Subgroups — #32978747

Order 18688.

540 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609784	C₂	1	2	yes	maximal, complemented
2	C₂×#17988327	C₂	1	2	yes	maximal
3	#22607812	C₂	1	2	yes	maximal, complemented
4	C₄×#124×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	—	2	2	no	—
6	C₂×#14556502	C₂²	1	4	yes	—
7	#17988342	—	2	2	no	—
8	#17988327	—	2	4	no	—
9	C₂×#124×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₇₃	C₄	1	1	yes	characteristic
12	#391×C₇₃	C₄	1	1	yes	characteristic
13	#390×C₇₃	C₄	1	1	yes	characteristic
14	#388×C₇₃	—	2	2	no	—
15	#382×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₇₃	—	2	2	no	—
17	C₄³×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×#10596831	—	2	4	no	—
19	C₄⋊#10596831	—	2	8	no	—
20	#14556502	—	2	4	no	—
21	#14556502	—	2	4	no	—
22	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₇₃	—	2	1	no	—
25	#124×C₇₃	—	2	2	no	—
26	#124×C₇₃	D₄	1	2	yes	—
27	#124×C₇₃	Q₈	1	2	yes	—
28	C₄×Q₈×C₇₃	—	4	2	no	—
29	C₄×D₄×C₇₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
36	C₂×C₄²×C₇₃	—	2	1	no	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	D₄	1	2	yes	—
40	C₂×C₄²×C₇₃	D₄	1	2	yes	—
41	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	#95×C₇₃	—	2	2	no	—
43	#95×C₇₃	—	2	2	no	—
44	C₂×#10596831	—	2	8	no	—
45	C₂×#10596831	—	4	4	no	—
46	C₂×#10596831	—	4	4	no	—
47	C₂⁴×C₇₃	—	2	1	no	—
48	C₂×Q₈×C₇₃	—	2	1	no	—
49	C₂×Q₈×C₇₃	—	4	1	no	—
50	C₂×Q₈×C₇₃	—	4	1	no	—
51	C₂×Q₈×C₇₃	C₄⋊C₄	1	2	yes	—
52	C₂×D₄×C₇₃	—	2	1	no	—
53	C₂×D₄×C₇₃	—	4	1	no	—
54	C₂×D₄×C₇₃	—	4	1	no	—
55	C₂×D₄×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₃	—	2	1	no	—
57	C₂²×C₄×C₇₃	—	2	1	no	—
58	C₂²×C₄×C₇₃	—	2	1	no	—
59	C₂²×C₄×C₇₃	—	4	2	no	—
60	C₂²×C₄×C₇₃	—	4	4	no	—
61	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₇₃	—	4	1	no	—
67	C₄⋊C₄×C₇₃	—	4	1	no	—
68	C₄⋊C₄×C₇₃	—	4	2	no	—
69	C₂²⋊C₄×C₇₃	—	2	4	no	—
70	C₂²⋊C₄×C₇₃	—	2	4	no	—
71	C₂²⋊C₄×C₇₃	—	4	4	no	—
72	C₂²⋊C₄×C₇₃	—	4	2	no	—
73	C₄²×C₇₃	—	2	2	no	—
74	C₄²×C₇₃	—	2	4	no	—
75	C₄²×C₇₃	—	4	1	no	—
76	C₄²×C₇₃	—	4	2	no	—
77	C₄²×C₇₃	—	4	2	no	—
78	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₇₃	#41	1	2	yes	—
81	C₂³×C₇₃	—	2	2	no	—
82	C₂³×C₇₃	—	4	1	no	—
83	C₂³×C₇₃	—	4	1	no	—
84	C₂³×C₇₃	—	4	2	no	—
85	C₂³×C₇₃	—	4	2	no	—
86	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₇₃	—	4	2	no	—
88	Q₈×C₇₃	—	4	2	no	—
89	D₄×C₇₃	—	4	2	no	—
90	D₄×C₇₃	—	4	2	no	—
91	C₂×C₄×C₇₃	—	2	1	no	—
92	C₂×C₄×C₇₃	—	2	1	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	4	1	no	—
102	C₂×C₄×C₇₃	—	4	1	no	—
103	C₂×C₄×C₇₃	—	4	1	no	—
104	C₂×C₄×C₇₃	—	4	1	no	—
105	C₂×C₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	—	8	4	no	—
108	C₂×C₄×C₇₃	#100	1	2	yes	—
109	C₂×C₄×C₇₃	#104	1	4	yes	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#121	1	2	yes	—
113	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
114	#10596831	—	4	8	no	—
115	C₂²×C₇₃	—	4	2	no	—
116	C₂²×C₇₃	—	4	4	no	—
117	C₂²×C₇₃	—	8	1	no	—
118	C₂²×C₇₃	—	8	1	no	—
119	C₂²×C₇₃	—	8	2	no	—
120	C₂²×C₇₃	#380	1	1	yes	characteristic
121	C₂²×C₇₃	C₂×#100	1	1	yes	characteristic
122	C₂²×C₇₃	#413	1	1	yes	characteristic
123	C₂²×C₇₃	C₂×#104	1	2	yes	—
124	C₂²×C₇₃	#421	1	2	yes	—
125	C₄×C₇₃	—	2	1	no	—
126	C₄×C₇₃	—	2	1	no	—
127	C₄×C₇₃	—	2	2	no	—
128	C₄×C₇₃	—	2	2	no	—
129	C₄×C₇₃	—	4	1	no	—
130	C₄×C₇₃	—	4	2	no	—
131	C₄×C₇₃	—	4	2	no	—
132	C₄×C₇₃	—	8	1	no	—
133	C₄×C₇₃	—	8	1	no	—
