Subgroups — #33084122

Order 18688.

1222 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988640	C₂	1	2	yes	maximal
2	#22609784	C₂	1	4	yes	maximal, complemented
3	#22609693	C₂	1	4	yes	maximal
4	#22609677	C₂	1	2	yes	maximal, complemented
5	C₄×#14556502	C₂	1	2	yes	maximal, complemented
6	C₂×C₄³×C₇₃	C₂	1	1	yes	maximal, characteristic
7	#17988640	C₄	1	16	yes	complemented
8	C₂×C₄⋊#10596831	C₂²	1	4	yes	—
9	C₂×C₂²⋊#10596831	C₂²	1	4	yes	—
10	C₂×#14556502	C₂²	1	8	yes	—
11	C₂×C₄×#10596831	C₂²	1	4	yes	—
12	#17988342	C₂²	1	8	yes	—
13	C₂²×C₄²×C₇₃	C₄	1	2	yes	—
14	C₂²×C₄²×C₇₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₄³×C₇₃	C₄	1	2	yes	—
18	C₄³×C₇₃	C₄	1	2	yes	—
19	C₄³×C₇₃	C₂²	1	1	yes	characteristic
20	C₄³×C₇₃	C₂²	1	2	yes	—
21	C₄³×C₇₃	C₂²	1	1	yes	characteristic
22	C₂²×#10596831	C₂³	1	8	yes	—
23	C₄⋊#10596831	C₂×C₄	1	16	yes	complemented
24	C₂²⋊#10596831	C₂×C₄	1	16	yes	—
25	#14556502	—	2	16	no	—
26	C₄×#10596831	C₂×C₄	1	16	yes	complemented
27	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	—	2	8	no	—
32	C₂×C₄²×C₇₃	—	2	8	no	—
33	C₂×C₄²×C₇₃	—	2	8	no	—
34	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂×C₄	1	8	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
48	C₂×#10596831	—	2	16	no	—
49	C₂×#10596831	C₂²×C₄	1	16	yes	—
50	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	—	2	8	no	—
52	C₂²×C₄×C₇₃	—	2	8	no	—
53	C₂²×C₄×C₇₃	—	2	8	no	—
54	C₂²×C₄×C₇₃	—	2	4	no	—
55	C₂²×C₄×C₇₃	—	2	4	no	—
56	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
57	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₇₃	M₁₆	1	8	yes	—
59	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₇₃	—	2	8	no	—
69	C₄²×C₇₃	—	2	8	no	—
70	C₄²×C₇₃	—	2	4	no	—
71	C₄²×C₇₃	—	2	8	no	—
72	C₄²×C₇₃	—	2	4	no	—
73	C₄²×C₇₃	—	2	4	no	—
74	C₄²×C₇₃	—	2	4	no	—
75	C₄²×C₇₃	C₄²	1	4	yes	—
76	C₄²×C₇₃	C₄²	1	4	yes	—
77	C₄²×C₇₃	M₁₆	1	8	yes	—
78	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
79	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₇₃	#41	1	8	yes	—
81	C₂³×C₇₃	—	2	2	no	—
82	C₂³×C₇₃	—	2	2	no	—
83	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
84	C₂³×C₇₃	C₂×M₁₆	1	4	yes	—
85	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₃	—	2	32	no	—
87	C₂×C₄×C₇₃	—	2	4	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	—	2	4	no	—
91	C₂×C₄×C₇₃	—	2	4	no	—
92	C₂×C₄×C₇₃	—	2	4	no	—
93	C₂×C₄×C₇₃	—	2	8	no	—
94	C₂×C₄×C₇₃	—	2	8	no	—
95	C₂×C₄×C₇₃	#97	1	16	yes	complemented
96	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₇₃	#117	1	8	yes	—
102	C₂×C₄×C₇₃	C₂×M₁₆	1	8	yes	—
103	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
105	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
108	#10596831	—	2	16	no	—
109	C₂²×C₇₃	—	2	2	no	—
110	C₂²×C₇₃	—	2	8	no	—
111	C₂²×C₇₃	—	2	2	no	—
112	C₂²×C₇₃	C₂×#97	1	4	yes	—
113	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×#117	1	2	yes	—
115	C₂²×C₇₃	C₂²×M₁₆	1	2	yes	—
116	C₂²×C₇₃	C₂²×M₁₆	1	2	yes	—
117	C₄×C₇₃	—	2	2	no	—
118	C₄×C₇₃	—	2	8	no	—
119	C₄×C₇₃	—	2	8	no	—
120	C₄×C₇₃	—	2	2	no	—
121	C₄×C₇₃	C₂×#97	1	4	yes	—
122	C₄×C₇₃	C₂×#97	1	4	yes	—
123	C₄×C₇₃	C₄×M₁₆	1	4	yes	—
124	C₄×C₇₃	C₄×#41	1	4	yes	—
125	#20463	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	—	2	4	no	—
127	C₂×C₇₃	C₄×#117	1	1	yes	characteristic
128	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
129	C₂×C₇₃	C₂×C₄×M₁₆	1	1	yes	characteristic
130	C₂×C₇₃	#2920	1	4	yes	—
131	C₂×#431	—	73	2	no	—
132	C₂×C₄³	D₇₃	1	1	yes	characteristic
133	#1767	—	73	4	no	—
134	#1749	—	73	4	no	—
135	#1744	—	73	2	no	—
136	C₄×#97	—	73	2	no	—
137	C₇₃	#20463	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄²	Dic₇₃	1	2	yes	—
139	C₂²×C₄²	Dic₇₃	1	2	yes	—
140	C₂²×C₄²	C₂×D₇₃	1	2	yes	—
141	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
142	#431	—	73	16	no	—
143	C₂×C₄⋊C₈	—	73	4	no	—
