Subgroups — #34004013

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556660	C₂	1	1	yes	maximal, characteristic
2	#22625728	C₂	1	2	yes	maximal
3	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
4	#22624765	C₂	1	2	yes	maximal
5	#335×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556660	C₄	1	2	yes	—
7	C₂×#14556645	—	2	2	no	—
8	C₄×#11924036	C₂²	1	1	yes	characteristic
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊#64408	C₂²	1	2	yes	—
11	#17989461	C₄	1	2	yes	—
12	#17989452	—	2	2	no	—
13	C₈×#10596833	C₄	1	2	yes	complemented
14	C₈×#10596833	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988340	C₄	1	1	yes	characteristic, complemented
18	#17988339	C₄	1	2	yes	complemented
19	#335×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂×#11924036	—	2	1	no	—
21	C₂×#11924036	D₄	1	1	yes	characteristic
22	C₂×#11924036	D₄	1	1	yes	characteristic
23	C₂×C₄⋊#64408	C₂×C₄	1	4	yes	—
24	C₂×C₄×#64408	—	2	2	no	—
25	#14556660	—	2	4	no	—
26	#14556645	—	4	4	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	C₂×C₄	1	2	yes	—
29	C₄×#10596833	C₂×C₄	1	2	yes	—
30	C₄×#10596833	Q₈	1	2	yes	—
31	(C₂×#64408)⋊C₄	—	2	2	no	—
32	C₂×C₈×D₇₃	—	2	2	no	—
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊#10596831	C₂×C₄	1	2	yes	—
37	#14556520	—	2	2	no	—
38	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#64408	—	2	4	no	—
44	#11924036	—	2	4	no	—
45	#11924036	—	2	1	no	—
46	#11924036	—	2	1	no	—
47	#11924036	D₈	1	2	yes	—
48	#11924036	Q₁₆	1	2	yes	—
49	C₂×#10596833	—	2	2	no	—
50	C₂×#10596833	C₄⋊C₄	1	2	yes	—
51	C₂×#10596833	C₄⋊C₄	1	2	yes	—
52	C₄⋊#64408	—	2	4	no	—
53	C₄⋊#64408	—	2	4	no	—
54	C₄×#64408	—	4	2	no	—
55	C₄×#64408	—	4	2	no	—
56	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
59	C₂×#10596831	—	2	2	no	—
60	C₂×#10596831	C₄²	1	1	yes	characteristic
61	C₂×#10596831	C₄²	1	1	yes	characteristic
62	C₈×D₇₃	—	2	2	no	—
63	C₈×D₇₃	—	4	2	no	—
64	C₈×D₇₃	—	4	2	no	—
65	C₈×D₇₃	M₁₆	1	4	yes	—
66	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₇₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
72	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
74	C₂×#64408	—	4	4	no	—
75	C₂×#64408	—	4	4	no	—
76	C₂×#64408	—	4	4	no	—
77	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×Dic₇₃	#117	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
83	C₄×D₇₃	—	2	2	no	—
84	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
85	C₄×D₇₃	#131	1	2	yes	—
86	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
87	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₄×D₇₃	#135	1	1	yes	characteristic
89	C₄×D₇₃	#135	1	1	yes	characteristic
90	#10596833	—	2	4	no	—
91	#10596833	—	2	2	no	—
92	#10596833	#107	1	4	yes	complemented
93	C₈×C₇₃	—	2	1	no	—
94	C₈×C₇₃	—	4	2	no	—
95	C₈×C₇₃	#97	1	2	yes	complemented
96	#10596831	—	2	1	no	—
97	#10596831	—	4	2	no	—
98	#10596831	#97	1	2	yes	complemented
99	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂×D₇₃	#436	1	2	yes	—
101	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
102	C₂×D₇₃	C₄×Q₁₆	1	1	yes	characteristic
103	C₂×D₇₃	#443	1	2	yes	—
104	#64408	—	8	4	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
107	C₄×C₇₃	#405	1	1	yes	characteristic
108	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
109	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	C₂×#107	1	1	yes	characteristic
112	Dic₇₃	#427	1	1	yes	characteristic
113	Dic₇₃	#446	1	2	yes	—
114	#8282	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	#1770	1	1	yes	characteristic
116	C₂×C₇₃	#1772	1	1	yes	characteristic
117	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
118	D₇₃	#1589	1	2	yes	—
119	D₇₃	#1591	1	2	yes	—
120	C₄×#107	—	73	1	no	—
