Subgroups — #32984701

Order 18688.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
2	#22608707	C₂	1	2	yes	maximal
3	#22608679	C₂	1	1	yes	maximal, characteristic
4	#22607832	C₂	1	2	yes	maximal
5	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	—	2	1	no	—
9	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988341	—	2	1	no	—
11	#17988340	C₄	1	2	yes	complemented
12	#17988339	—	2	4	no	—
13	#17988338	C₄	1	2	yes	complemented
14	C₂×#108×C₇₃	—	2	1	no	—
15	C₂×#108×C₇₃	—	2	1	no	—
16	#423×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×#10596831	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	—	2	1	no	—
20	C₂×C₄×Dic₇₃	—	2	1	no	—
21	C₄⋊#10596831	—	2	1	no	—
22	C₄⋊#10596831	C₂×C₄	1	4	yes	—
23	C₄⋊#10596831	D₄	1	2	yes	—
24	C₂²⋊#10596831	—	2	2	no	—
25	C₄×#10596831	—	2	2	no	—
26	C₈×Dic₇₃	—	4	2	no	—
27	#14556498	Q₈	1	2	yes	—
28	C₂×M₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	#108×C₇₃	—	4	2	no	—
33	#108×C₇₃	—	4	2	no	—
34	C₄⋊C₈×C₇₃	—	2	2	no	—
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
40	C₂²×Dic₇₃	—	2	2	no	—
41	C₂×#10596831	—	2	4	no	—
42	C₂×#10596831	—	2	2	no	—
43	C₂×#10596831	—	4	1	no	—
44	C₂×#10596831	—	4	1	no	—
45	C₄⋊Dic₇₃	—	2	4	no	—
46	C₄×Dic₇₃	—	4	1	no	—
47	C₄×Dic₇₃	—	4	1	no	—
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
51	M₁₆×C₇₃	—	4	2	no	—
52	M₁₆×C₇₃	—	4	2	no	—
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	4	2	no	—
56	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	D₈	1	2	yes	—
61	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
62	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇₃	#41	1	1	yes	characteristic
66	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	4	1	no	—
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
79	C₂×Dic₇₃	—	2	4	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	—	4	2	no	—
82	C₈×C₇₃	—	2	1	no	—
83	C₈×C₇₃	—	2	1	no	—
84	C₈×C₇₃	—	8	2	no	—
85	C₈×C₇₃	#107	1	2	yes	—
86	C₈×C₇₃	#107	1	2	yes	—
87	#10596831	—	4	4	no	—
88	#10596831	—	4	2	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	#389	1	1	yes	characteristic
92	C₂²×C₇₃	#474	1	1	yes	characteristic
93	C₂²×C₇₃	#477	1	1	yes	characteristic
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	—	2	1	no	—
96	C₄×C₇₃	—	2	1	no	—
97	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
98	C₄×C₇₃	#428	1	1	yes	characteristic
99	C₄×C₇₃	#443	1	2	yes	—
100	C₄×C₇₃	#459	1	1	yes	characteristic
101	C₄×C₇₃	#462	1	1	yes	characteristic
102	Dic₇₃	—	4	4	no	—
103	#9627	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	#1945	1	1	yes	characteristic
107	C₂×C₇₃	#1947	1	1	yes	characteristic
108	C₂×C₇₃	#1953	1	1	yes	characteristic
109	#1848	D₇₃	1	1	yes	characteristic
110	C₈×C₄⋊C₄	—	73	1	no	—
111	#1571	—	73	2	no	—
112	#1564	—	73	1	no	—
113	#1303	—	73	2	no	—
114	C₇₃	#9627	1	1	yes	characteristic, complemented, Sylow
115	#431	—	73	2	no	—
116	C₂×#108	—	2	1	no	—
117	C₂×#108	—	2	1	no	—
118	#423	C₂×D₇₃	1	2	yes	—
119	C₂×C₄⋊C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	146	1	no	—
121	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	73	1	no	—
123	#336	—	146	1	no	—
124	#335	—	73	2	no	—
125	#335	—	146	4	no	—
126	#334	—	73	2	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	—	146	1	no	—
129	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	146	1	no	—
132	C₂×C₄²	—	146	1	no	—
133	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
134	#108	—	4	2	no	—
135	#108	—	4	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	73	4	no	—
139	C₄⋊C₈	—	73	2	no	—
140	C₄⋊C₈	—	146	1	no	—
141	C₂²⋊C₈	—	146	2	no	—
142	C₄×C₈	—	292	2	no	—
143	C₄×C₈	—	146	2	no	—
144	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
146	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
147	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
148	#95	—	73	2	no	—
149	C₂²×C₄	—	146	2	no	—
150	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
152	C₂²×C₄	#11924029	1	1	yes	characteristic
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	292	1	no	—
159	C₂×C₈	—	292	1	no	—
160	C₂×C₈	—	146	4	no	—
161	C₂×C₈	—	146	2	no	—
162	C₂×C₈	#11924007	1	1	yes	characteristic
163	C₂×C₈	#11924007	1	1	yes	characteristic
164	C₂×C₈	#11924007	1	1	yes	characteristic
165	C₂×C₈	#11924007	1	1	yes	characteristic
166	C₂×C₈	#11924013	1	2	yes	—
167	C₂×C₈	#11924018	1	2	yes	—
168	C₄⋊C₄	—	146	4	no	—
169	C₄²	—	292	1	no	—
170	C₄²	—	292	1	no	—
171	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
172	C₄²	D₄×D₇₃	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
174	C₄²	#11924027	1	1	yes	characteristic
175	C₂³	#14556567	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	292	2	no	—
179	C₂×C₄	—	292	2	no	—
180	C₂×C₄	—	146	4	no	—
181	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
182	C₂×C₄	#14556553	1	1	yes	characteristic
183	C₂×C₄	#14556556	1	1	yes	characteristic
184	C₂×C₄	#14556562	1	1	yes	characteristic
185	C₂×C₄	#14556582	1	1	yes	characteristic
186	C₂×C₄	#14556590	1	1	yes	characteristic
187	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
188	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
189	C₂×C₄	#14556617	1	1	yes	characteristic
190	C₂×C₄	#14556622	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	8	2	no	—
194	C₈	—	292	4	no	—
195	C₈	—	292	2	no	—
196	C₈	#14556524	1	2	yes	—
197	C₈	#14556524	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#17988483	1	1	yes	characteristic
201	C₂²	#17988864	1	1	yes	characteristic
202	C₂²	#17988879	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	292	4	no	—
207	C₄	C₂×#14556524	1	1	yes	characteristic
208	C₄	#17988627	1	1	yes	characteristic
209	C₄	#17988703	1	2	yes	—
210	C₄	#17988779	1	1	yes	characteristic
211	C₄	#17988794	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#22610677	1	1	yes	characteristic
215	C₂	#22610687	1	1	yes	characteristic
216	C₂	#22610719	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556562
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	#389
prime radical	C₂³×C₇₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#9627
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
11. #17988340	C₄	semidirect
13. #17988338	C₄	semidirect
114. C₇₃	#9627	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₇₃	2
#22608707	2
#22608679	2
#22607832	2
C₇₃×#1848	2
#9627	73