Subgroups — #32977223

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556523	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×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556523	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	C₈²×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	#14556523	—	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	1	yes	characteristic
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₃	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₃	Q₈	1	2	yes	—
37	C₂²×Dic₇₃	—	2	4	no	—
38	C₂×#10596831	—	2	4	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄⋊Dic₇₃	—	2	4	no	—
43	C₄×Dic₇₃	—	4	2	no	—
44	C₄×Dic₇₃	—	4	2	no	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	2	4	no	—
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	—	2	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	2	yes	—
55	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
57	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
58	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₇₃	#41	1	1	yes	characteristic
62	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₇₃	#131	1	2	yes	—
72	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₇₃	#135	1	2	yes	—
74	C₂×Dic₇₃	—	4	4	no	—
75	C₂×Dic₇₃	—	4	4	no	—
76	C₂×Dic₇₃	—	4	4	no	—
77	C₈×C₇₃	—	2	4	no	—
78	C₈×C₇₃	—	2	2	no	—
79	C₈×C₇₃	—	2	2	no	—
80	C₈×C₇₃	#97	1	4	yes	complemented
81	C₈×C₇₃	#106	1	4	yes	complemented
82	#10596831	—	4	4	no	—
83	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₃	#436	1	2	yes	—
85	C₂²×C₇₃	C₄×SD₁₆	1	2	yes	—
86	C₂²×C₇₃	#443	1	2	yes	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
90	C₄×C₇₃	#405	1	1	yes	characteristic
91	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
92	C₄×C₇₃	#427	1	1	yes	characteristic
93	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
94	C₄×C₇₃	#446	1	2	yes	—
95	Dic₇₃	—	8	4	no	—
96	#8283	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	#1590	1	4	yes	—
98	C₂×C₇₃	#1770	1	1	yes	characteristic
99	C₂×C₇₃	#1772	1	1	yes	characteristic
100	C₂×C₇₃	C₄×#106	1	1	yes	characteristic
101	C₄×#106	—	73	1	no	—
102	#1770	—	73	2	no	—
103	C₂×#334	—	73	1	no	—
104	C₂×C₈²	D₇₃	1	1	yes	characteristic
105	#1279	—	73	2	no	—
106	C₇₃	#8283	1	1	yes	characteristic, complemented, Sylow
107	C₂×#106	—	73	2	no	—
108	C₂×C₄⋊C₈	—	73	2	no	—
109	C₂×#97	—	146	2	no	—
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	73	2	no	—
114	#340	—	73	2	no	—
115	#336	—	146	2	no	—
116	#334	—	73	4	no	—
117	C₈²	Dic₇₃	1	4	yes	complemented
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	146	2	no	—
120	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
122	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
123	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	73	4	no	—
125	C₂×C₄²	—	146	2	no	—
126	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	#106	—	146	4	no	—
128	C₄⋊C₈	—	73	4	no	—
129	C₄⋊C₈	—	146	2	no	—
130	#97	—	292	4	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	Dic₁₄₆	1	2	yes	—
133	C₄×C₈	C₄×D₇₃	1	2	yes	—
134	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
135	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
136	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
137	#95	—	146	2	no	—
138	C₂²×C₄	—	146	4	no	—
139	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
141	C₂²×C₄	#11924028	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	292	2	no	—
147	C₂×C₈	—	292	2	no	—
148	C₂×C₈	—	146	4	no	—
149	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
150	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
151	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
152	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
153	C₂×C₈	#11924011	1	4	yes	—
154	C₂×C₈	#11924017	1	4	yes	—
155	C₄⋊C₄	—	146	4	no	—
156	C₄⋊C₄	—	146	4	no	—
157	C₄²	—	292	2	no	—
158	C₄²	—	292	2	no	—
159	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
160	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
161	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
162	C₄²	#11924028	1	1	yes	characteristic
163	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	292	4	no	—
166	C₂×C₄	—	292	4	no	—
167	C₂×C₄	—	292	4	no	—
168	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
170	C₂×C₄	#14556550	1	1	yes	characteristic
171	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
172	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
173	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#11924011	1	2	yes	—
175	C₂×C₄	#14556597	1	2	yes	—
176	C₂×C₄	C₂×#11924017	1	2	yes	—
177	C₂×C₄	#14556613	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	292	4	no	—
182	C₈	#14556501	1	4	yes	complemented
183	C₈	#14556523	1	4	yes	complemented
184	C₂²	C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
185	C₂²	#17988666	1	2	yes	—
186	C₂²	C₄×#11924017	1	2	yes	—
187	C₂²	#17988705	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	584	4	no	—
191	C₄	C₂×#14556501	1	1	yes	characteristic
192	C₄	#17988542	1	1	yes	characteristic
193	C₄	C₂×#14556523	1	1	yes	characteristic
194	C₄	#17988625	1	1	yes	characteristic
195	C₄	C₄×#11924017	1	2	yes	—
196	C₄	#17988717	1	2	yes	—
197	C₂	#22608823	1	4	yes	—
198	C₂	#22609798	1	1	yes	characteristic
199	C₂	#22609810	1	1	yes	characteristic
200	C₂	C₄×#14556523	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×C₈²×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	#8283
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. #17988338	C₄	semidirect
16. C₈²×C₇₃	C₄	semidirect
80. C₈×C₇₃	#97	semidirect
81. C₈×C₇₃	#106	semidirect
106. C₇₃	#8283	semidirect
117. C₈²	Dic₇₃	semidirect
182. C₈	#14556501	semidirect
183. C₈	#14556523	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556523	2
#22609798	2
C₂×#17988338	2
#22607771	2
C₂×C₈²×C₇₃	2
#8283	73