Subgroups — #34007478

Order 18688.

382 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	#22626028	C₂	1	2	yes	maximal
2	C₂×#17989448	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989428	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#107×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989448	—	2	2	no	—
12	#17989428	—	2	2	no	—
13	C₂×#14556525	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988349	C₄	1	2	yes	—
17	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	2	yes	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	4	4	no	—
24	C₄×#10596833	—	2	1	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	Q₈	1	2	yes	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556525	—	2	2	no	—
33	C₈×Dic₇₃	D₄	1	2	yes	—
34	C₈×Dic₇₃	Q₈	1	2	yes	—
35	#14556498	—	2	2	no	—
36	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
38	#107×C₇₃	C₈	1	4	yes	complemented
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	4	1	no	—
44	C₂×#10596833	—	4	1	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	—	4	2	no	—
47	C₂×#10596833	—	4	2	no	—
48	C₂²×Dic₇₃	—	2	2	no	—
49	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₇₃	#41	1	1	yes	characteristic
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	C₄⋊C₄	1	2	yes	—
54	C₄⋊Dic₇₃	—	2	2	no	—
55	C₄⋊Dic₇₃	—	2	2	no	—
56	C₄×Dic₇₃	—	4	2	no	—
57	C₄×Dic₇₃	—	4	2	no	—
58	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₇₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
67	C₄⋊C₄×C₇₃	—	2	2	no	—
68	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
69	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	4	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	—	4	2	no	—
82	C₂×Dic₇₃	—	4	2	no	—
83	C₂×Dic₇₃	—	4	2	no	—
84	C₂×Dic₇₃	#121	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₇₃	#125	1	1	yes	characteristic
87	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×Dic₇₃	#137	1	2	yes	—
90	#10596833	—	4	2	no	—
91	#10596833	—	4	2	no	—
92	#10596833	—	8	2	no	—
93	C₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
94	#10596831	—	2	2	no	—
95	C₂²×C₇₃	#389	1	1	yes	characteristic
96	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
97	C₂²×C₇₃	#436	1	1	yes	characteristic
98	C₂²×C₇₃	#441	1	1	yes	characteristic
99	C₂²×C₇₃	#442	1	1	yes	characteristic
100	C₂²×C₇₃	#444	1	1	yes	characteristic
101	C₂²×C₇₃	#444	1	1	yes	characteristic
102	C₄×C₇₃	—	4	4	no	—
103	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₇₃	#424	1	1	yes	characteristic
105	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
106	C₄×C₇₃	#446	1	1	yes	characteristic
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	—	2	1	no	—
109	Dic₇₃	—	8	2	no	—
110	Dic₇₃	#468	1	2	yes	—
111	Dic₇₃	#501	1	2	yes	—
112	#9595	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	#1579	1	1	yes	characteristic
114	C₂×C₇₃	#1580	1	1	yes	characteristic
115	C₂×C₇₃	#1593	1	1	yes	characteristic
116	C₂×C₇₃	#1594	1	1	yes	characteristic
117	C₂×C₇₃	#1939	1	1	yes	characteristic
118	C₂×C₇₃	#1943	1	1	yes	characteristic
119	C₂×C₇₃	#1946	1	1	yes	characteristic
120	#1939	—	73	2	no	—
121	C₄×#107	—	73	1	no	—
122	C₂×#334	—	73	1	no	—
123	C₂×#322	—	73	1	no	—
124	#1279	—	73	2	no	—
125	C₇₃	#9595	1	1	yes	characteristic, complemented, Sylow
126	C₂×#107	—	73	1	no	—
127	C₂×#107	#64408	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	73	2	no	—
129	C₂×C₄⋊C₈	—	146	2	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₂×C₄×C₈	—	73	1	no	—
133	C₄×C₄⋊C₄	—	73	2	no	—
134	#340	—	73	2	no	—
135	#336	—	146	2	no	—
136	#334	—	146	2	no	—
137	#322	—	146	2	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	2	no	—
140	C₂²×C₈	—	146	2	no	—
141	C₂²×C₈	—	146	1	no	—
142	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	73	2	no	—
144	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	146	2	no	—
147	#107	—	146	2	no	—
148	#107	#10596833	1	4	yes	complemented
149	C₄⋊C₈	—	292	4	no	—
150	C₄⋊C₈	—	146	2	no	—
151	C₄⋊C₈	—	146	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	146	1	no	—
156	C₄×C₈	—	146	1	no	—
157	C₄×C₈	—	146	1	no	—
158	#95	—	146	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	146	2	no	—
163	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
164	C₂×C₈	—	292	1	no	—
165	C₂×C₈	—	292	1	no	—
166	C₂×C₈	—	292	2	no	—
167	C₂×C₈	—	292	2	no	—
168	C₂×C₈	—	292	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	146	2	no	—
171	C₂×C₈	—	146	2	no	—
172	C₂×C₈	—	146	2	no	—
173	C₂×C₈	—	146	2	no	—
174	C₂×C₈	—	146	1	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	C₄⋊#64408	1	2	yes	—
177	C₂×C₈	C₂×#10596833	1	2	yes	—
178	C₂×C₈	#11924038	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	146	2	no	—
181	C₄⋊C₄	—	146	2	no	—
182	C₄⋊C₄	C₂×#10596833	1	4	yes	—
183	C₄²	—	292	2	no	—
184	C₄²	—	292	2	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	1	no	—
188	C₄²	—	73	1	no	—
189	C₂³	D₄×#64408	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	292	2	no	—
194	C₂×C₄	—	292	2	no	—
195	C₂×C₄	—	292	2	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	2	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	146	1	no	—
203	C₂×C₄	—	146	1	no	—
204	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
205	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
206	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
207	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
208	C₂×C₄	#14556678	1	1	yes	characteristic
209	C₂×C₄	#14556679	1	1	yes	characteristic
210	C₈	—	292	2	no	—
211	C₈	—	292	2	no	—
212	C₈	—	584	2	no	—
213	C₈	—	146	2	no	—
214	C₈	C₄⋊#10596833	1	4	yes	complemented
215	C₂²	#17989538	1	1	yes	characteristic
216	C₂²	D₄×#10596833	1	1	yes	characteristic
217	C₂²	#17989621	1	1	yes	characteristic
218	C₂²	#17989627	1	1	yes	characteristic
219	C₂²	#17989628	1	1	yes	characteristic
220	C₂²	#17989633	1	1	yes	characteristic
221	C₂²	#17989634	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	—	584	2	no	—
224	C₄	—	73	2	no	—
225	C₄	—	73	2	no	—
226	C₄	—	146	1	no	—
227	C₄	—	146	1	no	—
228	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
229	C₄	#17989594	1	1	yes	characteristic
230	C₄	Q₈×#10596833	1	1	yes	characteristic
231	C₄	#17989637	1	1	yes	characteristic
232	C₂	#22625524	1	1	yes	characteristic
233	C₂	#22625527	1	1	yes	characteristic
234	C₂	#22625577	1	1	yes	characteristic
235	C₂	#22625580	1	1	yes	characteristic
236	C₂	#22626028	1	1	yes	characteristic
237	C₂	#22626036	1	1	yes	characteristic
238	C₂	#22626040	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#107×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#107	#64408
hypocenter	C₇₃	#9595
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
38. #107×C₇₃	C₈	semidirect
93. C₈×C₇₃	C₄⋊C₈	semidirect
125. C₇₃	#9595	semidirect
148. #107	#10596833	semidirect
214. 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
#22626028	2
C₂×#17989448	2
C₂×#17989428	2
#22624767	2
#107×Dic₇₃	2
#9595	73