Subgroups — #34003986

Order 18688.

446 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10596833	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989448	C₂	1	1	yes	maximal, characteristic
4	#22624761	C₂	1	2	yes	maximal
5	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	—	2	2	no	—
9	#17989453	—	2	2	no	—
10	#17989450	C₄	1	4	yes	complemented
11	#17989448	C₄	1	4	yes	complemented
12	C₂×#14556502	C₄	1	1	yes	characteristic
13	C₂×#14556501	C₄	1	1	yes	characteristic
14	C₂×#14556501	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	—	2	2	no	—
20	C₂²×#10596833	—	2	2	no	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	C₂×C₄	1	4	yes	—
24	C₄⋊#10596833	C₂×C₄	1	4	yes	—
25	C₄×#10596833	—	4	4	no	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556502	—	2	2	no	—
33	#14556501	—	2	2	no	—
34	#14556501	C₈	1	4	yes	complemented
35	C₈×Dic₇₃	—	2	1	no	—
36	C₈×Dic₇₃	—	2	1	no	—
37	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
38	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	#97×C₇₃	C₈	1	4	yes	complemented
43	C₂×#10596833	—	2	4	no	—
44	C₂×#10596833	—	2	4	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	—	4	2	no	—
47	C₂×#10596833	—	4	2	no	—
48	C₂×#10596833	—	4	2	no	—
49	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
50	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	—	2	1	no	—
55	C₂×#10596831	—	2	1	no	—
56	C₂×#10596831	C₄²	1	2	yes	—
57	C₂×#10596831	C₂×C₈	1	2	yes	—
58	C₄×Dic₇₃	—	2	2	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	—	4	2	no	—
61	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
62	C₄×Dic₇₃	M₁₆	1	2	yes	—
63	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₇₃	#41	1	1	yes	characteristic
66	C₄×Dic₇₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	—	2	1	no	—
70	C₂×C₈×C₇₃	—	2	1	no	—
71	C₂×C₈×C₇₃	C₄²	1	2	yes	—
72	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
75	C₄²×C₇₃	C₂×C₈	1	2	yes	—
76	C₄²×C₇₃	M₁₆	1	2	yes	—
77	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×Dic₇₃	—	2	2	no	—
88	C₂×Dic₇₃	—	2	2	no	—
89	C₂×Dic₇₃	—	2	2	no	—
90	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
91	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
95	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×Dic₇₃	#137	1	2	yes	—
97	#10596833	—	4	4	no	—
98	#10596833	—	4	4	no	—
99	C₈×C₇₃	—	2	2	no	—
100	C₈×C₇₃	C₄×C₈	1	4	yes	complemented
101	#10596831	—	2	2	no	—
102	#10596831	—	2	2	no	—
103	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₇₃	#428	1	2	yes	—
107	C₂²×C₇₃	#430	1	2	yes	—
108	C₄×C₇₃	—	2	4	no	—
109	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₇₃	#431	1	1	yes	characteristic
112	C₄×C₇₃	#431	1	1	yes	characteristic
113	Dic₇₃	—	2	4	no	—
114	Dic₇₃	—	2	1	no	—
115	Dic₇₃	—	2	1	no	—
116	Dic₇₃	#440	1	2	yes	—
117	Dic₇₃	#441	1	1	yes	characteristic
118	Dic₇₃	#442	1	1	yes	characteristic
119	#8275	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	#1565	1	2	yes	—
121	C₂×C₇₃	#1565	1	2	yes	—
122	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
123	C₂×C₇₃	#1776	1	1	yes	characteristic
124	C₂×C₇₃	#1778	1	1	yes	characteristic
125	C₄×C₄⋊C₈	—	73	2	no	—
126	C₄×#97	—	73	1	no	—
127	C₂×#335	—	73	1	no	—
128	C₂×#334	—	73	1	no	—
129	#1277	—	73	2	no	—
130	C₇₃	#8275	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊C₈	—	73	2	no	—
132	C₂×C₄⋊C₈	—	73	2	no	—
133	C₂×#97	—	73	1	no	—
134	C₂×#97	—	73	1	no	—
135	C₂×#97	—	73	1	no	—
136	C₂×#97	#64408	1	1	yes	characteristic
137	C₂×C₄×C₈	—	73	1	no	—
138	C₂×C₄×C₈	—	73	1	no	—
139	C₂×C₄×C₈	—	146	2	no	—
