₉	#41	1	4	yes	—
82	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
83	Q₈×C₈₉	C₄⋊Q₈	1	16	yes	complemented
84	C₂×C₄×C₈₉	—	2	6	no	—
85	C₂×C₄×C₈₉	—	2	6	no	—
86	C₂×C₄×C₈₉	—	2	6	no	—
87	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
89	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
91	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
92	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
93	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
95	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
96	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
97	C₂×Dic₈₉	—	2	4	no	—
98	C₂×Dic₈₉	—	2	12	no	—
99	C₂×Dic₈₉	—	2	12	no	—
100	C₂×Dic₈₉	—	2	12	no	—
101	C₂×Dic₈₉	—	2	8	no	—
102	Dic₁₇₈	—	4	24	no	—
103	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
104	C₂²×C₈₉	C₂³×Q₈	1	2	yes	—
105	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
106	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
108	C₄×C₈₉	—	2	6	no	—
109	C₄×C₈₉	—	2	12	no	—
110	C₄×C₈₉	C₂×C₄⋊Q₈	1	12	yes	—
111	C₄×C₈₉	#550	1	4	yes	—
112	C₄×C₈₉	#557	1	8	yes	complemented
113	Dic₈₉	—	4	12	no	—
114	Dic₈₉	—	4	4	no	—
115	#36593	—	89	1	no	maximal, Sylow
116	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₈₉	C₂×#550	1	1	yes	characteristic
118	C₂×C₈₉	C₂×#557	1	2	yes	—
119	C₂×C₈₉	#3499	1	1	yes	characteristic
120	C₂×C₈₉	#3503	1	2	yes	—
121	C₂×#557	—	89	2	no	—
122	#2868	—	89	3	no	—
123	#2552	—	89	12	no	—
124	#2546	—	89	6	no	—
125	Q₈×C₄⋊C₄	—	89	4	no	—
126	C₄×C₄⋊Q₈	—	89	3	no	—
127	C₄²×Q₈	D₈₉	1	1	yes	characteristic
128	C₈₉	#36593	1	1	yes	characteristic, complemented, Sylow
129	#557	—	178	16	no	—
130	C₂×C₄⋊Q₈	—	89	6	no	—
131	C₂×C₄⋊Q₈	—	89	3	no	—
132	C₂×C₄⋊Q₈	—	89	6	no	—
133	C₂×#125	—	89	12	no	—
134	C₂×C₄×Q₈	—	89	4	no	—
135	C₂×C₄×Q₈	—	89	6	no	—
136	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
137	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
138	#398	—	89	24	no	—
139	#391	—	89	6	no	—
140	#389	—	89	24	no	—
141	#389	—	89	12	no	—
142	#384	—	89	1	no	—
143	#384	—	89	12	no	—
144	#384	—	89	12	no	—
145	C₄×C₄⋊C₄	—	89	12	no	—
146	C₄×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
147	C₄×C₄⋊C₄	C₂×D₈₉	1	6	yes	—
148	C₄³	C₂×D₈₉	1	3	yes	—
149	C₂²×Q₈	—	89	6	no	—
150	C₂²×Q₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₄⋊Q₈	—	356	12	no	—
152	C₄⋊Q₈	—	178	24	no	—
153	C₄⋊Q₈	—	178	12	no	—
154	#125	—	178	48	no	—
155	C₄×Q₈	—	178	16	no	—
156	C₄×Q₈	—	178	24	no	—
157	C₄×Q₈	Dic₁₇₈	1	16	yes	complemented
158	C₄×Q₈	D₃₅₆	1	8	yes	—
159	C₂×C₄⋊C₄	—	89	24	no	—
160	C₂×C₄⋊C₄	—	89	2	no	—
161	C₂×C₄⋊C₄	—	89	12	no	—
162	C₂×C₄⋊C₄	—	89	12	no	—
163	C₂×C₄⋊C₄	—	89	12	no	—
164	C₂×C₄⋊C₄	—	89	12	no	—
165	C₂×C₄⋊C₄	—	89	4	no	—
166	C₂×C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
167	C₂×C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
168	C₂×C₄²	—	89	12	no	—
