Subgroups — #32973170

Order 18688.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610091	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556525	—	2	1	no	—
5	C₂×#14556525	C₄	1	2	yes	—
6	C₂×C₄⋊#10596831	—	2	1	no	—
7	#17988457	—	2	1	no	—
8	#17988340	—	2	1	no	—
9	#17988340	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₃	—	2	1	no	—
13	C₄⋊C₁₆×C₇₃	C₄	1	2	yes	complemented
14	C₂²×#10596831	—	4	1	no	—
15	C₂×C₄⋊Dic₇₃	—	2	2	no	—
16	C₂×C₄⋊Dic₇₃	—	4	1	no	—
17	C₂×C₄⋊Dic₇₃	—	4	1	no	—
18	#14556525	—	4	2	no	—
19	#14556525	—	4	1	no	—
20	#14556525	—	4	1	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	—	4	1	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	—	2	2	no	—
27	C₂×C₁₆×C₇₃	—	4	1	no	—
28	C₂×C₁₆×C₇₃	—	4	1	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₃	—	4	2	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₂×#10596831	—	8	1	no	—
36	C₂×#10596831	—	8	1	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₄⋊Dic₇₃	—	8	2	no	—
40	C₄⋊Dic₇₃	—	8	1	no	—
41	C₄⋊Dic₇₃	—	8	1	no	—
42	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
55	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
59	C₁₆×C₇₃	—	4	2	no	—
60	C₂³×C₇₃	#102	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	1	no	—
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
73	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
76	C₂×C₄×C₇₃	Q₃₂	1	1	yes	characteristic
77	C₂×Dic₇₃	—	8	2	no	—
78	C₂×Dic₇₃	—	8	2	no	—
79	C₂×Dic₇₃	—	8	2	no	—
80	C₈×C₇₃	—	2	1	no	—
81	C₈×C₇₃	—	2	1	no	—
82	C₈×C₇₃	#102	1	1	yes	characteristic
83	C₈×C₇₃	#103	1	1	yes	characteristic
84	C₈×C₇₃	#106	1	1	yes	characteristic
85	C₈×C₇₃	#107	1	1	yes	characteristic
86	#10596831	—	8	2	no	—
87	C₂²×C₇₃	#331	1	1	yes	characteristic
88	C₂²×C₇₃	#332	1	1	yes	characteristic
89	C₂²×C₇₃	#340	1	1	yes	characteristic
90	C₂²×C₇₃	#357	1	1	yes	characteristic
91	C₂²×C₇₃	#358	1	1	yes	characteristic
92	C₂²×C₇₃	#361	1	1	yes	characteristic
93	C₂²×C₇₃	#362	1	1	yes	characteristic
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	—	2	1	no	—
96	C₄×C₇₃	#332	1	1	yes	characteristic
97	C₄×C₇₃	#333	1	1	yes	characteristic
98	C₄×C₇₃	#340	1	1	yes	characteristic
99	C₄×C₇₃	#343	1	1	yes	characteristic
100	C₄×C₇₃	#357	1	1	yes	characteristic
101	C₄×C₇₃	#358	1	1	yes	characteristic
102	C₄×C₇₃	#366	1	1	yes	characteristic
103	C₄×C₇₃	#367	1	1	yes	characteristic
104	Dic₇₃	—	16	2	no	—
105	#7274	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	#1296	1	1	yes	characteristic
107	C₂×C₇₃	#1361	1	1	yes	characteristic
108	C₂×C₇₃	#1362	1	1	yes	characteristic
109	C₂×C₇₃	#1364	1	1	yes	characteristic
110	C₂×C₇₃	#1365	1	1	yes	characteristic
111	C₂×C₇₃	#1380	1	1	yes	characteristic
112	C₂×C₇₃	#1387	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	D₇₃	1	1	yes	characteristic
114	#1845	—	73	1	no	—
115	C₂×#335	—	73	1	no	—
116	C₇₃	#7274	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₁₆	—	2	1	no	—
118	C₂×#107	—	73	2	no	—
119	C₂×#107	—	146	1	no	—
120	C₂×C₄⋊C₈	—	146	1	no	—
121	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
122	#384	—	146	1	no	—
123	C₄⋊C₁₆	—	2	1	no	—
124	C₄⋊C₁₆	Dic₇₃	1	2	yes	complemented
125	#335	—	73	2	no	—
126	#335	—	146	1	no	—
127	C₂²×C₈	—	292	1	no	—
128	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	292	1	no	—
131	C₂×C₄⋊C₄	—	292	1	no	—
132	C₂×C₄⋊C₄	—	146	2	no	—
133	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	4	1	no	—
136	C₂×C₁₆	—	4	1	no	—
137	#107	—	292	1	no	—
138	#107	—	292	1	no	—
