Subgroups — #33086589

Order 18688.

842 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612329	C₂	1	2	yes	maximal
2	#22610095	C₂	1	2	yes	maximal, complemented
3	#22610062	C₂	1	2	yes	maximal, complemented
4	#22609964	C₂	1	4	yes	maximal
5	#22609828	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556525	C₂²	1	4	yes	—
8	C₂×#14556523	C₂²	1	4	yes	—
9	#17988455	—	2	2	no	—
10	#17988453	C₂²	1	4	yes	—
11	#17988450	—	2	2	no	—
12	#17988447	C₂²	1	4	yes	—
13	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
14	C₄×C₂²⋊Dic₇₃	—	2	2	no	—
15	#17988433	—	2	2	no	—
16	#17988350	C₂²	1	8	yes	—
17	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₇₃	C₄	1	8	yes	complemented
20	C₂×#97×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇₃	C₂³	1	8	yes	—
24	C₂×C₂²⋊Dic₇₃	—	2	4	no	—
25	C₂×C₄×Dic₇₃	—	2	4	no	—
26	#14556525	—	2	8	no	—
27	#14556523	—	2	8	no	—
28	#14556498	—	2	4	no	—
29	#14556498	—	2	4	no	—
30	#14556498	—	2	4	no	—
31	#14556498	—	2	4	no	—
32	C₂³×C₄×C₇₃	D₄	1	2	yes	—
33	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₇₃	—	2	4	no	—
35	C₂×M₁₆×C₇₃	—	2	4	no	—
36	C₂×M₁₆×C₇₃	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇₃	D₄	1	2	yes	—
41	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	#97×C₇₃	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
47	C₂²×Dic₇₃	—	2	8	no	—
48	C₄⋊Dic₇₃	—	2	8	no	—
49	C₄⋊Dic₇₃	—	2	8	no	—
50	C₂²⋊Dic₇₃	—	4	8	no	—
51	C₄×Dic₇₃	—	4	8	no	—
52	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	—	2	2	no	—
55	C₂²×C₄×C₇₃	—	4	4	no	—
56	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₇₃	—	2	16	no	—
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₇₃	#41	1	4	yes	—
69	C₂×C₈×C₇₃	#41	1	4	yes	—
70	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
71	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
75	C₂³×C₇₃	—	2	2	no	—
76	C₂³×C₇₃	—	2	4	no	—
77	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	—	2	4	no	—
81	C₂×C₄×C₇₃	—	2	4	no	—
82	C₂×C₄×C₇₃	—	2	4	no	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	—	4	8	no	—
86	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
93	C₂×Dic₇₃	—	4	8	no	—
94	C₂×Dic₇₃	—	4	8	no	—
95	C₂×Dic₇₃	—	4	8	no	—
96	C₈×C₇₃	—	2	4	no	—
97	C₈×C₇₃	#117	1	8	yes	—
98	C₂²×C₇₃	—	2	4	no	—
99	C₂²×C₇₃	—	2	2	no	—
100	C₂²×C₇₃	—	4	4	no	—
101	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₇₃	#532	1	2	yes	—
103	C₂²×C₇₃	#575	1	4	yes	—
104	C₄×C₇₃	—	2	4	no	—
105	C₄×C₇₃	—	2	4	no	—
106	C₄×C₇₃	—	2	2	no	—
107	C₄×C₇₃	C₂×#117	1	2	yes	—
108	C₄×C₇₃	#533	1	1	yes	characteristic
109	C₄×C₇₃	#533	1	1	yes	characteristic
110	Dic₇₃	—	8	8	no	—
111	#20825	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	—	4	2	no	—
113	C₂×C₇₃	#2285	1	1	yes	characteristic
114	C₂×C₇₃	#2911	1	2	yes	—
115	C₂×C₇₃	#3183	1	4	yes	—
116	C₂×C₄×M₁₆	D₇₃	1	1	yes	characteristic
117	#2283	—	73	2	no	—
118	#1846	—	73	2	no	—
119	#1835	—	73	2	no	—
120	#1809	—	73	4	no	—
121	#1776	—	73	4	no	—
122	C₇₃	#20825	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
125	C₂×#107	—	73	4	no	—
126	C₂×#106	—	73	4	no	—
127	C₄×M₁₆	Dic₇₃	1	8	yes	complemented
128	C₂×#97	C₂×D₇₃	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
131	#383	—	146	2	no	—
132	#382	—	73	4	no	—
133	#381	—	146	2	no	—
134	#380	—	73	4	no	—
135	C₄×C₄⋊C₄	—	73	4	no	—
136	C₄×C₂²⋊C₄	—	146	2	no	—
137	#376	—	146	2	no	—
138	#340	—	73	8	no	—
139	C₂³×C₄	D₂₉₂	1	2	yes	—
140	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	C₂×Dic₇₃	1	8	yes	—
144	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
145	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
146	C₂×C₄⋊C₄	—	73	8	no	—
147	C₂×C₂²⋊C₄	—	146	4	no	—
148	C₂×C₄²	—	146	4	no	—
149	C₂×C₄²	Dic₁₄₆	1	4	yes	—
150	C₂×C₄²	D₂₉₂	1	2	yes	—
151	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
152	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
154	#107	—	146	8	no	—
155	#106	—	146	8	no	—
156	#97	C₂×Dic₇₃	1	8	yes	—
157	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
158	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
159	#95	—	146	4	no	—
160	#95	—	146	4	no	—
161	#95	—	146	4	no	—
162	#95	—	146	4	no	—
163	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	—	146	8	no	—
168	C₂²×C₄	C₄⋊Dic₇₃	1	8	yes	—
169	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
170	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
172	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
173	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
174	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
175	M₁₆	—	2	16	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
179	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
180	C₂×C₈	#11924028	1	4	yes	—
181	C₂×C₈	#11924028	1	4	yes	—
182	C₄⋊C₄	—	146	8	no	—
183	C₄⋊C₄	—	146	8	no	—
184	C₂²⋊C₄	—	292	8	no	—
185	C₄²	—	292	8	no	—
186	C₄²	C₄⋊Dic₇₃	1	8	yes	—
187	C₄²	C₂²×Dic₇₃	1	2	yes	—
188	C₄²	C₂²×Dic₇₃	1	2	yes	—
189	C₄²	C₂×Dic₁₄₆	1	2	yes	—
190	C₄²	C₂×Dic₁₄₆	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
194	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	292	8	no	—
203	C₂×C₄	—	292	8	no	—
204	C₂×C₄	—	292	8	no	—
205	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
208	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
210	C₂×C₄	C₂×#11924028	1	2	yes	—
211	C₂×C₄	C₂×#11924028	1	2	yes	—
212	C₈	—	2	4	no	—
213	C₈	#14556550	1	8	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
218	C₂²	#17989137	1	2	yes	—
219	C₂²	#17989387	1	4	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	584	8	no	—
224	C₄	C₂×#14556550	1	2	yes	—
225	C₄	#17989144	1	1	yes	characteristic
226	C₄	#17989144	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	#22612338	1	1	yes	characteristic
229	C₂	#22618521	1	2	yes	—
230	C₂	#22620660	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₄×M₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇₃
hypocenter	C₇₃	#20825
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22610095	C₂	semidirect
3. #22610062	C₂	semidirect
5. #22609828	C₂	semidirect
19. C₄×M₁₆×C₇₃	C₄	semidirect
122. C₇₃	#20825	semidirect
127. C₄×M₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612329	2
#22610095	2
#22610062	2
#22609964	2
#22609828	2
C₂×C₄×M₁₆×C₇₃	2
#20825	73