Subgroups — #32976552

Order 18688.

518 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556508	C₂	1	1	yes	maximal, characteristic
2	#22608252	C₂	1	1	yes	maximal, characteristic
3	#22608220	C₂	1	2	yes	maximal, complemented
4	#22607781	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556508	C₄	1	2	yes	—
8	C₄×#11924012	C₂²	1	2	yes	—
9	#17988356	C₄	1	2	yes	—
10	#17988311	C₄	1	4	yes	—
11	#17988309	—	2	2	no	—
12	#17988308	—	2	2	no	—
13	C₂³×C₈×C₇₃	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₇₃	—	2	2	no	—
16	C₈×D₄×C₇₃	—	2	2	no	—
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₇₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×#11924012	C₂×C₄	1	4	yes	—
21	C₄⋊#10596831	—	2	2	no	—
22	#14556508	—	2	4	no	—
23	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
24	#14556502	—	2	2	no	—
25	C₄×#10596831	—	2	2	no	—
26	C₄×#10596831	—	2	2	no	—
27	C₂²×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₇₃	—	2	2	no	—
30	C₂²×C₈×C₇₃	—	2	2	no	—
31	C₂²×C₈×C₇₃	—	4	4	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₇₃	—	2	2	no	—
36	C₄×D₄×C₇₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₇₃	—	2	2	no	—
41	C₂²⋊C₈×C₇₃	—	2	4	no	—
42	C₄×C₈×C₇₃	D₄	1	2	yes	—
43	C₄×C₈×C₇₃	D₄	1	2	yes	—
44	#11924012	—	2	4	no	—
45	#11924012	—	2	4	no	—
46	C₂×#10596831	—	2	4	no	—
47	C₂×#10596831	—	2	4	no	—
48	C₂⁴×C₇₃	C₄²	1	2	yes	—
49	C₂×D₄×C₇₃	—	2	2	no	—
50	C₂×D₄×C₇₃	—	2	2	no	—
51	C₂×D₄×C₇₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₇₃	—	2	2	no	—
53	C₂²×C₄×C₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	—	4	2	no	—
55	C₂²×C₄×C₇₃	—	4	2	no	—
56	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	—	2	2	no	—
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	—	4	1	no	—
63	C₂×C₈×C₇₃	—	4	4	no	—
64	C₂×C₈×C₇₃	—	4	4	no	—
65	C₂×C₈×C₇₃	—	4	2	no	—
66	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₇₃	—	2	2	no	—
70	C₂²⋊C₄×C₇₃	—	2	4	no	—
71	C₄²×C₇₃	C₂×D₄	1	2	yes	—
72	C₄²×C₇₃	#41	1	2	yes	—
73	C₂³×C₇₃	—	2	2	no	—
74	C₂³×C₇₃	—	2	2	no	—
75	C₂³×C₇₃	—	4	2	no	—
76	C₂³×C₇₃	—	4	2	no	—
77	C₂³×C₇₃	#97	1	4	yes	—
78	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
79	D₄×C₇₃	—	2	8	no	—
80	C₂×C₄×C₇₃	—	2	1	no	—
81	C₂×C₄×C₇₃	—	2	1	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	—	4	1	no	—
84	C₂×C₄×C₇₃	—	4	4	no	—
85	C₂×C₄×C₇₃	—	4	2	no	—
86	C₂×C₄×C₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	—	4	2	no	—
88	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₇₃	#131	1	2	yes	—
94	C₈×C₇₃	—	2	1	no	—
95	C₈×C₇₃	—	2	2	no	—
96	C₈×C₇₃	—	4	2	no	—
97	C₈×C₇₃	#100	1	2	yes	complemented
98	#10596831	—	4	4	no	—
99	#10596831	—	4	4	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	—	2	1	no	—
102	C₂²×C₇₃	—	2	4	no	—
103	C₂²×C₇₃	—	4	4	no	—
104	C₂²×C₇₃	—	4	4	no	—
105	C₂²×C₇₃	—	4	2	no	—
106	C₂²×C₇₃	—	4	2	no	—
107	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
109	C₂²×C₇₃	#405	1	1	yes	characteristic
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	2	2	no	—
112	C₄×C₇₃	—	4	2	no	—
113	C₄×C₇₃	C₂×#100	1	1	yes	characteristic
114	C₄×C₇₃	#413	1	1	yes	characteristic
115	C₄×C₇₃	#435	1	4	yes	—
116	#8164	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	—	2	1	no	—
118	C₂×C₇₃	—	2	1	no	—
119	C₂×C₇₃	—	4	4	no	—
120	C₂×C₇₃	#1752	1	1	yes	characteristic
121	C₂×C₇₃	C₄×#100	1	1	yes	characteristic
122	C₂×C₇₃	#1757	1	1	yes	characteristic
123	C₂×C₈×D₄	D₇₃	1	1	yes	characteristic
