Subgroups — #32984554

Order 18688.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610647	C₂	1	2	yes	maximal
2	C₄×#14556523	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988309	C₂	1	1	yes	maximal, characteristic
4	#22607770	C₂	1	2	yes	maximal
5	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556523	C₄	1	2	yes	—
7	C₂×#14556520	—	2	2	no	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	#17988350	C₄	1	1	yes	characteristic
12	#17988350	C₄	1	1	yes	characteristic
13	#17988341	—	2	2	no	—
14	#17988309	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#335×C₇₃	—	2	2	no	—
18	C₂²×#10596831	—	2	1	no	—
19	C₂²×#10596831	D₄	1	1	yes	characteristic
20	C₂²×#10596831	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₇₃	—	2	2	no	—
24	#14556523	—	2	4	no	—
25	#14556520	—	4	4	no	—
26	C₄×#10596831	—	2	1	no	—
27	C₄×#10596831	—	2	1	no	—
28	C₄×#10596831	—	2	1	no	—
29	C₄×#10596831	Q₈	1	2	yes	—
30	#14556498	—	2	2	no	—
31	C₂²×C₈×C₇₃	—	2	2	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	C₄⋊C₈×C₇₃	—	2	2	no	—
37	C₄×C₈×C₇₃	D₄	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	—	2	2	no	—
42	C₂×#10596831	—	2	2	no	—
43	C₂×#10596831	—	2	2	no	—
44	C₂×#10596831	—	2	1	no	—
45	C₂×#10596831	—	2	1	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₄⋊Dic₇₃	—	2	2	no	—
49	C₄⋊Dic₇₃	—	2	2	no	—
50	C₄⋊Dic₇₃	—	2	2	no	—
51	C₄⋊Dic₇₃	M₁₆	1	4	yes	—
52	C₄×Dic₇₃	—	4	2	no	—
53	C₄×Dic₇₃	—	4	2	no	—
54	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	4	2	no	—
60	C₂×C₈×C₇₃	—	4	2	no	—
61	C₂×C₈×C₇₃	—	4	2	no	—
62	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₃	#41	1	1	yes	characteristic
68	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	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	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₇₃	#137	1	2	yes	—
81	C₂×Dic₇₃	—	2	4	no	—
82	C₂×Dic₇₃	—	4	2	no	—
83	C₂×Dic₇₃	—	4	2	no	—
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	—	4	2	no	—
86	C₂×Dic₇₃	—	4	2	no	—
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	—	2	2	no	—
89	C₈×C₇₃	—	8	2	no	—
90	#10596831	—	2	4	no	—
91	#10596831	—	4	1	no	—
92	#10596831	—	4	1	no	—
93	C₂²×C₇₃	#389	1	1	yes	characteristic
94	C₂²×C₇₃	#435	1	2	yes	—
95	C₂²×C₇₃	#440	1	2	yes	—
96	C₂²×C₇₃	#443	1	1	yes	characteristic
97	C₂²×C₇₃	#443	1	1	yes	characteristic
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	#423	1	1	yes	characteristic
101	C₄×C₇₃	#424	1	1	yes	characteristic
102	C₄×C₇₃	#446	1	2	yes	—
103	C₄×C₇₃	#469	1	1	yes	characteristic
104	C₄×C₇₃	#470	1	1	yes	characteristic
105	C₄×C₇₃	#501	1	2	yes	—
106	Dic₇₃	—	4	4	no	—
107	Dic₇₃	—	8	2	no	—
108	#9600	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	#1585	1	2	yes	—
110	C₂×C₇₃	#1588	1	2	yes	—
111	C₂×C₇₃	#1940	1	1	yes	characteristic
112	C₂×C₇₃	#1944	1	1	yes	characteristic
113	C₂×C₇₃	#1945	1	1	yes	characteristic
114	#1939	—	73	2	no	—
115	C₄×#106	—	73	1	no	—
116	C₂×#335	D₇₃	1	1	yes	characteristic
117	C₂×#322	—	73	1	no	—
118	#1279	—	73	2	no	—
119	C₇₃	#9600	1	1	yes	characteristic, complemented, Sylow
120	C₂×#106	—	73	2	no	—
121	C₂×C₄⋊C₈	—	146	2	no	—
122	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	73	2	no	—
127	#340	—	73	1	no	—