121	#1770	—	73	2	no	—
122	C₂×#335	—	73	1	no	—
123	C₂×C₈²	—	73	1	no	—
124	#1279	—	73	2	no	—
125	C₇₃	#8282	1	1	yes	characteristic, complemented, Sylow
126	C₂×#107	—	73	2	no	—
127	C₂×C₄⋊C₈	—	73	2	no	—
128	C₂×#97	—	146	2	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₄×C₄⋊C₄	—	73	2	no	—
133	#340	—	73	2	no	—
134	#336	—	146	2	no	—
135	#335	—	73	2	no	—
136	#335	—	73	1	no	—
137	#335	#64408	1	1	yes	characteristic, complemented
138	C₈²	—	73	2	no	—
139	C₈²	—	73	2	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂²×C₈	—	73	1	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	73	1	no	—
144	C₂²×C₈	—	146	2	no	—
145	C₂²×C₈	—	146	1	no	—
146	C₂×C₄⋊C₄	—	73	4	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₂×C₄²	—	146	2	no	—
149	#107	—	146	4	no	—
150	C₄⋊C₈	—	73	2	no	—
151	C₄⋊C₈	—	146	2	no	—
152	C₄⋊C₈	C₂×#64408	1	2	yes	—
153	#97	—	292	4	no	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	73	2	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	73	1	no	—
158	C₄×C₈	—	73	2	no	—
159	C₄×C₈	—	73	1	no	—
160	C₄×C₈	—	146	1	no	—
161	C₄×C₈	C₂×#64408	1	1	yes	characteristic
162	#95	—	146	2	no	—
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	73	1	no	—
165	C₂²×C₄	—	73	1	no	—
166	C₂²×C₄	—	146	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	292	2	no	—
169	C₂×C₈	—	292	2	no	—
170	C₂×C₈	—	73	2	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	4	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	1	no	—
177	C₂×C₈	—	146	2	no	—
178	C₂×C₈	—	146	2	no	—
179	C₂×C₈	—	146	2	no	—
180	C₂×C₈	—	146	1	no	—
181	C₂×C₈	—	146	1	no	—
182	C₂×C₈	—	146	4	no	—
183	C₂×C₈	C₄×#64408	1	1	yes	characteristic
184	C₂×C₈	C₄×#64408	1	1	yes	characteristic
185	C₄⋊C₄	—	146	4	no	—
186	C₄⋊C₄	—	146	4	no	—
187	C₄²	—	292	2	no	—
188	C₄²	—	292	2	no	—
189	C₄²	—	73	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	1	no	—
192	C₄²	C₂²×#64408	1	1	yes	characteristic
193	C₂³	—	73	1	no	—
194	C₂×C₄	—	292	4	no	—
195	C₂×C₄	—	292	4	no	—
196	C₂×C₄	—	292	4	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	2	no	—
200	C₂×C₄	—	73	2	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	1	no	—
203	C₂×C₄	—	73	1	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	146	2	no	—
207	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
208	C₂×C₄	D₄×#64408	1	1	yes	characteristic
209	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	292	2	no	—
213	C₈	—	73	4	no	—
214	C₈	—	73	2	no	—
215	C₈	—	146	2	no	—
216	C₈	—	146	4	no	—
217	C₈	—	146	1	no	—
218	C₈	#14556645	1	2	yes	complemented
219	C₂²	—	73	2	no	—
220	C₂²	—	73	2	no	—
221	C₂²	—	73	1	no	—
222	C₂²	—	73	1	no	—
223	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	584	4	no	—
226	C₄	—	73	2	no	—
227	C₄	—	73	1	no	—
228	C₄	—	73	1	no	—
229	C₄	—	146	1	no	—
230	C₄	C₂×#14556645	1	1	yes	characteristic
231	C₄	#17989553	1	1	yes	characteristic
232	C₄	D₈×#64408	1	1	yes	characteristic
233	C₄	Q₁₆×#64408	1	1	yes	characteristic
234	C₂	—	73	2	no	—
235	C₂	—	73	2	no	—
236	C₂	#22625728	1	1	yes	characteristic
237	C₂	#22625736	1	1	yes	characteristic
238	C₂	#107×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄²×D₇₃	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	#8282
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
13. C₈×#10596833	C₄	semidirect
14. C₈×#10596833	C₄	semidirect
17. #17988340	C₄	semidirect
18. #17988339	C₄	semidirect
19. #335×C₇₃	C₄	semidirect
92. #10596833	#107	semidirect
95. C₈×C₇₃	#97	semidirect
98. #10596831	#97	semidirect
125. C₇₃	#8282	semidirect
137. #335	#64408	semidirect
218. C₈	#14556645	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556660	2
#22625728	2
C₈×#11924036	2
#22624765	2
#335×D₇₃	2
#8282	73