Subgroups — #32977243

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
2	#22609798	C₂	1	2	yes	maximal
3	C₂×#17988338	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×#14556501	—	2	2	no	—
9	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	#17988350	C₄	1	2	yes	—
11	#17988341	—	2	2	no	—
12	#17988338	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#322×C₇₃	C₄	1	4	yes	complemented
17	C₂²×#10596831	—	2	2	no	—
18	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₇₃	—	2	2	no	—
20	#14556525	—	2	4	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	C₂×C₄	1	4	yes	—
23	#14556501	—	4	4	no	—
24	#14556498	—	2	2	no	—
25	C₂²×C₈×C₇₃	—	2	1	no	—
26	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	D₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₇₃	—	2	1	no	—
31	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	Q₈	1	2	yes	—
36	C₂²×Dic₇₃	—	2	4	no	—
37	C₂×#10596831	—	2	4	no	—
38	C₂×#10596831	—	4	2	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₄⋊Dic₇₃	—	2	4	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄×Dic₇₃	—	4	2	no	—
43	C₄×Dic₇₃	—	4	2	no	—
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	—	4	2	no	—
52	C₂×C₈×C₇₃	C₄²	1	2	yes	—
53	C₂×C₈×C₇₃	C₄²	1	2	yes	—
54	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
56	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₃	#41	1	1	yes	characteristic
60	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₇₃	#131	1	2	yes	—
70	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₇₃	#137	1	2	yes	—
72	C₂×Dic₇₃	—	4	4	no	—
73	C₂×Dic₇₃	—	4	4	no	—
74	C₂×Dic₇₃	—	4	4	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	—	2	4	no	—
77	C₈×C₇₃	—	4	2	no	—
78	C₈×C₇₃	#97	1	4	yes	complemented
79	#10596831	—	4	4	no	—
80	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₇₃	#436	1	2	yes	—
82	C₂²×C₇₃	#441	1	1	yes	characteristic
83	C₂²×C₇₃	#442	1	1	yes	characteristic
84	C₂²×C₇₃	#444	1	2	yes	—
85	C₄×C₇₃	—	2	1	no	—
86	C₄×C₇₃	—	2	1	no	—
87	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
88	C₄×C₇₃	#405	1	1	yes	characteristic
89	C₄×C₇₃	#428	1	2	yes	—
90	C₄×C₇₃	#440	1	2	yes	—
91	C₄×C₇₃	#446	1	2	yes	—
92	Dic₇₃	—	8	4	no	—
93	#8287	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	#1593	1	2	yes	—
95	C₂×C₇₃	#1594	1	2	yes	—
96	C₂×C₇₃	#1770	1	1	yes	characteristic
97	C₂×C₇₃	#1773	1	1	yes	characteristic
98	C₂×C₇₃	#1776	1	1	yes	characteristic
99	C₄×#107	—	73	1	no	—
100	#1770	—	73	2	no	—
101	C₂×#334	—	73	1	no	—
102	C₂×#322	D₇₃	1	1	yes	characteristic
103	#1279	—	73	2	no	—
104	C₇₃	#8287	1	1	yes	characteristic, complemented, Sylow
105	C₂×#107	—	73	2	no	—
106	C₂×C₄⋊C₈	—	73	2	no	—
107	C₂×#97	—	146	2	no	—
108	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	73	2	no	—
112	#340	—	73	2	no	—
113	#336	—	146	2	no	—
114	#334	—	73	4	no	—
115	#322	Dic₇₃	1	4	yes	complemented
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	146	2	no	—
118	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
120	C₂²×C₈	D₂₉₂	1	2	yes	—
121	C₂×C₄⋊C₄	—	73	4	no	—
122	C₂×C₄²	—	146	2	no	—
123	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
124	#107	—	146	4	no	—
125	C₄⋊C₈	—	73	4	no	—
126	C₄⋊C₈	—	146	2	no	—
127	#97	—	292	4	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	Dic₁₄₆	1	2	yes	—
130	C₄×C₈	C₄×D₇₃	1	2	yes	—
131	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
132	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
133	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
134	#95	—	146	2	no	—
135	C₂²×C₄	—	146	4	no	—
136	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
138	C₂²×C₄	#11924028	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	292	2	no	—
145	C₂×C₈	—	292	2	no	—
146	C₂×C₈	—	146	4	no	—
147	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
148	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
150	C₂×C₈	#11924011	1	4	yes	—
151	C₄⋊C₄	—	146	4	no	—
152	C₄⋊C₄	—	146	4	no	—
153	C₄²	—	292	2	no	—
154	C₄²	—	292	2	no	—
155	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
156	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
157	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
158	C₄²	#11924028	1	1	yes	characteristic
159	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	292	4	no	—
162	C₂×C₄	—	292	4	no	—
163	C₂×C₄	—	292	4	no	—
164	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
166	C₂×C₄	#14556550	1	1	yes	characteristic
167	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
169	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
170	C₂×C₄	C₂×#11924011	1	2	yes	—
171	C₂×C₄	#14556597	1	2	yes	—
172	C₂×C₄	#14556619	1	2	yes	—
173	C₂×C₄	#14556624	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	2	2	no	—
176	C₈	—	4	2	no	—
177	C₈	—	292	4	no	—
178	C₈	#14556501	1	4	yes	complemented
179	C₂²	C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
180	C₂²	#17988666	1	2	yes	—
181	C₂²	#17988695	1	1	yes	characteristic
182	C₂²	#17988700	1	1	yes	characteristic
183	C₂²	#17988710	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	584	4	no	—
187	C₄	C₂×#14556501	1	1	yes	characteristic
188	C₄	#17988542	1	1	yes	characteristic
189	C₄	#17988629	1	2	yes	—
190	C₄	#17988689	1	2	yes	—
191	C₄	#17988717	1	2	yes	—
192	C₂	#22608841	1	2	yes	—
193	C₂	#22608846	1	2	yes	—
194	C₂	#22609798	1	1	yes	characteristic
195	C₂	#22609815	1	1	yes	characteristic
196	C₂	#22609828	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
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	#8287
LCS	G ▷ C₂×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
12. #17988338	C₄	semidirect
16. #322×C₇₃	C₄	semidirect
78. C₈×C₇₃	#97	semidirect
104. C₇₃	#8287	semidirect
115. #322	Dic₇₃	semidirect
178. C₈	#14556501	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556525	2
#22609798	2
C₂×#17988338	2
#22607771	2
C₂×#322×C₇₃	2
#8287	73