Subgroups — #34004063

Order 18688.

358 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625478	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#10596833	C₂	1	2	yes	maximal
3	C₂×#17989428	C₂	1	1	yes	maximal, characteristic
4	#22624764	C₂	1	2	yes	maximal
5	#108×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989603	C₂²	1	2	yes	—
7	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	#17989450	C₄	1	2	yes	—
10	#17989448	C₄	1	2	yes	—
11	#17989428	—	2	2	no	—
12	#17989428	C₄	1	2	yes	—
13	#17989428	C₄	1	2	yes	—
14	C₈×#10596833	—	2	2	no	—
15	C₂×#14556527	C₄	1	1	yes	characteristic
16	M₁₆×Dic₇₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	#17988351	C₄	1	2	yes	—
19	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	—	2	1	no	—
21	C₂²×#10596833	Q₈	1	2	yes	—
22	C₄⋊#10596833	C₂×C₄	1	4	yes	—
23	C₂²⋊#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	C₂×C₄	1	2	yes	—
27	C₄×#10596833	C₂×C₄	1	2	yes	—
28	C₄×#10596833	D₄	1	2	yes	—
29	C₂×#11924012	C₂×C₄	1	2	yes	—
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556527	C₈	1	4	yes	complemented
33	#14556501	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	#108×C₇₃	C₈	1	4	yes	complemented
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	2	4	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	C₄⋊C₄	1	4	yes	—
47	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×Dic₇₃	#41	1	1	yes	characteristic
49	#11924012	—	2	2	no	—
50	#11924012	C₂×C₈	1	4	yes	—
51	C₂×#10596831	—	2	2	no	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	C₄²	1	1	yes	characteristic
55	C₂×#10596831	C₄²	1	1	yes	characteristic
56	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₇₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₇₃	—	2	2	no	—
62	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
68	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×Dic₇₃	#117	1	1	yes	characteristic
79	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
81	#10596833	—	2	4	no	—
82	#10596833	—	4	2	no	—
83	#10596833	—	4	4	no	—
84	C₈×C₇₃	—	2	4	no	—
85	C₈×C₇₃	C₄×C₈	1	2	yes	—
86	C₈×C₇₃	C₄×C₈	1	2	yes	—
87	#10596831	—	2	1	no	—
88	#10596831	—	2	1	no	—
89	#10596831	—	2	4	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
94	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
95	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₇₃	#405	1	1	yes	characteristic
97	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
98	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
99	Dic₇₃	—	2	1	no	—
100	Dic₇₃	—	2	1	no	—
101	Dic₇₃	#430	1	2	yes	—
102	Dic₇₃	#444	1	2	yes	—
103	#8294	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₇₃	#1773	1	1	yes	characteristic
108	C₂×C₇₃	#1778	1	1	yes	characteristic
109	C₄×#108	—	73	1	no	—
110	#1564	—	73	1	no	—
111	C₈×M₁₆	—	73	2	no	—
112	C₂×#322	—	73	1	no	—
113	#1278	—	73	2	no	—
114	C₇₃	#8294	1	1	yes	characteristic, complemented, Sylow
115	#431	—	73	2	no	—
116	C₂×#108	—	73	1	no	—
117	C₂×#108	#64408	1	1	yes	characteristic
118	C₄×M₁₆	—	73	2	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	#341	—	73	2	no	—
123	#335	—	73	2	no	—
124	#334	—	73	2	no	—
125	#322	—	73	2	no	—
126	#322	—	73	2	no	—
127	#322	—	146	2	no	—
128	C₈²	—	146	2	no	—
129	C₂×M₁₆	—	73	2	no	—
130	C₂×M₁₆	C₂×#64408	1	2	yes	—
131	C₂²×C₈	—	73	2	no	—
132	C₂²×C₈	—	73	1	no	—
133	C₂²×C₈	—	146	1	no	—
134	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
135	C₂×C₄²	—	73	1	no	—
136	#108	—	73	4	no	—
137	#108	#10596833	1	4	yes	complemented
138	C₄⋊C₈	—	73	4	no	—
139	C₂²⋊C₈	—	146	2	no	—
140	#97	—	146	2	no	—
141	C₄×C₈	—	73	2	no	—
142	C₄×C₈	—	73	2	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	1	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	146	2	no	—
149	C₄×C₈	—	146	2	no	—
150	C₄×C₈	—	146	1	no	—
151	C₂²×C₄	—	73	1	no	—
152	C₂²×C₄	—	73	1	no	—
153	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	73	4	no	—
156	M₁₆	—	146	2	no	—
157	M₁₆	C₂×#10596833	1	4	yes	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	292	2	no	—
160	C₂×C₈	—	292	2	no	—
161	C₂×C₈	—	73	4	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	4	no	—
166	C₂×C₈	—	146	1	no	—
167	C₂×C₈	—	146	1	no	—
168	C₂×C₈	—	146	2	no	—
169	C₂×C₈	C₄×#64408	1	1	yes	characteristic
170	C₂×C₈	C₄×#64408	1	1	yes	characteristic
171	C₂×C₈	C₂×#10596833	1	2	yes	—
172	C₂×C₈	#11924036	1	2	yes	—
173	C₄²	—	73	1	no	—
174	C₄²	—	73	1	no	—
175	C₄²	—	73	1	no	—
176	C₄²	—	73	1	no	—
177	C₂³	Q₈×#64408	1	1	yes	characteristic
178	C₂×C₄	—	73	1	no	—
179	C₂×C₄	—	73	1	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	73	1	no	—
182	C₂×C₄	—	146	2	no	—
183	C₂×C₄	—	146	2	no	—
184	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
185	C₂×C₄	D₄×#64408	1	1	yes	characteristic
186	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
187	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
188	C₂×C₄	#14556678	1	1	yes	characteristic
189	C₂×C₄	#14556679	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	292	4	no	—
192	C₈	—	292	2	no	—
193	C₈	—	146	4	no	—
194	C₈	—	146	1	no	—
195	C₈	—	146	1	no	—
196	C₈	—	146	4	no	—
197	C₈	C₄×#10596833	1	2	yes	—
198	C₈	C₄×#10596833	1	2	yes	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
202	C₂²	Q₈×#10596833	1	1	yes	characteristic
203	C₂²	#17989636	1	1	yes	characteristic
204	C₄	—	73	2	no	—
205	C₄	—	73	2	no	—
206	C₄	—	146	1	no	—
207	C₄	—	146	1	no	—
208	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
209	C₄	#17989552	1	1	yes	characteristic
210	C₄	D₄×#10596833	1	1	yes	characteristic
211	C₄	#17989615	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₄⋊C₄×#10596833	1	1	yes	characteristic
215	C₂	#22625740	1	1	yes	characteristic
216	C₂	#22625752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#108×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#108	#64408
hypocenter	C₇₃	#8294
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
32. #14556527	C₈	semidirect
41. #108×C₇₃	C₈	semidirect
114. C₇₃	#8294	semidirect
137. #108	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625478	2
M₁₆×#10596833	2
C₂×#17989428	2
#22624764	2
#108×Dic₇₃	2
#8294	73