124	C₄×#100	—	73	1	no	—
125	#1460	—	73	1	no	—
126	#1451	—	73	2	no	—
127	#1283	—	73	2	no	—
128	C₇₃	#8164	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₈	Dic₇₃	1	2	yes	—
130	C₂×C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
131	C₈×D₄	—	2	2	no	—
132	C₈×D₄	—	2	2	no	—
133	#431	—	73	2	no	—
134	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
135	C₂×#100	—	73	2	no	—
136	C₂×C₂²⋊C₈	Dic₇₃	1	2	yes	—
137	C₄×M₁₆	—	73	2	no	—
138	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
139	#343	—	73	2	no	—
140	#323	—	73	4	no	—
141	#322	—	146	2	no	—
142	#322	—	146	2	no	—
143	C₂²×D₄	C₄×D₇₃	1	1	yes	characteristic
144	C₂³×C₄	C₂×Dic₇₃	1	2	yes	—
145	C₂×M₁₆	—	73	4	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	4	4	no	—
149	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
151	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₄×D₇₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₄×D₇₃	1	2	yes	—
156	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	146	2	no	—
159	#100	—	146	4	no	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	—	73	4	no	—
162	#97	—	146	2	no	—
163	C₄×C₈	—	146	2	no	—
164	C₄×C₈	—	146	2	no	—
165	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
166	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
167	C₂⁴	C₄×Dic₇₃	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	#11924011	1	4	yes	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
176	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
179	M₁₆	—	146	4	no	—
180	M₁₆	—	146	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	146	4	no	—
188	C₂×C₈	—	146	4	no	—
189	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
190	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
195	C₄²	#11924028	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	#14556501	1	4	yes	—
201	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
202	D₄	—	2	8	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
212	C₂×C₄	#14556550	1	1	yes	characteristic
213	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
214	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
215	C₂×C₄	C₂×#11924011	1	2	yes	—
216	C₂×C₄	#14556597	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₈	—	292	4	no	—
221	C₈	—	292	4	no	—
222	C₈	#14556508	1	2	yes	complemented
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₄×C₂²⋊Dic₇₃	1	1	yes	characteristic
231	C₂²	C₂×#14556501	1	1	yes	characteristic
232	C₂²	#17988542	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂×#14556508	1	1	yes	characteristic
237	C₄	#17988571	1	1	yes	characteristic
238	C₄	#17988658	1	4	yes	—
239	C₂	—	2	1	no	—
240	C₂	—	2	1	no	—
241	C₂	—	4	4	no	—
242	C₂	#22609707	1	1	yes	characteristic
243	C₂	C₄×#14556508	1	1	yes	characteristic
244	C₂	#22609731	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₃
commutator	C₂³×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×C₈×D₄×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₇₃	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈×D₄	D₇₃
hypocenter	C₇₃	#8164
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. #22608220	C₂	semidirect
4. #22607781	C₂	semidirect
97. C₈×C₇₃	#100	semidirect
128. C₇₃	#8164	semidirect
222. C₈	#14556508	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556508	2
#22608252	2
#22608220	2
#22607781	2
C₂×C₈×D₄×C₇₃	2
#8164	73