144	C₂×C₂²⋊C₈	—	73	4	no	—
145	C₂×#97	—	73	8	no	—
146	C₂×C₄×C₈	—	73	4	no	—
147	C₄³	Dic₇₃	1	2	yes	—
148	C₄³	Dic₇₃	1	2	yes	—
149	C₄³	C₂×D₇₃	1	2	yes	—
150	C₄³	C₂×D₇₃	1	1	yes	characteristic
151	C₄³	C₂×D₇₃	1	1	yes	characteristic
152	#336	—	73	8	no	—
153	C₂³×C₄	C₂×Dic₇₃	1	4	yes	—
154	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
156	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
157	C₂²×C₈	—	73	8	no	—
158	C₂×C₄²	—	2	8	no	—
159	C₂×C₄²	—	2	8	no	—
160	C₂×C₄²	—	2	8	no	—
161	C₂×C₄²	C₄×D₇₃	1	8	yes	—
162	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
163	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
164	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
165	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
166	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
167	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
168	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
171	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
172	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
174	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
175	C₄⋊C₈	—	73	16	no	—
176	C₂²⋊C₈	—	73	16	no	—
177	#97	—	146	16	no	—
178	C₄×C₈	—	73	16	no	—
179	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
186	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
187	C₂²×C₄	#11924012	1	8	yes	—
188	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
189	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
192	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
197	C₂×C₈	—	73	16	no	—
198	C₂×C₈	—	146	16	no	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	8	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	8	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	C₄×Dic₇₃	1	4	yes	—
207	C₄²	C₄×Dic₇₃	1	4	yes	—
208	C₄²	#11924012	1	8	yes	—
209	C₄²	C₂×C₄×D₇₃	1	4	yes	—
210	C₄²	C₂×C₄×D₇₃	1	4	yes	—
211	C₄²	#11924028	1	8	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
215	C₂³	C₂×#11924012	1	4	yes	—
216	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
217	C₂×C₄	—	2	32	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	#14556502	1	16	yes	complemented
227	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
230	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
231	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
232	C₂×C₄	#14556550	1	8	yes	—
233	C₂×C₄	C₂×#11924012	1	8	yes	—
234	C₂×C₄	C₂×#11924012	1	4	yes	—
235	C₂×C₄	C₂×#11924012	1	4	yes	—
236	C₂×C₄	C₂×#11924012	1	4	yes	—
237	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
238	C₂×C₄	C₂×#11924028	1	4	yes	—
239	C₈	—	146	16	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₂×#14556502	1	4	yes	—
244	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
245	C₂²	C₂×#14556550	1	2	yes	—
246	C₂²	C₂²×#11924012	1	2	yes	—
247	C₂²	C₂²×#11924012	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	8	no	—
251	C₄	—	2	2	no	—
252	C₄	C₂×#14556502	1	4	yes	—
253	C₄	C₂×#14556502	1	4	yes	—
254	C₄	C₄×#11924012	1	4	yes	—
255	C₄	C₄×#11924028	1	4	yes	—
256	C₂	—	2	4	no	—
257	C₂	C₄×#14556550	1	1	yes	characteristic
258	C₂	C₂²×#14556502	1	1	yes	characteristic
259	C₂	C₂×C₄×#11924012	1	1	yes	characteristic
260	C₂	#22618564	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₄³×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄³	D₇₃
hypocenter	C₇₃	#20463
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22609784	C₂	semidirect
4. #22609677	C₂	semidirect
5. C₄×#14556502	C₂	semidirect
7. #17988640	C₄	semidirect
23. C₄⋊#10596831	C₂×C₄	semidirect
26. C₄×#10596831	C₂×C₄	semidirect
95. C₂×C₄×C₇₃	#97	semidirect
137. C₇₃	#20463	semidirect
226. C₂×C₄	#14556502	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988640	2
#22609784	2
#22609693	2
#22609677	2
C₄×#14556502	2
C₂×C₄³×C₇₃	2
#20463	73