Subgroups — #34033348

Order 18688.

422 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	#22627073	C₂	1	2	yes	maximal
2	C₂×#17989677	C₂	1	2	yes	maximal
3	C₂×#17989676	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596842	C₂²	1	4	yes	—
6	(C₂×#10596842)⋊C₄	—	2	4	no	—
7	#17989677	—	2	4	no	—
8	#17989676	—	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	—	2	4	no	—
17	C₄⋊#10596842	—	2	4	no	—
18	C₄⋊#10596842	—	2	4	no	—
19	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×#64408	D₄	1	2	yes	—
23	C₂×C₄×#64408	Q₈	1	2	yes	—
24	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
25	C₄×#10596833	—	2	2	no	—
26	C₈×#64408	—	2	2	no	—
27	C₈×#64408	—	2	2	no	—
28	C₈×#64408	D₄	1	2	yes	—
29	C₈×#64408	D₄	1	2	yes	—
30	C₈×#64408	Q₈	1	2	yes	—
31	C₈×#64408	Q₈	1	2	yes	—
32	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596831	C₈	1	1	yes	characteristic, complemented
36	C₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
37	C₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
38	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
39	C₂×#10596842	—	4	4	no	—
40	C₂×#10596842	—	4	8	no	—
41	C₂²×#64408	C₄⋊C₄	1	2	yes	—
42	C₂²×#64408	C₂×D₄	1	1	yes	characteristic
43	C₂²×#64408	C₂×Q₈	1	1	yes	characteristic
44	#11924036	—	2	1	no	—
45	#11924036	—	2	1	no	—
46	#11924036	C₄⋊C₄	1	2	yes	—
47	#11924036	C₄⋊C₄	1	2	yes	—
48	C₂×#10596833	—	2	1	no	—
49	C₂×#10596833	—	2	1	no	—
50	C₄×#64408	—	2	4	no	—
51	C₄×#64408	—	2	4	no	—
52	C₄×#64408	—	2	1	no	—
53	C₄×#64408	—	2	1	no	—
54	C₄×#64408	C₂×D₄	1	1	yes	characteristic
55	C₄×#64408	C₂×D₄	1	1	yes	characteristic
56	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
57	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
61	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
62	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
63	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
64	C₈×D₇₃	M₁₆	1	2	yes	—
65	C₈×D₇₃	M₁₆	1	2	yes	—
66	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
67	C₄×Dic₇₃	M₁₆	1	2	yes	—
68	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×#64408	—	2	2	no	—
73	C₂×#64408	—	2	2	no	—
74	C₂×#64408	—	2	2	no	—
75	C₂×#64408	#108	1	4	yes	—
76	C₂×#64408	#127	1	1	yes	characteristic
77	C₂×#64408	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×#64408	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#64408	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×#64408	#135	1	4	yes	—
81	#10596842	—	8	4	no	—
82	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
83	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
85	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
86	C₄×D₇₃	—	2	2	no	—
87	C₄×D₇₃	—	2	2	no	—
88	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
92	#10596833	—	2	2	no	—
93	#10596833	—	2	2	no	—
94	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
95	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
96	#10596831	C₄⋊C₈	1	2	yes	—
97	#10596831	C₄⋊C₈	1	2	yes	—
98	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₂×D₇₃	#384	1	1	yes	characteristic
100	C₂×D₇₃	#423	1	1	yes	characteristic
101	C₂×D₇₃	#427	1	2	yes	—
102	C₂×D₇₃	C₂×#108	1	2	yes	—
103	#64408	—	2	4	no	—
104	#64408	—	2	1	no	—
105	#64408	—	2	1	no	—
106	#64408	#495	1	2	yes	—
107	#64408	#499	1	2	yes	—
108	C₄×C₇₃	—	2	2	no	—
109	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	Dic₇₃	—	2	2	no	—
112	Dic₇₃	#423	1	1	yes	characteristic
113	Dic₇₃	#423	1	1	yes	characteristic
114	#8807	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	#1564	1	2	yes	—
116	C₂×C₇₃	#1842	1	1	yes	characteristic
117	D₇₃	#1568	1	2	yes	—
118	D₇₃	#1846	1	1	yes	characteristic
119	D₇₃	#1848	1	1	yes	characteristic
120	#1831	—	73	2	no	—
121	C₄²×C₈	—	73	1	no	—
122	C₂×#335	—	73	2	no	—
123	C₂×#334	—	73	2	no	—
124	C₇₃	#8807	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊C₈	—	73	4	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	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	C₄³	—	73	1	no	—
133	#336	—	146	4	no	—
134	#335	—	146	4	no	—
135	#334	—	146	4	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	146	4	no	—
141	C₂×C₄²	—	73	1	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₂×C₄²	—	73	2	no	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	73	2	no	—
146	C₄⋊C₈	—	146	4	no	—
147	C₄⋊C₈	—	146	4	no	—
148	C₄×C₈	—	73	2	no	—
149	C₄×C₈	—	73	2	no	—
150	C₄×C₈	—	73	2	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	1	no	—
153	C₄×C₈	—	73	1	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	146	2	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	#10596842	1	1	yes	characteristic, complemented
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	2	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	—	73	2	no	—
164	C₂×C₈	—	292	4	no	—
165	C₂×C₈	—	292	8	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	73	2	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	2	no	—
174	C₂×C₈	—	146	1	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	1	no	—
177	C₂×C₈	—	146	1	no	—
178	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
179	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	73	1	no	—
183	C₄²	—	73	1	no	—
184	C₄²	—	73	2	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	146	1	no	—
187	C₄²	—	146	1	no	—
188	C₄²	—	146	4	no	—
189	C₄²	—	146	4	no	—
190	C₄²	C₂×#10596842	1	1	yes	characteristic
191	C₂³	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	4	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	4	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	—	146	2	no	—
205	C₂×C₄	—	146	2	no	—
206	C₂×C₄	—	146	2	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	—	146	2	no	—
209	C₂×C₄	C₄⋊#10596842	1	2	yes	—
210	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
211	C₈	—	584	4	no	—
212	C₈	—	73	2	no	—
213	C₈	—	73	2	no	—
214	C₈	—	146	2	no	—
215	C₈	—	146	2	no	—
216	C₈	C₄⋊#10596842	1	2	yes	—
217	C₈	C₄⋊#10596842	1	2	yes	—
218	C₂²	—	73	1	no	—
219	C₂²	—	73	1	no	—
220	C₂²	—	73	2	no	—
221	C₂²	—	73	2	no	—
222	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	73	2	no	—
225	C₄	—	73	2	no	—
226	C₄	—	73	1	no	—
227	C₄	—	73	1	no	—
228	C₄	—	146	4	no	—
229	C₄	—	146	1	no	—
230	C₄	—	146	1	no	—
231	C₄	—	146	2	no	—
232	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
233	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
234	C₂	—	73	1	no	—
235	C₂	—	73	2	no	—
236	C₂	—	73	1	no	—
237	C₂	#22627055	1	2	yes	—
238	C₂	#22627075	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10596842
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×#64408	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#8807
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×#10596831	C₈	semidirect
36. C₈×Dic₇₃	C₈	semidirect
37. C₈×Dic₇₃	C₈	semidirect
38. C₄×C₈×C₇₃	C₈	semidirect
124. C₇₃	#8807	semidirect
158. C₄×C₈	#10596842	semidirect

Maximal subgroups

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

subgroup	index
#22627073	2
C₂×#17989677	2
C₂×#17989676	2
C₄×C₈×#64408	2
#8807	73