Subgroups — #32997372

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611730	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608690	C₂	1	1	yes	maximal, characteristic, complemented
3	#22607977	C₂	1	4	yes	maximal, complemented
4	C₂×#493×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17988611	—	2	1	no	—
6	C₄⋊#11924003	C₂²	1	2	yes	complemented
7	C₂²⋊#11924003	—	2	1	no	—
8	C₄×#11924003	—	2	1	no	—
9	#17988340	—	2	1	no	—
10	#17988340	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₇₃	—	2	1	no	—
12	C₂²×D₈×C₇₃	C₄	1	2	yes	—
13	C₂×#127×C₇₃	—	2	1	no	—
14	#493×C₇₃	—	2	1	no	—
15	#493×C₇₃	—	2	1	no	—
16	#493×C₇₃	C₄	1	2	yes	—
17	#493×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×#11924012	—	4	1	no	—
20	C₂×#11924003	—	2	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	—	4	1	no	—
23	C₂²×D₄×C₇₃	—	2	2	no	—
24	C₂²×D₄×C₇₃	—	4	1	no	—
25	C₂²×D₄×C₇₃	—	4	1	no	—
26	C₂×D₈×C₇₃	—	2	2	no	—
27	C₂×D₈×C₇₃	—	2	2	no	—
28	C₂×D₈×C₇₃	—	2	2	no	—
29	C₂×D₈×C₇₃	—	4	4	no	—
30	C₂×D₈×C₇₃	—	4	4	no	—
31	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	#127×C₇₃	—	2	2	no	—
35	#127×C₇₃	—	2	2	no	—
36	#127×C₇₃	—	4	1	no	—
37	#127×C₇₃	—	4	1	no	—
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
43	#11924012	—	8	2	no	—
44	C₂×#10596831	—	4	2	no	—
45	#11924003	—	4	2	no	—
46	C₂⁴×C₇₃	—	4	2	no	—
47	C₂×D₄×C₇₃	—	2	4	no	—
48	C₂×D₄×C₇₃	—	4	4	no	—
49	C₂×D₄×C₇₃	—	4	4	no	—
50	C₂×D₄×C₇₃	—	4	2	no	—
51	C₂×D₄×C₇₃	—	4	2	no	—
52	C₂×D₄×C₇₃	—	4	2	no	—
53	C₂×D₄×C₇₃	—	4	2	no	—
54	C₂×D₄×C₇₃	—	4	2	no	—
55	C₂×D₄×C₇₃	—	8	4	no	—
56	C₂×D₄×C₇₃	—	8	2	no	—
57	C₂×D₄×C₇₃	—	8	2	no	—
58	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
61	D₈×C₇₃	—	4	4	no	—
62	D₈×C₇₃	—	4	4	no	—
63	D₈×C₇₃	—	4	4	no	—
64	D₈×C₇₃	—	4	4	no	—
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	—	2	2	no	—
67	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₇₃	D₈	1	1	yes	characteristic
73	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
74	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₇₃	—	4	4	no	—
76	C₂³×C₇₃	—	8	4	no	—
77	C₂³×C₇₃	—	8	4	no	—
78	C₂³×C₇₃	—	8	4	no	—
79	C₂³×C₇₃	#102	1	1	yes	characteristic
80	D₄×C₇₃	—	4	4	no	—
81	D₄×C₇₃	—	4	4	no	—
82	D₄×C₇₃	—	8	4	no	—
83	D₄×C₇₃	—	8	4	no	—
84	D₄×C₇₃	—	8	4	no	—
85	D₄×C₇₃	—	8	4	no	—
86	D₄×C₇₃	—	8	2	no	—
87	D₄×C₇₃	—	8	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
95	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
96	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₇₃	—	2	1	no	—
100	C₈×C₇₃	—	2	1	no	—
101	C₈×C₇₃	#100	1	2	yes	—
102	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
103	#10596831	—	8	2	no	—
104	C₂²×C₇₃	—	2	1	no	—
105	C₂²×C₇₃	—	2	1	no	—
106	C₂²×C₇₃	—	8	4	no	—
107	C₂²×C₇₃	—	8	4	no	—
108	C₂²×C₇₃	—	8	4	no	—
109	C₂²×C₇₃	—	16	8	no	—
110	C₂²×C₇₃	#357	1	2	yes	—
111	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
112	C₄×C₇₃	—	2	1	no	—
113	C₄×C₇₃	—	2	1	no	—
114	C₄×C₇₃	#357	1	2	yes	—
115	C₄×C₇₃	#361	1	2	yes	—
116	C₄×C₇₃	C₂×#100	1	1	yes	characteristic
117	C₄×C₇₃	#418	1	1	yes	characteristic
118	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
119	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
120	#11961	—	73	1	no	maximal, Sylow
121	C₂×C₇₃	—	2	2	no	—
122	C₂×C₇₃	—	16	4	no	—
123	C₂×C₇₃	#1545	1	1	yes	characteristic
124	C₂×C₇₃	C₂×#357	1	1	yes	characteristic
125	C₂×C₇₃	C₂×#361	1	1	yes	characteristic
126	C₂×#493	D₇₃	1	1	yes	characteristic
127	#2162	—	73	1	no	—
128	#1567	—	73	1	no	—
129	#1361	—	73	4	no	—
