Subgroups — #34033283

Order 18688.

502 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	C₄×C₄⋊#10596842	C₂	1	2	yes	maximal
2	C₂×#17989671	C₂	1	2	yes	maximal
3	(C₂×#10596842)⋊C₈	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596842	C₄	1	4	yes	—
6	C₂×C₄×#10596842	C₂²	1	4	yes	—
7	(C₂×#10596842)⋊C₄	C₄	1	4	yes	—
8	#17989671	—	2	4	no	—
9	C₄×#11924036	C₄	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	C₄²×#64408	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
16	C₂²×#10596842	C₂×C₄	1	8	yes	—
17	C₄⋊#10596842	—	2	8	no	—
18	C₄×#10596842	—	2	4	no	—
19	C₄×#10596842	—	2	4	no	—
20	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
23	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
24	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
26	C₄×#10596833	C₈	1	2	yes	complemented
27	C₄×#10596833	C₈	1	2	yes	complemented
28	C₈×#64408	—	2	2	no	—
29	C₈×#64408	—	2	2	no	—
30	C₈×#64408	—	2	2	no	—
31	C₈×#64408	D₄	1	2	yes	—
32	C₈×#64408	Q₈	1	2	yes	—
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×#10596831	C₈	1	1	yes	characteristic, complemented
37	C₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
38	C₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
39	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
40	C₂×#10596842	—	2	8	no	—
41	C₂×#10596842	—	2	8	no	—
42	C₂×#10596842	—	2	8	no	—
43	C₂²×#64408	C₄²	1	2	yes	—
44	C₂²×#64408	C₂²×C₄	1	2	yes	—
45	#11924036	—	2	1	no	—
46	#11924036	—	2	1	no	—
47	#11924036	C₄⋊C₄	1	2	yes	—
48	#11924036	M₁₆	1	2	yes	—
49	C₂×#10596833	C₂×C₈	1	2	yes	—
50	C₂×#10596833	C₂×C₈	1	2	yes	—
51	C₄×#64408	—	2	4	no	—
52	C₄×#64408	—	2	1	no	—
53	C₄×#64408	—	2	1	no	—
54	C₄×#64408	M₁₆	1	8	yes	—
55	C₄×#64408	C₂×D₄	1	1	yes	characteristic
56	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
57	C₄×#64408	#41	1	2	yes	—
58	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
60	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
62	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
63	C₈×D₇₃	—	2	1	no	—
64	C₈×D₇₃	—	2	1	no	—
65	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
66	C₈×D₇₃	M₁₆	1	2	yes	—
67	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
68	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
69	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×#64408	—	2	2	no	—
75	C₂×#64408	—	2	2	no	—
76	C₂×#64408	#97	1	4	yes	—
77	C₂×#64408	C₄×D₄	1	2	yes	—
78	C₂×#64408	C₄×Q₈	1	2	yes	—
79	C₂×#64408	C₂×M₁₆	1	4	yes	—
80	C₂×#64408	#131	1	4	yes	—
81	#10596842	—	4	8	no	—
82	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
86	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₄×D₇₃	—	2	2	no	—
89	C₄×D₇₃	#97	1	4	yes	—
90	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×D₇₃	#117	1	1	yes	characteristic
92	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
93	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
94	#10596833	—	2	2	no	—
95	#10596833	C₄⋊C₈	1	4	yes	complemented
96	C₈×C₇₃	—	2	1	no	—
97	C₈×C₇₃	C₄⋊C₈	1	2	yes	complemented
98	#10596831	—	2	1	no	—
99	#10596831	C₄⋊C₈	1	2	yes	complemented
100	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
101	C₂×D₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₇₃	C₂×#97	1	2	yes	—
103	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
104	C₂×D₇₃	#405	1	2	yes	—
105	#64408	—	2	4	no	—
106	#64408	#436	1	4	yes	—
107	#64408	#446	1	4	yes	—
108	C₄×C₇₃	—	2	1	no	—
109	C₄×C₇₃	#322	1	2	yes	complemented
110	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₇₃	#431	1	1	yes	characteristic
112	Dic₇₃	—	2	1	no	—
113	Dic₇₃	#322	1	2	yes	complemented
114	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	Dic₇₃	#431	1	1	yes	characteristic
116	#8260	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
118	C₂×C₇₃	#1450	1	1	yes	characteristic
119	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