140	C₄³	—	73	1	no	—
141	#336	—	146	2	no	—
142	#335	—	73	4	no	—
143	#334	—	73	4	no	—
144	C₂²×C₈	—	73	1	no	—
145	C₂²×C₈	—	73	1	no	—
146	C₂²×C₈	—	146	2	no	—
147	C₂²×C₈	—	146	2	no	—
148	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
149	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
150	C₂×C₄²	—	73	2	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₄²	—	73	2	no	—
154	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
155	C₄⋊C₈	—	73	4	no	—
156	C₄⋊C₈	—	73	4	no	—
157	C₄⋊C₈	—	146	2	no	—
158	C₄⋊C₈	—	146	2	no	—
159	#97	—	73	4	no	—
160	#97	—	146	2	no	—
161	#97	—	146	2	no	—
162	#97	#10596833	1	4	yes	complemented
163	C₄×C₈	—	292	4	no	—
164	C₄×C₈	—	73	2	no	—
165	C₄×C₈	—	73	2	no	—
166	C₄×C₈	—	146	1	no	—
167	C₄×C₈	—	146	1	no	—
168	C₂²×C₄	—	73	1	no	—
169	C₂²×C₄	—	73	2	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	C₄⋊#64408	1	2	yes	—
172	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	292	2	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	292	2	no	—
178	C₂×C₈	—	292	2	no	—
179	C₂×C₈	—	73	2	no	—
180	C₂×C₈	—	73	2	no	—
181	C₂×C₈	—	146	4	no	—
182	C₂×C₈	—	146	4	no	—
183	C₂×C₈	—	146	1	no	—
184	C₂×C₈	—	146	1	no	—
185	C₂×C₈	—	146	1	no	—
186	C₂×C₈	—	146	1	no	—
187	C₂×C₈	C₄×#64408	1	2	yes	—
188	C₂×C₈	C₂×#10596833	1	2	yes	—
189	C₂×C₈	C₂×#10596833	1	2	yes	—
190	C₂×C₈	#11924036	1	2	yes	—
191	C₄²	—	292	2	no	—
192	C₄²	—	73	2	no	—
193	C₄²	—	73	2	no	—
194	C₄²	—	73	1	no	—
195	C₄²	—	73	1	no	—
196	C₄²	—	73	1	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	146	2	no	—
199	C₄²	—	146	2	no	—
200	C₄²	C₂×#10596833	1	2	yes	—
201	C₄²	#11924038	1	2	yes	—
202	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	73	1	no	—
207	C₂×C₄	—	73	2	no	—
208	C₂×C₄	—	73	2	no	—
209	C₂×C₄	—	73	1	no	—
210	C₂×C₄	—	73	1	no	—
211	C₂×C₄	—	73	4	no	—
212	C₂×C₄	—	146	2	no	—
213	C₂×C₄	—	146	2	no	—
214	C₂×C₄	—	146	2	no	—
215	C₂×C₄	C₄⋊#10596833	1	4	yes	—
216	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
217	C₂×C₄	#14556670	1	1	yes	characteristic
218	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
219	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
220	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
221	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	292	4	no	—
224	C₈	—	292	4	no	—
225	C₈	—	146	2	no	—
226	C₈	—	146	2	no	—
227	C₈	C₄×#10596833	1	4	yes	complemented
228	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
229	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
230	C₂²	#17989588	1	1	yes	characteristic
231	C₂²	#17989598	1	2	yes	—
232	C₂²	#17989602	1	2	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	73	1	no	—
235	C₄	—	73	2	no	—
236	C₄	—	73	1	no	—
237	C₄	—	146	4	no	—
238	C₄	—	146	1	no	—
239	C₄	—	146	1	no	—
240	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
241	C₄	C₄×#11924038	1	1	yes	characteristic
242	C₄	#17989603	1	1	yes	characteristic
243	C₄	#17989606	1	1	yes	characteristic
244	C₂	#22625482	1	2	yes	—
245	C₂	#22625483	1	2	yes	—
246	C₂	C₄×C₄⋊#10596833	1	1	yes	characteristic
247	C₂	#22625745	1	1	yes	characteristic
248	C₂	#22625751	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#8275
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
10. #17989450	C₄	semidirect
11. #17989448	C₄	semidirect
34. #14556501	C₈	semidirect
42. #97×C₇₃	C₈	semidirect
100. C₈×C₇₃	C₄×C₈	semidirect
130. C₇₃	#8275	semidirect
162. #97	#10596833	semidirect
227. C₈	C₄×#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596833	2
C₂×#17989450	2
C₂×#17989448	2
#22624761	2
#97×Dic₇₃	2
#8275	73