Subgroups — #32985743

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
2	#22610892	C₂	1	2	yes	maximal
3	#22610733	C₂	1	2	yes	maximal
4	#22610091	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	2	yes	—
7	#17988992	—	2	2	no	—
8	#17988992	C₄	1	4	yes	complemented
9	C₂×#14556525	—	2	2	no	—
10	C₂×#14556525	—	2	2	no	—
11	C₂×#14556516	—	2	4	no	—
12	#17988457	C₂²	1	2	yes	—
13	#17988350	—	2	4	no	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₈²×C₇₃	—	2	2	no	—
18	C₂²×Dic₁₄₆	—	2	2	no	—
19	#14556591	—	2	2	no	—
20	#14556591	—	2	2	no	—
21	#14556591	—	2	2	no	—
22	#14556591	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₇₃	—	2	2	no	—
24	C₂×C₄⋊Dic₇₃	—	2	2	no	—
25	C₂×C₄⋊Dic₇₃	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	2	no	—
28	#14556525	—	2	8	no	—
29	#14556525	—	4	4	no	—
30	#14556516	—	4	8	no	—
31	C₂²×C₈×C₇₃	D₄	1	2	yes	—
32	C₂²×C₈×C₇₃	D₄	1	2	yes	—
33	C₂²×C₈×C₇₃	D₄	1	2	yes	—
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₇₃	—	2	1	no	—
36	C₄×C₈×C₇₃	—	2	1	no	—
37	C₄×C₈×C₇₃	—	2	1	no	—
38	C₄×C₈×C₇₃	—	2	1	no	—
39	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₃	D₄	1	2	yes	—
41	C₂×Dic₁₄₆	—	2	4	no	—
42	C₂×Dic₁₄₆	—	4	4	no	—
43	C₂×Dic₁₄₆	—	4	4	no	—
44	C₂×Dic₁₄₆	—	4	2	no	—
45	C₂²×Dic₇₃	—	4	2	no	—
46	C₂²×Dic₇₃	—	4	2	no	—
47	C₄⋊Dic₇₃	—	2	4	no	—
48	C₄⋊Dic₇₃	—	2	4	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₄⋊Dic₇₃	—	4	4	no	—
52	C₄⋊Dic₇₃	—	4	4	no	—
53	C₄⋊Dic₇₃	—	4	4	no	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	—	2	2	no	—
61	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₇₃	D₈	1	4	yes	—
63	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
64	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
65	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
66	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₄²×C₇₃	#41	1	1	yes	characteristic
70	C₂³×C₇₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₇₃	#135	1	2	yes	—
82	C₂×C₄×C₇₃	#135	1	2	yes	—
83	C₂×Dic₇₃	—	4	4	no	—
84	C₂×Dic₇₃	—	8	2	no	—
85	C₂×Dic₇₃	—	8	2	no	—
86	C₂×Dic₇₃	—	8	2	no	—
87	C₂×Dic₇₃	—	8	2	no	—
88	C₂×Dic₇₃	—	8	2	no	—
89	Dic₁₄₆	—	4	8	no	—
90	Dic₁₄₆	—	8	4	no	—
91	C₈×C₇₃	—	2	4	no	—
92	C₈×C₇₃	—	2	4	no	—
93	C₈×C₇₃	#103	1	8	yes	complemented
94	C₂²×C₇₃	#390	1	1	yes	characteristic
95	C₂²×C₇₃	#492	1	1	yes	characteristic
96	C₂²×C₇₃	#493	1	1	yes	characteristic
97	C₂²×C₇₃	#494	1	1	yes	characteristic
98	C₂²×C₇₃	#494	1	1	yes	characteristic
99	C₂²×C₇₃	#495	1	1	yes	characteristic
100	C₂²×C₇₃	#495	1	1	yes	characteristic
101	C₄×C₇₃	C₂×#103	1	2	yes	—
102	C₄×C₇₃	C₄×D₈	1	2	yes	—
103	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
104	C₄×C₇₃	#466	1	2	yes	—
105	C₄×C₇₃	#467	1	2	yes	—
106	C₄×C₇₃	#487	1	2	yes	—
107	Dic₇₃	—	8	4	no	—
108	Dic₇₃	—	16	2	no	—
109	#9823	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	#1706	1	1	yes	characteristic
111	C₂×C₇₃	#1707	1	1	yes	characteristic
112	C₂×C₇₃	#1710	1	1	yes	characteristic
113	C₂×C₇₃	#1713	1	1	yes	characteristic
114	C₂×C₇₃	#1956	1	1	yes	characteristic
115	C₂×C₇₃	#1969	1	1	yes	characteristic
116	C₂×C₇₃	#1970	1	1	yes	characteristic
117	C₂×#500	—	73	1	no	—
118	#1990	—	73	2	no	—
119	#1956	—	73	2	no	—
120	#1845	—	73	1	no	—
121	C₂×C₈²	D₇₃	1	1	yes	characteristic
122	C₇₃	#9823	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄⋊Q₈	—	73	2	no	—
124	#500	—	73	4	no	—
125	#500	—	146	2	no	—
126	C₂×#107	—	146	2	no	—
127	C₂×#107	—	146	2	no	—
128	C₂×#103	—	146	4	no	—
129	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