134	C₄×C₇₃	#329	1	4	yes	complemented
135	#8579	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	—	8	2	no	—
137	C₂×C₇₃	C₂×#329	1	1	yes	characteristic
138	C₂×C₇₃	#1529	1	2	yes	—
139	C₂×C₇₃	#1536	1	1	yes	characteristic
140	C₂×C₇₃	#1800	1	1	yes	characteristic
141	C₂×C₇₃	#1806	1	2	yes	—
142	C₄×#124	D₇₃	1	1	yes	characteristic
143	#1767	—	73	2	no	—
144	C₂×#329	—	73	2	no	—
145	#1295	—	73	2	no	—
146	C₇₃	#8579	1	1	yes	characteristic, complemented, Sylow
147	C₂×#124	C₂×D₇₃	1	1	yes	characteristic
148	C₂×C₄×Q₈	Dic₇₃	1	1	yes	characteristic
149	C₂×C₄×D₄	Dic₇₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	146	2	no	—
151	C₂×#97	—	73	4	no	—
152	#391	Dic₇₃	1	1	yes	characteristic
153	#390	Dic₇₃	1	1	yes	characteristic
154	#388	—	2	2	no	—
155	#382	C₂×D₇₃	1	1	yes	characteristic
156	C₄×C₂²⋊C₄	—	2	2	no	—
157	C₄³	C₂×D₇₃	1	1	yes	characteristic
158	#336	—	146	2	no	—
159	#329	—	146	4	no	—
160	C₂²×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
161	C₂²×D₄	C₂×Dic₇₃	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂²×C₈	—	146	4	no	—
164	#124	—	2	2	no	—
165	#124	Dic₁₄₆	1	2	yes	—
166	#124	D₂₉₂	1	2	yes	—
167	C₄×Q₈	—	4	2	no	—
168	C₄×D₄	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
179	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
180	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
181	C₄⋊C₈	—	146	8	no	—
182	#97	—	146	4	no	—
183	#97	—	146	4	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	C₂⁴	—	2	1	no	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	C₄⋊Dic₇₃	1	2	yes	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	C₄⋊Dic₇₃	1	2	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
201	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
205	C₂×C₈	—	292	4	no	—
206	C₂×C₈	—	292	4	no	—
207	C₂×C₈	—	146	8	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	2	no	—
220	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
221	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
222	C₄²	#11924028	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	#14556508	1	2	yes	—
251	C₂×C₄	#14556518	1	4	yes	—
252	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
253	C₂×C₄	#14556550	1	1	yes	characteristic
254	C₂×C₄	#14556565	1	2	yes	—
255	C₂×C₄	#14556570	1	2	yes	—
256	C₈	—	292	8	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	1	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	#17988447	1	1	yes	characteristic
263	C₂²	C₂×#14556508	1	1	yes	characteristic
264	C₂²	#17988571	1	1	yes	characteristic
265	C₂²	C₂×#14556518	1	2	yes	—
266	C₂²	#17988604	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	#17988327	1	4	yes	complemented
277	C₂	—	8	2	no	—
278	C₂	C₂×#17988327	1	1	yes	characteristic
279	C₂	#22608538	1	2	yes	—
280	C₂	#22608566	1	1	yes	characteristic
281	C₂	#22609923	1	1	yes	characteristic
282	C₂	#22609951	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#124×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₇₃	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#124	D₇₃
hypocenter	C₇₃	#8579
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #22609784	C₂	semidirect
3. #22607812	C₂	semidirect
134. C₄×C₇₃	#329	semidirect
146. C₇₃	#8579	semidirect
276. C₄	#17988327	semidirect

Maximal subgroups

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

subgroup	index
#22609784	2
C₂×#17988327	2
#22607812	2
C₄×#124×C₇₃	2
#8579	73