Subgroups — #34005424

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625848	C₂	1	2	yes	maximal
2	#22624761	C₂	1	2	yes	maximal
3	#22624761	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596833	C₂²	1	4	yes	—
6	#17989453	—	2	2	no	—
7	#17989453	—	2	2	no	—
8	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596833	—	2	4	no	—
16	C₄⋊#10596833	—	2	4	no	—
17	C₄⋊#10596833	—	2	4	no	—
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
26	C₄×#10596831	—	2	2	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₈×Dic₇₃	—	2	2	no	—
29	C₈×Dic₇₃	—	2	2	no	—
30	C₈×Dic₇₃	C₈	1	4	yes	complemented
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	—	2	2	no	—
35	C₂×#10596833	—	4	4	no	—
36	C₂×#10596833	—	4	4	no	—
37	C₂×#10596833	—	4	4	no	—
38	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	—	2	1	no	—
45	C₂×#10596831	—	2	1	no	—
46	C₂×#10596831	C₂×C₈	1	2	yes	—
47	C₂×#10596831	M₁₆	1	2	yes	—
48	C₄×Dic₇₃	—	2	2	no	—
49	C₄×Dic₇₃	—	2	2	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	—	2	2	no	—
52	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
53	C₄×Dic₇₃	M₁₆	1	2	yes	—
54	C₄×Dic₇₃	#41	1	2	yes	—
55	C₄×Dic₇₃	#41	1	2	yes	—
56	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	—	2	1	no	—
60	C₂×C₈×C₇₃	—	2	1	no	—
61	C₂×C₈×C₇₃	—	2	1	no	—
62	C₂×C₈×C₇₃	—	2	1	no	—
63	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
65	C₄²×C₇₃	—	2	1	no	—
66	C₄²×C₇₃	—	2	1	no	—
67	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	#108	1	2	yes	—
73	C₂×C₄×C₇₃	#108	1	2	yes	—
74	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	—	2	1	no	—
82	C₂×Dic₇₃	—	2	1	no	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	—	2	1	no	—
86	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
87	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
88	C₂×Dic₇₃	#124	1	2	yes	—
89	C₂×Dic₇₃	#125	1	2	yes	—
90	C₂×Dic₇₃	#135	1	2	yes	—
91	C₂×Dic₇₃	#135	1	2	yes	—
92	#10596833	—	8	4	no	—
93	C₈×C₇₃	—	2	2	no	—
94	C₈×C₇₃	—	2	2	no	—
95	#10596831	—	2	2	no	—
96	#10596831	—	2	2	no	—
97	C₂²×C₇₃	#382	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₇₃	#423	1	1	yes	characteristic
100	C₂²×C₇₃	#427	1	1	yes	characteristic
101	C₂²×C₇₃	#427	1	1	yes	characteristic
102	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
103	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
104	C₄×C₇₃	—	2	2	no	—
105	C₄×C₇₃	—	2	2	no	—
106	C₄×C₇₃	#431	1	1	yes	characteristic
107	C₄×C₇₃	#431	1	1	yes	characteristic
108	C₄×C₇₃	#432	1	1	yes	characteristic
109	C₄×C₇₃	#432	1	1	yes	characteristic
110	Dic₇₃	—	2	2	no	—
111	Dic₇₃	—	2	2	no	—
112	Dic₇₃	—	2	2	no	—
113	Dic₇₃	#488	1	4	yes	—
114	#8785	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	#1564	1	1	yes	characteristic
116	C₂×C₇₃	#1564	1	1	yes	characteristic
117	C₂×C₇₃	#1568	1	1	yes	characteristic
118	C₂×C₇₃	#1568	1	1	yes	characteristic
119	C₂×C₇₃	#1831	1	1	yes	characteristic
120	C₂×C₇₃	#1835	1	1	yes	characteristic
121	C₂×C₇₃	#1837	1	1	yes	characteristic
122	#1831	—	73	2	no	—
123	C₄²×C₈	—	73	1	no	—
124	#1277	—	73	2	no	—
125	#1277	—	73	2	no	—
126	C₇₃	#8785	1	1	yes	characteristic, complemented, Sylow
127	C₂×C₄⋊C₈	—	73	4	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₂×C₄×C₈	—	73	1	no	—
133	C₂×C₄×C₈	#64408	1	1	yes	characteristic
134	C₄³	—	73	1	no	—
135	#336	—	146	2	no	—
136	#336	—	146	2	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	146	4	no	—
140	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
141	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
142	C₂×C₄²	—	73	1	no	—
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	73	1	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
149	C₄⋊C₈	—	146	4	no	—
150	C₄⋊C₈	—	146	4	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	73	4	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	—	146	2	no	—
155	C₄×C₈	—	146	2	no	—
156	C₄×C₈	—	146	2	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
163	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	292	4	no	—
169	C₂×C₈	—	292	4	no	—
170	C₂×C₈	—	292	4	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	146	1	no	—
174	C₂×C₈	—	146	1	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	1	no	—
177	C₂×C₈	#11924036	1	2	yes	—
178	C₂×C₈	#11924037	1	2	yes	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	73	2	no	—
182	C₄²	—	73	2	no	—
183	C₄²	—	73	2	no	—
184	C₄²	—	73	2	no	—
185	C₄²	—	146	2	no	—
186	C₄²	—	146	2	no	—
187	C₄²	—	146	2	no	—
188	C₄²	—	146	2	no	—
189	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	73	2	no	—
195	C₂×C₄	—	73	2	no	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	73	2	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	2	no	—
200	C₂×C₄	—	146	1	no	—
201	C₂×C₄	—	146	1	no	—
202	C₂×C₄	—	146	1	no	—
203	C₂×C₄	—	146	1	no	—
204	C₂×C₄	—	146	1	no	—
205	C₂×C₄	—	146	1	no	—
206	C₂×C₄	#14556661	1	2	yes	—
207	C₂×C₄	#14556662	1	2	yes	—
208	C₂×C₄	#14556670	1	1	yes	characteristic
209	C₂×C₄	#14556670	1	1	yes	characteristic
210	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
211	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
212	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
213	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	584	4	no	—
217	C₈	—	146	2	no	—
218	C₈	—	146	2	no	—
219	C₂²	#17989531	1	1	yes	characteristic
220	C₂²	#17989588	1	1	yes	characteristic
221	C₂²	#17989590	1	1	yes	characteristic
222	C₂²	#17989597	1	1	yes	characteristic
223	C₂²	#17989597	1	1	yes	characteristic
224	C₂²	C₂×#14556661	1	1	yes	characteristic
225	C₂²	C₂×#14556662	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	73	4	no	—
229	C₄	—	146	2	no	—
230	C₄	—	146	2	no	—
231	C₄	—	146	2	no	—
232	C₄	#17989604	1	1	yes	characteristic
233	C₄	#17989605	1	1	yes	characteristic
234	C₄	#17989607	1	1	yes	characteristic
235	C₄	#17989610	1	1	yes	characteristic
236	C₂	#22625479	1	1	yes	characteristic
237	C₂	#22625481	1	1	yes	characteristic
238	C₂	#22625491	1	1	yes	characteristic
239	C₂	#22625493	1	1	yes	characteristic
240	C₂	#22625847	1	1	yes	characteristic
241	C₂	#22625853	1	1	yes	characteristic
242	C₂	#22625855	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#8785
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
30. C₈×Dic₇₃	C₈	semidirect
126. C₇₃	#8785	semidirect

Maximal subgroups

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

subgroup	index
#22625848	2
#22624761	2
C₄×C₈×Dic₇₃	2
#8785	73