132	#384	—	73	2	no	—
133	#340	—	146	4	no	—
134	C₈²	—	2	2	no	—
135	C₂²×Q₈	—	146	2	no	—
136	C₂²×C₈	D₂₉₂	1	2	yes	—
137	C₂²×C₈	D₂₉₂	1	2	yes	—
138	C₂²×C₈	D₂₉₂	1	2	yes	—
139	C₄⋊Q₈	—	73	4	no	—
140	C₄⋊Q₈	—	146	2	no	—
141	C₄⋊Q₈	—	146	2	no	—
142	C₄⋊Q₈	—	146	2	no	—
143	C₂×C₄⋊C₄	—	146	2	no	—
144	C₂×C₄⋊C₄	—	146	2	no	—
145	C₂×C₄⋊C₄	—	146	2	no	—
146	C₂×C₄⋊C₄	—	146	2	no	—
147	C₂×C₄⋊C₄	—	146	2	no	—
148	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
149	#107	—	292	4	no	—
150	#107	—	146	8	no	—
151	#103	—	292	8	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	C₄×D₇₃	1	2	yes	—
157	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
158	C₂×Q₈	—	292	2	no	—
159	C₂×Q₈	—	292	4	no	—
160	C₂×Q₈	—	292	4	no	—
161	C₂×Q₈	—	146	4	no	—
162	C₂²×C₄	—	292	2	no	—
163	C₂²×C₄	—	292	2	no	—
164	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	#11924005	1	4	yes	—
172	C₂×C₈	D₅₈₄	1	4	yes	—
173	C₂×C₈	#11924017	1	4	yes	—
174	C₂×C₈	Dic₂₉₂	1	4	yes	—
175	C₂×C₈	Dic₂₉₂	1	4	yes	—
176	C₄⋊C₄	—	292	2	no	—
177	C₄⋊C₄	—	292	2	no	—
178	C₄⋊C₄	—	292	4	no	—
179	C₄⋊C₄	—	292	4	no	—
180	C₄⋊C₄	—	292	4	no	—
181	C₄⋊C₄	—	146	4	no	—
182	C₄⋊C₄	—	146	4	no	—
183	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
185	C₄²	#11924028	1	1	yes	characteristic
186	C₄²	#11924028	1	1	yes	characteristic
187	C₂³	#14556588	1	1	yes	characteristic
188	Q₈	—	292	8	no	—
189	Q₈	—	584	4	no	—
190	C₂×C₄	—	292	4	no	—
191	C₂×C₄	—	584	2	no	—
192	C₂×C₄	—	584	2	no	—
193	C₂×C₄	—	584	2	no	—
194	C₂×C₄	—	584	2	no	—
195	C₂×C₄	—	584	2	no	—
196	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
197	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
198	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
199	C₂×C₄	#14556565	1	1	yes	characteristic
200	C₂×C₄	#14556565	1	1	yes	characteristic
201	C₂×C₄	#14556579	1	1	yes	characteristic
202	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
203	C₂×C₄	C₂×#11924017	1	2	yes	—
204	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
205	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
206	C₂×C₄	#14556613	1	2	yes	—
207	C₂×C₄	#14556613	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	2	4	no	—
210	C₈	#14556516	1	8	yes	complemented
211	C₂²	#17988489	1	1	yes	characteristic
212	C₂²	#17988960	1	1	yes	characteristic
213	C₂²	#17988964	1	1	yes	characteristic
214	C₂²	#17988967	1	1	yes	characteristic
215	C₂²	#17988967	1	1	yes	characteristic
216	C₂²	#17988971	1	1	yes	characteristic
217	C₂²	#17988971	1	1	yes	characteristic
218	C₄	—	584	4	no	—
219	C₄	—	1168	2	no	—
220	C₄	C₂×#14556516	1	2	yes	—
221	C₄	C₄×D₅₈₄	1	2	yes	—
222	C₄	C₄×Dic₂₉₂	1	2	yes	—
223	C₄	#17988823	1	2	yes	—
224	C₄	#17988828	1	2	yes	—
225	C₄	#17988938	1	2	yes	—
226	C₂	#22609523	1	1	yes	characteristic
227	C₂	#22609528	1	1	yes	characteristic
228	C₂	#22609541	1	1	yes	characteristic
229	C₂	#22609549	1	1	yes	characteristic
230	C₂	#22610733	1	1	yes	characteristic
231	C₂	#22610792	1	1	yes	characteristic
232	C₂	#22610797	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556588
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×C₈²×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#127
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	#9823
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
8. #17988992	C₄	semidirect
93. C₈×C₇₃	#103	semidirect
122. C₇₃	#9823	semidirect
210. C₈	#14556516	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988992	2
#22610892	2
#22610733	2
#22610091	2
C₂×C₈²×C₇₃	2
#9823	73