169	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
170	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
171	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
172	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
173	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
174	#95	—	89	24	no	—
175	#95	C₂²×D₈₉	1	6	yes	—
176	C₂×Q₈	—	178	12	no	—
177	C₂×Q₈	—	178	12	no	—
178	C₂×Q₈	—	178	12	no	—
179	C₂×Q₈	C₂×D₃₅₆	1	4	yes	—
180	C₂×Q₈	C₂×Dic₁₇₈	1	8	yes	—
181	C₂²×C₄	—	89	4	no	—
182	C₂²×C₄	—	89	12	no	—
183	C₂²×C₄	C₂³×D₈₉	1	3	yes	—
184	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
185	C₂²×C₄	C₂³×D₈₉	1	3	yes	—
186	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
187	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
188	C₄⋊C₄	—	356	24	no	—
189	C₄⋊C₄	—	178	4	no	—
190	C₄⋊C₄	—	178	24	no	—
191	C₄⋊C₄	—	178	24	no	—
192	C₄⋊C₄	—	178	24	no	—
193	C₄⋊C₄	—	178	24	no	—
194	C₄⋊C₄	—	178	8	no	—
195	C₄⋊C₄	C₂×D₃₅₆	1	12	yes	—
196	C₄⋊C₄	C₂×Dic₁₇₈	1	24	yes	—
197	C₄²	—	2	6	no	—
198	C₄²	—	2	12	no	—
199	C₄²	—	178	24	no	—
200	C₄²	C₂×D₃₅₆	1	12	yes	—
201	C₄²	C₂×Dic₁₇₈	1	24	yes	—
202	C₄²	#13083575	1	4	yes	—
203	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
204	Q₈	—	356	24	no	—
205	Q₈	#15668012	1	16	yes	complemented
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	6	no	—
209	C₂×C₄	—	178	4	no	—
210	C₂×C₄	—	178	12	no	—
211	C₂×C₄	—	178	12	no	—
212	C₂×C₄	—	178	12	no	—
213	C₂×C₄	—	178	8	no	—
214	C₂×C₄	C₂²×D₃₅₆	1	6	yes	—
215	C₂×C₄	C₂²×D₃₅₆	1	6	yes	—
216	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
217	C₂×C₄	C₂²×Dic₁₇₈	1	12	yes	—
218	C₂×C₄	C₂²×Dic₁₇₈	1	12	yes	—
219	C₂×C₄	C₂²×Dic₁₇₈	1	4	yes	—
220	C₂×C₄	C₂×#13083575	1	2	yes	—
221	C₂×C₄	C₂×#13083575	1	4	yes	—
222	C₂×C₄	#15668058	1	2	yes	—
223	C₂×C₄	#15668059	1	4	yes	—
224	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
225	C₂²	C₂³×Dic₁₇₈	1	2	yes	—
226	C₂²	C₂²×#13083575	1	1	yes	characteristic
227	C₂²	C₂×#15668058	1	1	yes	characteristic
228	C₂²	C₂×#15668059	1	2	yes	—
229	C₄	—	2	6	no	—
230	C₄	—	2	12	no	—
231	C₄	—	356	12	no	—
232	C₄	—	356	4	no	—
233	C₄	C₂×#15668012	1	12	yes	—
234	C₄	#19152270	1	4	yes	—
235	C₄	#19152327	1	8	yes	complemented
236	C₂	C₂²×#15668012	1	1	yes	characteristic
237	C₂	C₂×#19152270	1	1	yes	characteristic
238	C₂	C₂×#19152327	1	2	yes	—
239	C₂	#26034037	1	1	yes	characteristic
240	C₂	#26034071	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁵
Fitting	C₄²×Q₈×C₈₉	C₂
hypercenter	C₄²×Q₈	D₈₉
hypocenter	C₈₉	#36593
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
46. C₄×Q₈×C₈₉	Q₈	semidirect
83. Q₈×C₈₉	C₄⋊Q₈	semidirect
112. C₄×C₈₉	#557	semidirect
128. C₈₉	#36593	semidirect
157. C₄×Q₈	Dic₁₇₈	semidirect
205. Q₈	#15668012	semidirect
235. C₄	#19152327	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152327	2
#26028650	2
#26025283	2
#26025240	2
Q₈×C₄⋊Dic₈₉	2
C₄×#15668012	2
C₄²×Q₈×C₈₉	2
#36593	89