128	#340	—	73	1	no	—
129	#336	—	146	2	no	—
130	#335	—	2	2	no	—
131	#322	—	146	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	146	1	no	—
136	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
137	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	73	2	no	—
139	C₂×C₄⋊C₄	—	73	2	no	—
140	C₂×C₄²	—	146	2	no	—
141	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
142	#106	—	146	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	292	4	no	—
146	C₄×C₈	—	73	2	no	—
147	C₄×C₈	—	146	1	no	—
148	C₄×C₈	—	146	1	no	—
149	C₄×C₈	—	146	1	no	—
150	C₄×C₈	Dic₁₄₆	1	2	yes	—
151	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
152	#95	—	146	2	no	—
153	C₂²×C₄	—	146	2	no	—
154	C₂²×C₄	—	146	2	no	—
155	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
156	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
157	C₂²×C₄	#11924029	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	146	1	no	—
164	C₂×C₈	—	146	1	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	—	146	2	no	—
167	C₂×C₈	—	146	2	no	—
168	C₂×C₈	—	146	2	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	#11924007	1	2	yes	—
171	C₂×C₈	#11924007	1	2	yes	—
172	C₄⋊C₄	—	73	4	no	—
173	C₄⋊C₄	—	146	2	no	—
174	C₄⋊C₄	—	146	2	no	—
175	C₄⋊C₄	—	146	2	no	—
176	C₄²	—	292	2	no	—
177	C₄²	—	292	2	no	—
178	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
179	C₄²	Q₈×D₇₃	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
181	C₄²	#11924027	1	1	yes	characteristic
182	C₂³	#14556553	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	292	2	no	—
186	C₂×C₄	—	292	2	no	—
187	C₂×C₄	—	292	2	no	—
188	C₂×C₄	—	292	2	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	146	4	no	—
191	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
192	C₂×C₄	#14556556	1	1	yes	characteristic
193	C₂×C₄	#14556562	1	1	yes	characteristic
194	C₂×C₄	#14556567	1	1	yes	characteristic
195	C₂×C₄	#14556582	1	1	yes	characteristic
196	C₂×C₄	#14556590	1	1	yes	characteristic
197	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
198	C₂×C₄	#14556597	1	2	yes	—
199	C₂×C₄	#14556615	1	2	yes	—
200	C₂×C₄	#14556623	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	8	2	no	—
204	C₈	—	292	1	no	—
205	C₈	—	292	1	no	—
206	C₈	—	146	4	no	—
207	C₂²	#17988483	1	1	yes	characteristic
208	C₂²	#17988657	1	2	yes	—
209	C₂²	#17988686	1	2	yes	—
210	C₂²	#17988703	1	1	yes	characteristic
211	C₂²	#17988703	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	292	4	no	—
215	C₄	—	584	2	no	—
216	C₄	#17988610	1	1	yes	characteristic
217	C₄	#17988613	1	1	yes	characteristic
218	C₄	#17988716	1	2	yes	—
219	C₄	#17988837	1	1	yes	characteristic
220	C₄	#17988842	1	1	yes	characteristic
221	C₄	#17988995	1	2	yes	—
222	C₂	#22608787	1	2	yes	—
223	C₂	#22608808	1	2	yes	—
224	C₂	#22610652	1	1	yes	characteristic
225	C₂	#22610672	1	1	yes	characteristic
226	C₂	#22610677	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	#9600
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
119. C₇₃	#9600	semidirect

Maximal subgroups

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

subgroup	index
#22610647	2
C₄×#14556523	2
C₂×#17988309	2
#22607770	2
C₂×#335×C₇₃	2
#9600	73