Subgroups — #34005476

Order 18688.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625848	C₂	1	2	yes	maximal
2	C₂×#17989451	C₂	1	2	yes	maximal
3	C₂×#17989448	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596833	C₂²	1	4	yes	—
6	#17989453	—	2	4	no	—
7	#17989451	—	2	4	no	—
8	#17989448	—	2	4	no	—
9	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
16	C₂²×#10596833	—	2	4	no	—
17	C₄⋊#10596833	—	2	4	no	—
18	C₄⋊#10596833	—	2	4	no	—
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
24	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
25	C₄×#10596831	—	2	2	no	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	C₈	1	4	yes	complemented
28	C₈×Dic₇₃	D₄	1	2	yes	—
29	C₈×Dic₇₃	D₄	1	2	yes	—
30	C₈×Dic₇₃	Q₈	1	2	yes	—
31	C₈×Dic₇₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	—	2	2	no	—
36	C₂×#10596833	—	4	4	no	—
37	C₂×#10596833	—	4	8	no	—
38	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
39	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×#10596831	—	2	1	no	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	C₄⋊C₄	1	2	yes	—
44	C₂×#10596831	C₄⋊C₄	1	2	yes	—
45	C₂×#10596831	C₂×C₈	1	2	yes	—
46	C₂×#10596831	M₁₆	1	2	yes	—
47	C₄×Dic₇₃	—	2	4	no	—
48	C₄×Dic₇₃	—	2	4	no	—
49	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
50	C₄×Dic₇₃	M₁₆	1	2	yes	—
51	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
53	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
63	C₄²×C₇₃	—	2	1	no	—
64	C₄²×C₇₃	—	2	1	no	—
65	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	#108	1	4	yes	—
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	—	2	2	no	—
78	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
79	C₂×Dic₇₃	#127	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₇₃	#135	1	4	yes	—
84	#10596833	—	8	4	no	—
85	C₈×C₇₃	—	2	2	no	—
86	C₈×C₇₃	C₄⋊C₈	1	4	yes	—
87	#10596831	—	2	2	no	—
88	#10596831	C₄⋊C₈	1	4	yes	—
89	C₂²×C₇₃	#384	1	1	yes	characteristic
90	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₇₃	#423	1	1	yes	characteristic
92	C₂²×C₇₃	#427	1	2	yes	—
93	C₂²×C₇₃	C₂×#108	1	2	yes	—
94	C₄×C₇₃	—	2	4	no	—
95	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₇₃	#423	1	1	yes	characteristic
98	C₄×C₇₃	#423	1	1	yes	characteristic
99	Dic₇₃	—	2	4	no	—
100	Dic₇₃	—	2	1	no	—
101	Dic₇₃	—	2	1	no	—
102	Dic₇₃	#495	1	2	yes	—
103	Dic₇₃	#499	1	2	yes	—
104	#8807	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	#1564	1	2	yes	—
106	C₂×C₇₃	#1568	1	2	yes	—
107	C₂×C₇₃	#1842	1	1	yes	characteristic
108	C₂×C₇₃	#1846	1	1	yes	characteristic
109	C₂×C₇₃	#1848	1	1	yes	characteristic
110	#1831	—	73	2	no	—
111	C₄²×C₈	—	73	1	no	—
112	C₂×#335	—	73	2	no	—
113	C₂×#334	—	73	2	no	—
114	C₇₃	#8807	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄⋊C₈	—	73	4	no	—
116	C₂×C₄×C₈	—	73	1	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	1	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	C₂×C₄×C₈	#64408	1	1	yes	characteristic
122	C₄³	—	73	1	no	—
123	#336	—	146	4	no	—
124	#335	—	146	4	no	—
125	#334	—	146	4	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₂²×C₈	—	73	1	no	—
128	C₂²×C₈	—	146	4	no	—
129	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
130	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
131	C₂×C₄²	—	73	1	no	—
132	C₂×C₄²	—	73	1	no	—
133	C₂×C₄²	—	73	2	no	—
134	C₂×C₄²	—	73	2	no	—
135	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
136	C₄⋊C₈	—	146	4	no	—
137	C₄⋊C₈	—	146	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	73	2	no	—
140	C₄×C₈	—	73	2	no	—
141	C₄×C₈	—	73	2	no	—
142	C₄×C₈	—	73	2	no	—
143	C₄×C₈	—	73	4	no	—
144	C₄×C₈	—	146	2	no	—
145	C₄×C₈	—	146	2	no	—
146	C₂²×C₄	—	73	1	no	—
147	C₂²×C₄	—	73	2	no	—
148	C₂²×C₄	—	73	1	no	—
149	C₂²×C₄	C₄⋊#64408	1	2	yes	—
150	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	292	4	no	—
154	C₂×C₈	—	292	8	no	—
155	C₂×C₈	—	73	2	no	—
156	C₂×C₈	—	73	2	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	73	2	no	—
159	C₂×C₈	—	146	1	no	—
160	C₂×C₈	—	146	1	no	—
161	C₂×C₈	C₄⋊#64408	1	2	yes	—
162	C₂×C₈	C₄⋊#64408	1	2	yes	—
163	C₂×C₈	#11924036	1	2	yes	—
164	C₂×C₈	#11924037	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	73	2	no	—
168	C₄²	—	73	2	no	—
169	C₄²	—	73	1	no	—
170	C₄²	—	73	1	no	—
171	C₄²	—	73	1	no	—
172	C₄²	—	73	1	no	—
173	C₄²	—	146	4	no	—
174	C₄²	—	146	4	no	—
175	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	73	1	no	—
179	C₂×C₄	—	73	1	no	—
180	C₂×C₄	—	73	4	no	—
181	C₂×C₄	—	73	1	no	—
182	C₂×C₄	—	73	1	no	—
183	C₂×C₄	—	73	4	no	—
184	C₂×C₄	—	146	2	no	—
185	C₂×C₄	—	146	2	no	—
186	C₂×C₄	—	146	2	no	—
187	C₂×C₄	#14556661	1	4	yes	—
188	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
190	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
191	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
192	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
193	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	584	4	no	—
196	C₈	—	73	4	no	—
197	C₈	—	146	2	no	—
198	C₈	#14556658	1	4	yes	—
199	C₂²	#17989533	1	1	yes	characteristic
200	C₂²	#17989588	1	1	yes	characteristic
201	C₂²	#17989590	1	1	yes	characteristic
202	C₂²	#17989597	1	2	yes	—
203	C₂²	C₂×#14556661	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	73	2	no	—
206	C₄	—	73	2	no	—
207	C₄	—	146	1	no	—
208	C₄	—	146	1	no	—
209	C₄	—	146	4	no	—
210	C₄	#17989588	1	1	yes	characteristic
211	C₄	C₂×#14556658	1	1	yes	characteristic
212	C₄	#17989590	1	1	yes	characteristic
213	C₄	#17989592	1	1	yes	characteristic
214	C₂	#22625479	1	2	yes	—
215	C₂	#22625491	1	2	yes	—
216	C₂	#22625864	1	1	yes	characteristic
217	C₂	#22625868	1	1	yes	characteristic
218	C₂	#22625870	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#8807
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
27. C₈×Dic₇₃	C₈	semidirect
114. C₇₃	#8807	semidirect

Maximal subgroups

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

subgroup	index
#22625848	2
C₂×#17989451	2
C₂×#17989448	2
C₄×C₈×Dic₇₃	2
#8807	73