Subgroups — #32984546

Order 18688.

378 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	#22610649	C₂	1	2	yes	maximal
2	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
4	#22607771	C₂	1	2	yes	maximal
5	C₂×#322×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556525	C₄	1	2	yes	—
7	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
8	C₂×#14556520	—	2	2	no	—
9	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	#17988350	C₄	1	1	yes	characteristic
11	#17988350	C₄	1	1	yes	characteristic
12	#17988341	—	2	2	no	—
13	#17988340	—	2	2	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	#322×C₇₃	—	2	2	no	—
18	C₂²×#10596831	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₃	—	2	2	no	—
22	#14556525	—	2	4	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	#14556520	—	4	4	no	—
26	#14556498	—	2	2	no	—
27	C₂²×C₈×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₇₃	—	2	1	no	—
34	C₄×C₈×C₇₃	—	2	1	no	—
35	C₄×C₈×C₇₃	—	2	1	no	—
36	C₄×C₈×C₇₃	D₄	1	2	yes	—
37	C₄×C₈×C₇₃	Q₈	1	2	yes	—
38	C₄×C₈×C₇₃	Q₈	1	2	yes	—
39	C₂²×Dic₇₃	—	2	2	no	—
40	C₂²×Dic₇₃	—	2	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	—	4	2	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	M₁₆	1	4	yes	—
48	C₄×Dic₇₃	—	4	2	no	—
49	C₄×Dic₇₃	—	4	2	no	—
50	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	2	no	—
56	C₂×C₈×C₇₃	—	2	2	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	—	4	2	no	—
61	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₃	D₈	1	2	yes	—
64	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₇₃	#131	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
83	C₂×Dic₇₃	—	2	4	no	—
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	—	4	2	no	—
86	C₂×Dic₇₃	—	4	2	no	—
87	C₂×Dic₇₃	—	4	2	no	—
88	C₂×Dic₇₃	—	4	2	no	—
89	C₈×C₇₃	—	2	2	no	—
90	C₈×C₇₃	—	2	2	no	—
91	C₈×C₇₃	—	2	2	no	—
92	C₈×C₇₃	—	2	2	no	—
93	C₈×C₇₃	—	4	2	no	—
94	#10596831	—	8	2	no	—
95	C₂²×C₇₃	#389	1	1	yes	characteristic
96	C₂²×C₇₃	#435	1	2	yes	—
97	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
99	C₂²×C₇₃	#444	1	2	yes	—
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	#423	1	1	yes	characteristic
103	C₄×C₇₃	#424	1	1	yes	characteristic
104	C₄×C₇₃	#446	1	2	yes	—
105	C₄×C₇₃	#466	1	1	yes	characteristic
106	C₄×C₇₃	#467	1	1	yes	characteristic
107	C₄×C₇₃	#499	1	1	yes	characteristic
108	C₄×C₇₃	#500	1	1	yes	characteristic
109	Dic₇₃	—	4	4	no	—
110	Dic₇₃	—	8	2	no	—
111	#9598	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	#1581	1	2	yes	—
113	C₂×C₇₃	#1584	1	2	yes	—
114	C₂×C₇₃	#1940	1	1	yes	characteristic
115	C₂×C₇₃	#1942	1	1	yes	characteristic
116	C₂×C₇₃	#1946	1	1	yes	characteristic
117	#1939	—	73	2	no	—
118	C₄×#107	—	73	1	no	—
119	C₂×#335	—	73	1	no	—
120	C₂×#322	D₇₃	1	1	yes	characteristic
121	#1279	—	73	2	no	—
122	C₇₃	#9598	1	1	yes	characteristic, complemented, Sylow
123	C₂×#107	—	73	2	no	—
124	C₂×C₄⋊C₈	—	73	2	no	—
125	C₂×C₄⋊C₈	—	146	2	no	—
126	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	73	2	no	—
130	#340	—	73	1	no	—
131	#340	—	73	1	no	—
132	#336	—	146	2	no	—
133	#335	—	146	2	no	—
134	#322	—	2	2	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	146	2	no	—
137	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
138	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
139	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
140	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	73	2	no	—
142	C₂×C₄⋊C₄	—	73	2	no	—
143	C₂×C₄²	—	146	2	no	—
144	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	#107	—	146	4	no	—
146	C₄⋊C₈	—	292	4	no	—
147	C₄⋊C₈	—	146	2	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	Dic₁₄₆	1	2	yes	—
153	C₄×C₈	Dic₁₄₆	1	2	yes	—
154	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
155	#95	—	146	2	no	—
156	C₂²×C₄	—	146	2	no	—
157	C₂²×C₄	—	146	2	no	—
158	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
160	C₂²×C₄	#11924028	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	292	2	no	—
170	C₂×C₈	—	292	2	no	—
171	C₂×C₈	—	292	2	no	—
172	C₂×C₈	#11924007	1	2	yes	—
173	C₂×C₈	#11924007	1	2	yes	—
174	C₂×C₈	D₅₈₄	1	2	yes	—
175	C₂×C₈	Dic₂₉₂	1	2	yes	—
176	C₄⋊C₄	—	73	4	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄⋊C₄	—	146	2	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₄²	—	292	2	no	—
181	C₄²	—	292	2	no	—
182	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
183	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
184	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
185	C₄²	#11924028	1	1	yes	characteristic
186	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	292	2	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	292	2	no	—
193	C₂×C₄	—	146	4	no	—
194	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
195	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
196	C₂×C₄	#14556565	1	1	yes	characteristic
197	C₂×C₄	#14556570	1	1	yes	characteristic
198	C₂×C₄	#14556583	1	1	yes	characteristic
199	C₂×C₄	#14556591	1	1	yes	characteristic
200	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
201	C₂×C₄	#14556599	1	2	yes	—
202	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
203	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
204	C₂×C₄	#14556613	1	1	yes	characteristic
205	C₂×C₄	#14556613	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	584	2	no	—
212	C₂²	#17988485	1	1	yes	characteristic
213	C₂²	#17988660	1	2	yes	—
214	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
215	C₂²	C₄×Dic₂₉₂	1	1	yes	characteristic
216	C₂²	#17988710	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	292	4	no	—
220	C₄	—	584	2	no	—
221	C₄	#17988610	1	1	yes	characteristic
222	C₄	#17988615	1	1	yes	characteristic
223	C₄	#17988715	1	2	yes	—
224	C₄	#17988823	1	1	yes	characteristic
225	C₄	#17988828	1	1	yes	characteristic
226	C₄	#17988989	1	1	yes	characteristic
227	C₄	#17988992	1	1	yes	characteristic
228	C₂	#22608762	1	2	yes	—
229	C₂	#22608783	1	2	yes	—
230	C₂	#22610654	1	1	yes	characteristic
231	C₂	#22610664	1	1	yes	characteristic
232	C₂	#22610684	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#322×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	#9598
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
122. C₇₃	#9598	semidirect

Maximal subgroups

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

subgroup	index
#22610649	2
C₄×#14556525	2
C₂×#17988340	2
#22607771	2
C₂×#322×C₇₃	2
#9598	73