Subgroups — #34017987

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626846	C₂	1	2	yes	maximal
2	#22625482	C₂	1	6	yes	maximal, complemented
3	#22625478	C₂	1	3	yes	maximal, complemented
4	C₂×#17989450	C₂	1	3	yes	maximal, complemented
5	#131×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	—	2	2	no	—
7	D₄×#10596833	—	2	6	no	—
8	#17989603	C₂²	1	6	yes	—
9	C₂×C₄⋊#10596833	C₂²	1	6	yes	—
10	#17989450	C₂²	1	3	yes	—
11	#17989450	C₂²	1	6	yes	—
12	#17989450	C₂²	1	1	yes	characteristic
13	#17989448	C₂²	1	6	yes	—
14	C₂×#14556598	C₄	1	1	yes	characteristic
15	#41×Dic₇₃	C₂²	1	1	yes	characteristic
16	#17988545	C₄	1	3	yes	—
17	#17988542	C₄	1	3	yes	—
18	M₁₆×Dic₇₃	C₂²	1	3	yes	—
19	C₂×C₈×Dic₇₃	C₂²	1	3	yes	—
20	C₂×#131×C₇₃	C₄	1	1	yes	characteristic
21	C₂²×#10596833	—	2	6	no	—
22	C₄⋊#10596833	—	2	6	no	—
23	C₄⋊#10596833	C₂³	1	2	yes	—
24	C₄⋊#10596833	C₂³	1	6	yes	—
25	C₂²⋊#10596833	—	2	6	no	—
26	C₄×#10596833	—	2	6	no	—
27	#14556598	—	2	4	no	—
28	C₂×#11924012	C₂×C₄	1	3	yes	—
29	C₂²×#10596831	C₂×C₄	1	3	yes	—
30	Q₈×Dic₇₃	D₄	1	1	yes	characteristic
31	Q₈×Dic₇₃	Q₈	1	1	yes	characteristic
32	D₄×Dic₇₃	D₄	1	3	yes	—
33	D₄×Dic₇₃	Q₈	1	3	yes	—
34	#14556550	C₂×C₄	1	3	yes	—
35	C₂×C₄×Dic₇₃	C₂³	1	3	yes	—
36	#14556501	C₂×C₄	1	6	yes	—
37	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
38	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
39	C₈×Dic₇₃	C₂³	1	3	yes	—
40	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
41	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
42	#131×C₇₃	C₈	1	8	yes	complemented
43	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
45	C₂×#10596833	—	2	6	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	—	8	6	no	—
48	C₂²×Dic₇₃	C₂×D₄	1	3	yes	—
49	C₂²×Dic₇₃	C₂×Q₈	1	3	yes	—
50	#11924012	—	2	6	no	—
51	C₂×#10596831	—	2	6	no	—
52	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
53	C₂×#10596831	C₂²×C₄	1	3	yes	—
54	C₄⋊Dic₇₃	C₄⋊C₄	1	3	yes	—
55	C₄⋊Dic₇₃	C₄⋊C₄	1	3	yes	—
56	C₂²⋊Dic₇₃	C₄⋊C₄	1	6	yes	—
57	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
58	C₄×Dic₇₃	C₂×D₄	1	3	yes	—
59	C₄×Dic₇₃	C₂×Q₈	1	3	yes	—
60	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
61	#41×C₇₃	C₂×C₈	1	4	yes	—
62	#41×C₇₃	M₁₆	1	4	yes	—
63	C₂×Q₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×D₄×C₇₃	C₄⋊C₄	1	3	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
66	M₁₆×C₇₃	C₂×C₈	1	12	yes	complemented
67	C₂×C₈×C₇₃	C₂×C₈	1	12	yes	complemented
68	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
70	C₂³×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
71	Q₈×C₇₃	C₄⋊C₈	1	4	yes	—
72	D₄×C₇₃	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₇₃	—	2	3	no	—
78	C₂×Dic₇₃	—	2	3	no	—
79	C₂×Dic₇₃	—	4	3	no	—
80	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
82	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
84	#10596833	—	4	6	no	—
85	#10596833	—	4	2	no	—
86	C₈×C₇₃	C₂²×C₈	1	2	yes	—
87	C₈×C₇₃	C₂²×C₈	1	6	yes	—
88	#10596831	—	2	1	no	—
89	#10596831	—	2	3	no	—
90	C₂²×C₇₃	—	2	3	no	—
91	C₂²×C₇₃	—	2	3	no	—
92	C₂²×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
93	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
95	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
97	Dic₇₃	—	2	3	no	—
98	Dic₇₃	—	2	1	no	—
99	Dic₇₃	—	2	3	no	—
100	Dic₇₃	#576	1	1	yes	characteristic
101	Dic₇₃	#578	1	1	yes	characteristic
102	#17789	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	6	no	—
104	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₇₃	#2913	1	1	yes	characteristic
106	C₂×C₇₃	#2916	1	1	yes	characteristic