130	C₇₃	#11961	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₈	—	2	1	no	—
132	C₂²×D₈	Dic₇₃	1	2	yes	—
133	C₂×#127	—	2	1	no	—
134	#493	—	2	1	no	—
135	#493	—	2	1	no	—
136	#493	Dic₇₃	1	2	yes	—
137	#493	C₂×D₇₃	1	2	yes	—
138	#423	—	146	1	no	—
139	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
140	C₄⋊C₁₆	—	73	2	no	—
141	C₂²⋊C₁₆	—	146	1	no	—
142	C₄×C₁₆	—	146	1	no	—
143	#335	—	73	2	no	—
144	#335	—	146	1	no	—
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	—	4	1	no	—
147	C₂²×D₄	—	4	1	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	4	4	no	—
152	C₂×D₈	—	4	4	no	—
153	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
154	C₂×M₁₆	—	292	1	no	—
155	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	—	4	1	no	—
160	#127	—	4	1	no	—
161	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
162	C₂×C₁₆	—	146	2	no	—
163	C₄⋊C₈	—	292	1	no	—
164	C₄⋊C₈	—	146	2	no	—
165	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
169	C₂⁴	—	4	2	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	8	4	no	—
179	C₂×D₄	—	8	2	no	—
180	C₂×D₄	—	8	2	no	—
181	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
182	C₂²×C₄	#11924013	1	1	yes	characteristic
183	C₂²×C₄	#11924016	1	1	yes	characteristic
184	D₈	—	4	4	no	—
185	D₈	—	4	4	no	—
186	D₈	—	4	4	no	—
187	D₈	—	4	4	no	—
188	M₁₆	—	584	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	292	2	no	—
192	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
196	C₄²	C₂²⋊Dic₇₃	1	1	yes	characteristic
197	C₄²	#11924013	1	1	yes	characteristic
198	C₄²	#11924016	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
200	C₁₆	—	292	2	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	8	4	no	—
203	C₂³	—	8	4	no	—
204	C₂³	—	8	4	no	—
205	C₂³	#14556512	1	1	yes	characteristic
206	D₄	—	4	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	8	4	no	—
209	D₄	—	8	4	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	2	no	—
213	D₄	—	8	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	#14556512	1	1	yes	characteristic
218	C₂×C₄	#14556512	1	1	yes	characteristic
219	C₂×C₄	#14556512	1	1	yes	characteristic
220	C₂×C₄	#14556532	1	2	yes	—
221	C₂×C₄	#14556535	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
224	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	584	2	no	—
228	C₈	#14556508	1	2	yes	—
229	C₈	#14556617	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	16	8	no	—
236	C₂²	#17988388	1	2	yes	—
237	C₂²	C₂×#14556512	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	#17988388	1	2	yes	—
241	C₄	#17988400	1	2	yes	—
242	C₄	C₂×#14556508	1	1	yes	characteristic
243	C₄	#17988592	1	1	yes	characteristic
244	C₄	C₂×#14556532	1	1	yes	characteristic
245	C₄	C₂×#14556535	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	16	4	no	—
248	C₂	#22608599	1	1	yes	characteristic
249	C₂	C₂×#17988388	1	1	yes	characteristic
250	C₂	C₂×#17988400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556512
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting, prime radical	C₂×#493×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#102
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₂×#493	D₇₃
hypocenter	C₇₃	#11961
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #22611730	C₂	semidirect
2. #22608690	C₂	semidirect
3. #22607977	C₂	semidirect
6. C₄⋊#11924003	C₂²	semidirect
10. #17988340	C₂²	semidirect
130. C₇₃	#11961	semidirect

Maximal subgroups

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

subgroup	index
#22611730	2
#22608690	2
#22607977	2
C₂×#493×C₇₃	2
#11961	73