139	#107	—	292	2	no	—
140	C₄⋊C₈	—	292	1	no	—
141	C₄⋊C₈	—	146	2	no	—
142	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
143	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
146	C₂²×C₄	—	292	2	no	—
147	C₂²×C₄	#11924005	1	1	yes	characteristic
148	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
149	C₂²×C₄	#11924017	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	292	2	no	—
153	C₂×C₈	—	584	1	no	—
154	C₂×C₈	—	584	1	no	—
155	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
157	C₂×C₈	#11924007	1	1	yes	characteristic
158	C₂×C₈	#11924007	1	1	yes	characteristic
159	C₂×C₈	#11924013	1	1	yes	characteristic
160	C₂×C₈	#11924015	1	1	yes	characteristic
161	C₂×C₈	#11924016	1	1	yes	characteristic
162	C₂×C₈	#11924018	1	1	yes	characteristic
163	C₄⋊C₄	—	292	2	no	—
164	C₄⋊C₄	—	292	2	no	—
165	C₄⋊C₄	—	584	2	no	—
166	C₄⋊C₄	—	584	1	no	—
167	C₄⋊C₄	—	584	1	no	—
168	C₄²	#11924005	1	1	yes	characteristic
169	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
170	C₄²	#11924017	1	1	yes	characteristic
171	C₄²	Dic₂₉₂	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₂³	#14556513	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	584	2	no	—
177	C₂×C₄	—	584	2	no	—
178	C₂×C₄	—	584	2	no	—
179	C₂×C₄	#14556498	1	1	yes	characteristic
180	C₂×C₄	#14556507	1	1	yes	characteristic
181	C₂×C₄	#14556509	1	1	yes	characteristic
182	C₂×C₄	#14556511	1	1	yes	characteristic
183	C₂×C₄	#14556513	1	1	yes	characteristic
184	C₂×C₄	#14556514	1	1	yes	characteristic
185	C₂×C₄	#14556516	1	1	yes	characteristic
186	C₂×C₄	#14556516	1	1	yes	characteristic
187	C₂×C₄	#14556523	1	1	yes	characteristic
188	C₂×C₄	#14556525	1	1	yes	characteristic
189	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
190	C₂×C₄	#14556536	1	1	yes	characteristic
191	C₂×C₄	#14556536	1	1	yes	characteristic
192	C₂×C₄	Dic₅₈₄	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	584	2	no	—
196	C₈	#14556512	1	1	yes	characteristic
197	C₈	#14556515	1	1	yes	characteristic
198	C₈	#14556522	1	1	yes	characteristic
199	C₈	#14556524	1	1	yes	characteristic
200	C₂²	#17988330	1	1	yes	characteristic
201	C₂²	#17988332	1	1	yes	characteristic
202	C₂²	#17988350	1	1	yes	characteristic
203	C₂²	#17988389	1	1	yes	characteristic
204	C₂²	#17988392	1	1	yes	characteristic
205	C₂²	#17988401	1	1	yes	characteristic
206	C₂²	#17988404	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	1168	2	no	—
210	C₄	#17988334	1	1	yes	characteristic
211	C₄	#17988335	1	1	yes	characteristic
212	C₄	#17988349	1	1	yes	characteristic
213	C₄	#17988355	1	1	yes	characteristic
214	C₄	#17988389	1	1	yes	characteristic
215	C₄	#17988392	1	1	yes	characteristic
216	C₄	#17988413	1	1	yes	characteristic
217	C₄	#17988415	1	1	yes	characteristic
218	C₂	#22607815	1	1	yes	characteristic
219	C₂	#22607978	1	1	yes	characteristic
220	C₂	#22607981	1	1	yes	characteristic
221	C₂	#22607985	1	1	yes	characteristic
222	C₂	#22607988	1	1	yes	characteristic
223	C₂	#22608023	1	1	yes	characteristic
224	C₂	#22608040	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556513
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₂×C₄⋊C₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₃
hypocenter	C₇₃	#7274
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. #17988340	C₄	semidirect
13. C₄⋊C₁₆×C₇₃	C₄	semidirect
116. C₇₃	#7274	semidirect
124. C₄⋊C₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22610091	2
C₂×#17988340	2
C₂×C₄⋊C₁₆×C₇₃	2
#7274	73