107	#2988	—	73	2	no	—
108	C₄×#131	—	73	1	no	—
109	#1565	—	73	6	no	—
110	#1564	—	73	3	no	—
111	C₂×#335	—	73	3	no	—
112	C₇₃	#17789	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	—	73	1	no	—
114	C₂×#131	#64408	1	1	yes	characteristic
115	C₄×#41	—	73	1	no	—
116	C₈×Q₈	—	146	2	no	—
117	C₈×D₄	—	146	6	no	—
118	#431	—	73	6	no	—
119	C₂×C₄⋊C₈	—	73	6	no	—
120	#405	—	73	3	no	—
121	#405	—	73	3	no	—
122	C₄×M₁₆	—	73	3	no	—
123	C₂×C₄×C₈	—	73	3	no	—
124	#335	—	73	3	no	—
125	#335	—	73	6	no	—
126	#335	—	73	1	no	—
127	#334	—	73	6	no	—
128	C₂×#41	C₂×#64408	1	1	yes	characteristic
129	#131	—	146	4	no	—
130	#131	#10596833	1	8	yes	complemented
131	C₂×M₁₆	—	73	3	no	—
132	C₂×M₁₆	C₂×#64408	1	3	yes	—
133	C₂²×C₈	—	73	3	no	—
134	C₂²×C₈	—	146	6	no	—
135	C₂²×C₈	C₂×#64408	1	3	yes	—
136	C₄×Q₈	—	73	1	no	—
137	C₄×Q₈	—	73	1	no	—
138	C₄×D₄	—	73	3	no	—
139	C₄×D₄	—	73	3	no	—
140	#117	—	73	3	no	—
141	C₂×C₄²	—	73	3	no	—
142	C₄⋊C₈	—	73	2	no	—
143	C₄⋊C₈	—	73	6	no	—
144	C₄⋊C₈	—	146	6	no	—
145	C₂²⋊C₈	—	146	6	no	—
146	#97	—	73	6	no	—
147	C₄×C₈	—	73	3	no	—
148	C₄×C₈	—	73	1	no	—
149	C₄×C₈	—	73	3	no	—
150	C₄×C₈	—	73	3	no	—
151	C₄×C₈	—	146	6	no	—
152	#41	C₂×#10596833	1	4	yes	—
153	#41	#11924038	1	4	yes	—
154	C₂×Q₈	C₄⋊#64408	1	1	yes	characteristic
155	C₂×D₄	C₄⋊#64408	1	3	yes	—
156	C₂²×C₄	—	73	3	no	—
157	C₂²×C₄	—	73	3	no	—
158	C₂²×C₄	C₂²×#64408	1	3	yes	—
159	M₁₆	—	146	6	no	—
160	M₁₆	C₂×#10596833	1	12	yes	complemented
161	C₂×C₈	—	584	6	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	3	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	6	no	—
166	C₂×C₈	—	146	6	no	—
167	C₂×C₈	C₂×#10596833	1	12	yes	complemented
168	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
169	C₂×C₈	C₂²×#64408	1	3	yes	—
170	C₄⋊C₄	—	73	3	no	—
171	C₄⋊C₄	—	73	3	no	—
172	C₂²⋊C₄	—	73	6	no	—
173	C₄²	—	73	3	no	—
174	C₄²	—	73	1	no	—
175	C₄²	—	73	3	no	—
176	C₄²	—	73	3	no	—
177	C₂³	C₂×C₄⋊#64408	1	3	yes	—
178	Q₈	C₄⋊#10596833	1	4	yes	—
179	D₄	C₄⋊#10596833	1	12	yes	—
180	C₂×C₄	—	292	3	no	—
181	C₂×C₄	—	73	1	no	—
182	C₂×C₄	—	73	1	no	—
183	C₂×C₄	—	73	3	no	—
184	C₂×C₄	—	73	3	no	—
185	C₂×C₄	—	146	3	no	—
186	C₂×C₄	—	146	3	no	—
187	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
188	C₂×C₄	C₂²×#10596833	1	6	yes	—
189	C₂×C₄	C₂×#11924038	1	6	yes	—
190	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
191	C₈	—	292	6	no	—
192	C₈	—	292	2	no	—
193	C₈	—	146	1	no	—
194	C₈	—	146	3	no	—
195	C₈	C₂²×#10596833	1	2	yes	—
196	C₈	C₂²×#10596833	1	6	yes	—
197	C₂²	—	2	3	no	—
198	C₂²	—	2	3	no	—
199	C₂²	C₂×C₄⋊#10596833	1	6	yes	—
200	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
201	C₄	—	73	1	no	—
202	C₄	—	73	1	no	—
203	C₄	—	146	3	no	—
204	C₄	—	146	1	no	—
205	C₄	—	146	3	no	—
206	C₄	C₂×C₄⋊#10596833	1	6	yes	—
207	C₄	C₂³×#10596833	1	1	yes	characteristic
208	C₄	C₂²×#11924038	1	1	yes	characteristic
209	C₂	—	2	6	no	—
210	C₂	C₂²×C₄⋊#10596833	1	1	yes	characteristic
211	C₂	#22626586	1	1	yes	characteristic
212	C₂	#22626593	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×#131×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂×#131	#64408
hypocenter	C₇₃	#17789
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #22625482	C₂	semidirect
3. #22625478	C₂	semidirect
4. C₂×#17989450	C₂	semidirect
42. #131×C₇₃	C₈	semidirect
66. M₁₆×C₇₃	C₂×C₈	semidirect
67. C₂×C₈×C₇₃	C₂×C₈	semidirect
112. C₇₃	#17789	semidirect
130. #131	#10596833	semidirect
160. M₁₆	C₂×#10596833	semidirect
167. C₂×C₈	C₂×#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22626846	2
#22625482	2
#22625478	2
C₂×#17989450	2
#131×Dic₇₃	2
#17789	73