Subgroups — #34034122

Order 18688.

414 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	C₂×#17989690	C₂	1	2	yes	maximal
2	C₂×#17989681	C₂	1	2	yes	maximal
3	#22627001	C₂	1	2	yes	maximal
4	C₄²×#10596833	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924040	C₂²	1	4	yes	—
6	#17989690	C₄	1	8	yes	complemented
7	#17989681	—	2	4	no	—
8	C₂×C₄×#10596833	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₄	1	1	yes	characteristic
10	C₂×C₄×#10596833	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
14	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924040	—	2	4	no	—
16	C₂×#11924040	—	2	4	no	—
17	C₂×#11924040	C₂×C₄	1	8	yes	—
18	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596833	C₂³	1	1	yes	characteristic
22	C₄×#10596833	—	2	2	no	—
23	C₄×#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	C₈	1	4	yes	—
26	C₄×#10596833	C₂×C₄	1	4	yes	—
27	C₄×#10596833	D₄	1	2	yes	—
28	C₄×#10596833	Q₈	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
36	#11924040	—	2	8	no	—
37	C₂×#10596833	—	2	1	no	—
38	C₂×#10596833	—	2	1	no	—
39	C₂×#10596833	—	2	1	no	—
40	C₂×#10596833	—	2	1	no	—
41	C₂×#10596833	C₄²	1	2	yes	—
42	C₂×#10596833	C₄⋊C₄	1	2	yes	—
43	C₂×#10596833	C₂×C₈	1	2	yes	—
44	C₂×#10596833	C₂×C₈	1	2	yes	—
45	C₂×#10596833	M₁₆	1	2	yes	—
46	C₂×#10596833	C₂²×C₄	1	2	yes	—
47	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
48	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
49	C₂×#10596833	#41	1	2	yes	—
50	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₄×Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	—	2	2	no	—
56	C₄×Dic₇₃	—	2	2	no	—
57	C₄×Dic₇₃	—	2	1	no	—
58	C₄×Dic₇₃	—	2	1	no	—
59	C₄×Dic₇₃	C₄²	1	2	yes	—
60	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
61	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
62	C₄×Dic₇₃	M₁₆	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
82	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
83	C₂×Dic₇₃	M₃₂	1	2	yes	—
84	C₂×Dic₇₃	M₃₂	1	2	yes	—
85	C₂×Dic₇₃	M₃₂	1	2	yes	—
86	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₇₃	#117	1	1	yes	characteristic
89	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
92	#10596833	—	2	2	no	—
93	#10596833	—	2	2	no	—
94	#10596833	C₄⋊C₈	1	4	yes	—
95	#10596833	C₄×D₄	1	2	yes	—
96	#10596833	C₄×Q₈	1	2	yes	—
97	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₇₃	#431	1	1	yes	characteristic
100	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
101	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
102	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
103	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
104	C₄×C₇₃	—	2	2	no	—
105	C₄×C₇₃	—	2	2	no	—
106	C₄×C₇₃	—	2	2	no	—
107	Dic₇₃	—	2	2	no	—
108	Dic₇₃	—	2	2	no	—
109	Dic₇₃	#346	1	4	yes	complemented
110	Dic₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
111	Dic₇₃	C₄×M₁₆	1	1	yes	characteristic
112	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	Dic₇₃	#431	1	1	yes	characteristic
114	#12653	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
117	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
118	C₂×C₇₃	#2150	1	1	yes	characteristic
119	C₂×C₇₃	#2150	1	1	yes	characteristic
120	C₂×C₇₃	#2163	1	1	yes	characteristic
121	C₂×C₇₃	#2163	1	1	yes	characteristic
122	C₂×C₄⋊C₁₆	—	73	2	no	—
123	C₂×#346	—	73	2	no	—
124	C₄²×C₈	—	73	1	no	—
125	#1374	—	73	2	no	—
126	C₇₃	#12653	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₁₆	—	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	1	no	—
132	C₂×C₄×C₈	—	73	1	no	—
133	C₂×C₄×C₈	—	73	1	no	—
134	C₄³	—	73	1	no	—
135	C₄⋊C₁₆	—	73	8	no	—
136	#346	—	146	4	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂×C₄²	—	73	1	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	73	1	no	—
147	C₂×C₄²	#10596842	1	1	yes	characteristic
148	C₂×C₁₆	—	73	8	no	—
149	C₂×C₁₆	—	146	4	no	—
150	C₂×C₁₆	—	146	4	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	4	no	—
153	C₄×C₈	—	73	4	no	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	146	2	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	—	146	2	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
163	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
164	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
165	C₂×C₈	—	73	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	1	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	—	146	1	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	1	no	—
177	C₂×C₈	—	146	1	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	73	2	no	—
180	C₄²	—	73	2	no	—
181	C₄²	—	73	2	no	—
182	C₄²	—	73	2	no	—
183	C₄²	—	146	1	no	—
184	C₄²	—	146	1	no	—
185	C₄²	—	146	2	no	—
186	C₄²	—	146	2	no	—
187	C₄²	—	146	2	no	—
188	C₁₆	—	146	8	no	—
189	C₂³	C₂²×#10596842	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	1	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	—	73	2	no	—
205	C₂×C₄	—	73	2	no	—
206	C₂×C₄	—	73	2	no	—
207	C₂×C₄	—	146	1	no	—
208	C₂×C₄	—	146	1	no	—
209	C₂×C₄	—	146	1	no	—
210	C₂×C₄	—	146	1	no	—
211	C₂×C₄	C₄×#10596842	1	2	yes	—
212	C₂×C₄	C₄⋊#10596842	1	2	yes	—
213	C₂×C₄	#14556687	1	2	yes	—
214	C₈	—	73	2	no	—
215	C₈	—	73	4	no	—
216	C₈	—	73	2	no	—
217	C₈	—	146	2	no	—
218	C₈	—	146	2	no	—
219	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
220	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
221	C₂²	#17989700	1	1	yes	characteristic
222	C₂²	C₂×#14556687	1	1	yes	characteristic
223	C₂²	#17989706	1	1	yes	characteristic
224	C₂²	#17989706	1	1	yes	characteristic
225	C₂²	#17989706	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	73	4	no	—
230	C₄	—	73	1	no	—
231	C₄	—	73	1	no	—
232	C₄	—	73	1	no	—
233	C₄	—	73	1	no	—
234	C₄	—	146	2	no	—
235	C₄	—	146	2	no	—
236	C₂	C₄×C₄⋊#10596842	1	1	yes	characteristic
237	C₂	#22627087	1	1	yes	characteristic
238	C₂	C₄×#14556687	1	1	yes	characteristic
239	C₂	#22627119	1	1	yes	characteristic
240	C₂	#22627121	1	1	yes	characteristic
241	C₂	#22627132	1	1	yes	characteristic
242	C₂	#22627135	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×C₄²×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	C₂×C₄²	#10596842
hypocenter	C₇₃	#12653
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. #17989690	C₄	semidirect
109. Dic₇₃	#346	semidirect
126. C₇₃	#12653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989690	2
C₂×#17989681	2
#22627001	2
C₄²×#10596833	2
#12653	73