Subgroups — #32979806

Order 18688.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610103	C₂	1	1	yes	maximal, characteristic
2	#22610069	C₂	1	1	yes	maximal, characteristic
3	#22608690	C₂	1	1	yes	maximal, characteristic
4	#22608690	C₂	1	1	yes	maximal, characteristic
5	#22608687	C₂	1	1	yes	maximal, characteristic
6	#22608687	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556527	—	2	2	no	—
9	#17988611	C₂²	1	1	yes	characteristic
10	#17988611	C₂²	1	1	yes	characteristic
11	#17988611	C₂²	1	1	yes	characteristic
12	#17988611	C₂²	1	1	yes	characteristic
13	#17988352	—	2	2	no	—
14	#17988340	—	2	2	no	—
15	#17988340	—	2	2	no	—
16	#17988338	—	2	2	no	—
17	#17988338	—	2	2	no	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	#322×C₇₃	C₄	1	2	yes	—
22	#322×C₇₃	C₄	1	2	yes	—
23	C₂×#11924012	—	2	1	no	—
24	C₂×#11924012	—	2	1	no	—
25	C₂×#11924012	—	2	1	no	—
26	C₂×#11924012	—	2	1	no	—
27	#14556527	—	4	4	no	—
28	C₄⋊#10596831	—	2	1	no	—
29	C₄⋊#10596831	—	2	1	no	—
30	C₄⋊#10596831	—	2	1	no	—
31	C₄⋊#10596831	—	2	1	no	—
32	C₄⋊#10596831	—	2	1	no	—
33	C₄⋊#10596831	—	2	1	no	—
34	C₄⋊#10596831	—	2	1	no	—
35	C₄⋊#10596831	—	2	1	no	—
36	C₂²×C₈×C₇₃	D₄	1	2	yes	—
37	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
38	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₃	D₄	1	2	yes	—
45	C₄×C₈×C₇₃	D₄	1	2	yes	—
46	C₄×C₈×C₇₃	Q₈	1	2	yes	—
47	#11924012	—	4	2	no	—
48	#11924012	—	4	2	no	—
49	#11924012	—	4	2	no	—
50	#11924012	—	4	2	no	—
51	C₂×#10596831	—	4	1	no	—
52	C₂×#10596831	—	4	1	no	—
53	C₂×#10596831	—	4	1	no	—
54	C₂×#10596831	—	4	1	no	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	—	2	4	no	—
61	C₂×C₈×C₇₃	—	4	2	no	—
62	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
64	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₇₃	#137	1	2	yes	—
81	C₈×C₇₃	—	2	4	no	—
82	C₈×C₇₃	—	2	2	no	—
83	C₈×C₇₃	—	2	4	no	—
84	C₈×C₇₃	#108	1	4	yes	complemented
85	#10596831	—	8	1	no	—
86	#10596831	—	8	1	no	—
87	#10596831	—	8	1	no	—
88	#10596831	—	8	1	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	#384	1	1	yes	characteristic
92	C₂²×C₇₃	#501	1	2	yes	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	#428	1	2	yes	—
96	C₄×C₇₃	C₂×#108	1	2	yes	—
97	C₄×C₇₃	#430	1	2	yes	—
98	C₄×C₇₃	#496	1	1	yes	characteristic
99	C₄×C₇₃	#497	1	1	yes	characteristic
100	#8828	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	2	no	—
102	C₂×C₇₃	#1847	1	1	yes	characteristic
103	C₂×C₇₃	#1848	1	1	yes	characteristic
104	C₂×C₇₃	#1849	1	1	yes	characteristic
105	#1848	—	73	1	no	—
106	#1837	—	73	1	no	—
107	#1567	—	73	1	no	—
108	#1567	—	73	1	no	—
109	#1566	—	73	1	no	—
110	#1566	—	73	1	no	—
111	C₂×#322	D₇₃	1	1	yes	characteristic
112	C₇₃	#8828	1	1	yes	characteristic, complemented, Sylow
113	C₂×#108	—	146	2	no	—
114	#423	—	73	1	no	—
115	#423	—	73	1	no	—
116	#423	—	73	1	no	—
117	#423	—	73	1	no	—
118	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
121	#341	—	146	2	no	—
122	#335	—	146	2	no	—
123	#335	—	146	2	no	—
124	#334	—	146	2	no	—
125	#334	—	146	2	no	—
126	#322	Dic₇₃	1	2	yes	—
127	#322	Dic₇₃	1	2	yes	—
128	C₂×M₁₆	—	146	1	no	—
129	C₂×M₁₆	—	146	1	no	—
130	C₂×M₁₆	—	146	1	no	—
131	C₂×M₁₆	—	146	1	no	—
132	C₂²×C₈	Dic₁₄₆	1	2	yes	—
133	C₂²×C₈	Dic₁₄₆	1	2	yes	—
134	C₂²×C₈	D₂₉₂	1	2	yes	—
135	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
136	#108	—	292	4	no	—
137	C₄⋊C₈	—	146	1	no	—
138	C₄⋊C₈	—	146	1	no	—
139	C₄⋊C₈	—	146	1	no	—
140	C₄⋊C₈	—	146	1	no	—
141	C₄⋊C₈	—	146	1	no	—
142	C₄⋊C₈	—	146	1	no	—
143	C₄⋊C₈	—	146	1	no	—
144	C₄⋊C₈	—	146	1	no	—
145	C₄×C₈	Dic₁₄₆	1	2	yes	—
146	C₄×C₈	D₂₉₂	1	2	yes	—
147	C₄×C₈	D₂₉₂	1	2	yes	—
148	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
149	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
151	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
155	M₁₆	—	292	2	no	—
156	M₁₆	—	292	2	no	—
157	M₁₆	—	292	2	no	—
158	M₁₆	—	292	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	292	1	no	—
164	C₂×C₈	—	292	1	no	—
165	C₂×C₈	—	292	1	no	—
166	C₂×C₈	—	292	1	no	—
167	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
168	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
169	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
170	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
171	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
172	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
173	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
174	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
175	C₂³	#14556591	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
181	C₂×C₄	#14556588	1	1	yes	characteristic
182	C₂×C₄	#14556591	1	1	yes	characteristic
183	C₂×C₄	#14556591	1	1	yes	characteristic
184	C₂×C₄	#14556619	1	2	yes	—
185	C₂×C₄	#14556624	1	2	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	4	no	—
189	C₈	—	584	1	no	—
190	C₈	—	584	1	no	—
191	C₈	—	584	1	no	—
192	C₈	—	584	1	no	—
193	C₈	#14556527	1	4	yes	complemented
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#17988457	1	1	yes	characteristic
197	C₂²	#17988996	1	2	yes	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	#17988629	1	2	yes	—
201	C₄	C₂×#14556527	1	2	yes	—
202	C₄	#17988636	1	2	yes	—
203	C₄	#17988976	1	1	yes	characteristic
204	C₄	#17988981	1	1	yes	characteristic
205	C₂	—	2	2	no	—
206	C₂	#22610099	1	1	yes	characteristic
207	C₂	#22610103	1	1	yes	characteristic
208	C₂	#22610106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556588
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#322×C₇₃	C₂
socle	C₂²×C₇₃	#384
prime radical	C₂³×C₇₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	#8828
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
84. C₈×C₇₃	#108	semidirect
112. C₇₃	#8828	semidirect
193. C₈	#14556527	semidirect

Maximal subgroups

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

subgroup	index
#22610103	2
#22610069	2
#22608690	2
#22608687	2
C₂×#322×C₇₃	2
#8828	73