120	D₇₃	#1451	1	2	yes	—
121	D₇₃	#1770	1	2	yes	—
122	C₄×C₄⋊C₈	—	73	2	no	—
123	C₄²×C₈	—	73	1	no	—
124	C₂×#322	—	73	2	no	—
125	#1273	—	73	2	no	—
126	C₇₃	#8260	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	4	no	—
132	C₂×C₄×C₈	—	73	1	no	—
133	C₂×C₄×C₈	—	73	1	no	—
134	C₂×C₄×C₈	—	73	1	no	—
135	C₄³	—	73	1	no	—
136	#336	—	73	4	no	—
137	#322	—	146	4	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂²×C₈	—	73	1	no	—
142	C₂²×C₈	—	73	8	no	—
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	73	2	no	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	73	2	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₄⋊C₈	—	146	8	no	—
149	C₄×C₈	—	73	2	no	—
150	C₄×C₈	—	73	2	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	73	1	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	146	4	no	—
157	C₄×C₈	—	146	4	no	—
158	C₄×C₈	—	146	2	no	—
159	C₄×C₈	—	146	2	no	—
160	C₄×C₈	—	146	2	no	—
161	C₄×C₈	#10596842	1	1	yes	characteristic, complemented
162	C₂²×C₄	—	73	2	no	—
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	73	1	no	—
165	C₂²×C₄	—	73	2	no	—
166	C₂²×C₄	—	73	1	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	2	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	1	no	—
177	C₂×C₈	—	146	8	no	—
178	C₂×C₈	—	146	8	no	—
179	C₂×C₈	—	146	8	no	—
180	C₂×C₈	—	146	1	no	—
181	C₂×C₈	—	146	1	no	—
182	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
183	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
184	C₄²	—	73	8	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	2	no	—
188	C₄²	—	73	1	no	—
189	C₄²	—	73	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	146	1	no	—
192	C₄²	—	146	1	no	—
193	C₄²	—	146	4	no	—
194	C₄²	C₂×#10596842	1	1	yes	characteristic
195	C₂³	—	73	1	no	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	73	4	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	4	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	4	no	—
203	C₂×C₄	—	73	1	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	73	1	no	—
207	C₂×C₄	—	73	4	no	—
208	C₂×C₄	—	73	1	no	—
209	C₂×C₄	—	146	2	no	—
210	C₂×C₄	—	146	2	no	—
211	C₂×C₄	—	146	2	no	—
212	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
213	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
214	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	292	8	no	—
217	C₈	—	73	4	no	—
218	C₈	—	73	2	no	—
219	C₈	—	146	2	no	—
220	C₈	—	146	1	no	—
221	C₈	C₄⋊#10596842	1	2	yes	complemented
222	C₂²	—	73	2	no	—
223	C₂²	—	73	2	no	—
224	C₂²	—	73	1	no	—
225	C₂²	—	73	1	no	—
226	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	73	4	no	—
229	C₄	—	73	4	no	—
230	C₄	—	73	2	no	—
231	C₄	—	73	1	no	—
232	C₄	—	73	1	no	—
233	C₄	—	146	4	no	—
234	C₄	—	146	1	no	—
235	C₄	#17989671	1	2	yes	complemented
236	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
237	C₄	#17989700	1	1	yes	characteristic
238	C₂	—	73	2	no	—
239	C₂	—	73	2	no	—
240	C₂	C₂×#17989671	1	1	yes	characteristic
241	C₂	#22627040	1	1	yes	characteristic
242	C₂	C₄×C₄⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄×C₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂×C₄²
cosocle	C₂×C₄×#64408	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#8260
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
26. C₄×#10596833	C₈	semidirect
27. C₄×#10596833	C₈	semidirect
36. C₄×#10596831	C₈	semidirect
37. C₈×Dic₇₃	C₈	semidirect
38. C₈×Dic₇₃	C₈	semidirect
39. C₄×C₈×C₇₃	C₈	semidirect
95. #10596833	C₄⋊C₈	semidirect
97. C₈×C₇₃	C₄⋊C₈	semidirect
99. #10596831	C₄⋊C₈	semidirect
109. C₄×C₇₃	#322	semidirect
113. Dic₇₃	#322	semidirect
126. C₇₃	#8260	semidirect
161. C₄×C₈	#10596842	semidirect
221. C₈	C₄⋊#10596842	semidirect
235. C₄	#17989671	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596842	2
C₂×#17989671	2
(C₂×#10596842)⋊C₈	2
C₄×C₈×#64408	2
#8260	73