Subgroups — #32975563

Order 18688.

794 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556550	C₂	1	1	yes	maximal, characteristic
3	#22607770	C₂	1	4	yes	maximal, complemented
4	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	—	2	1	no	—
6	C₂×#14556550	C₄	1	2	yes	—
7	C₄×#11924012	—	2	2	no	—
8	C₄×#11924012	C₄	1	4	yes	complemented
9	C₂×#14556502	—	2	1	no	—
10	C₂×C₄×#10596831	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₇₃	C₄	1	2	yes	—
12	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	C₄×C₂²⋊Dic₇₃	—	2	1	no	—
14	C₄×C₂²⋊Dic₇₃	—	2	1	no	—
15	#17988433	—	2	1	no	—
16	C₄²×Dic₇₃	—	2	1	no	—
17	C₂²×M₁₆×C₇₃	—	2	1	no	—
18	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	#423×C₇₃	—	2	2	no	—
20	#423×C₇₃	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
22	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
23	C₂×#11924012	—	2	8	no	—
24	C₂×#11924012	—	2	4	no	—
25	C₂²×#10596831	—	2	2	no	—
26	#14556550	—	2	8	no	—
27	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
29	C₂×C₄×Dic₇₃	—	2	2	no	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	#14556502	—	2	4	no	—
33	C₄×#10596831	—	2	2	no	—
34	C₄×#10596831	C₂×C₄	1	4	yes	—
35	#14556498	—	2	2	no	—
36	#14556498	—	2	2	no	—
37	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₇₃	—	2	4	no	—
41	C₂×M₁₆×C₇₃	—	4	4	no	—
42	C₂²×C₈×C₇₃	—	2	2	no	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	D₄	1	2	yes	—
49	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
50	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₇₃	—	2	4	no	—
52	C₄⋊C₈×C₇₃	—	2	2	no	—
53	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
54	C₂²×Dic₇₃	—	2	4	no	—
55	#11924012	—	2	16	no	—
56	C₂×#10596831	—	2	8	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₄⋊Dic₇₃	—	2	4	no	—
59	C₄⋊Dic₇₃	M₁₆	1	8	yes	complemented
60	C₂²⋊Dic₇₃	—	2	8	no	—
61	C₄×Dic₇₃	—	2	8	no	—
62	C₄×Dic₇₃	—	4	4	no	—
63	C₄×Dic₇₃	—	4	2	no	—
64	C₄×Dic₇₃	—	4	2	no	—
65	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	—	2	4	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	—	2	2	no	—
70	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	M₁₆×C₇₃	—	4	8	no	—
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₂×C₈×C₇₃	—	4	4	no	—
85	C₄²×C₇₃	C₄²	1	2	yes	—
86	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₇₃	—	2	2	no	—
96	C₂³×C₇₃	—	2	2	no	—
97	C₂³×C₇₃	—	2	2	no	—
98	C₂³×C₇₃	#95	1	2	yes	—
99	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	—	2	4	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	4	no	—
110	C₂×C₄×C₇₃	#95	1	2	yes	—
111	C₂×C₄×C₇₃	#95	1	2	yes	—
112	C₂×C₄×C₇₃	#95	1	2	yes	—
113	C₂×C₄×C₇₃	#104	1	4	yes	—
114	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₇₃	#137	1	2	yes	—
123	C₂×Dic₇₃	—	2	8	no	—
124	C₂×Dic₇₃	—	4	4	no	—
125	C₂×Dic₇₃	—	4	4	no	—
126	C₈×C₇₃	—	4	4	no	—
127	#10596831	—	2	8	no	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	4	4	no	—
132	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
133	C₂²×C₇₃	#407	1	2	yes	—
134	C₂²×C₇₃	#417	1	2	yes	—
135	C₂²×C₇₃	C₂×#104	1	2	yes	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	#339	1	4	yes	—
141	C₄×C₇₃	C₄×M₁₆	1	2	yes	—
142	C₄×C₇₃	#418	1	1	yes	characteristic
143	C₄×C₇₃	#419	1	1	yes	characteristic
144	C₄×C₇₃	#423	1	2	yes	—
145	C₄×C₇₃	#428	1	2	yes	—
146	Dic₇₃	—	4	8	no	—
147	#7903	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	4	2	no	—
149	C₂×C₇₃	#1489	1	4	yes	—
150	C₂×C₇₃	#1728	1	1	yes	characteristic
151	C₂×C₇₃	C₂×#339	1	1	yes	characteristic
152	C₂×C₇₃	#1736	1	1	yes	characteristic
153	C₂×#423	D₇₃	1	1	yes	characteristic
154	C₂×C₄×M₁₆	—	73	1	no	—
155	C₄×#117	—	73	1	no	—
156	#1279	—	73	4	no	—
157	C₇₃	#7903	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	2	1	no	—
159	C₂²×M₁₆	—	146	1	no	—
160	C₂×#117	—	73	2	no	—
161	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
162	#423	—	2	2	no	—
163	#423	Dic₇₃	1	4	yes	complemented
164	C₂×C₄⋊C₈	—	2	1	no	—
165	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
166	C₄×M₁₆	—	73	4	no	—
167	C₄×M₁₆	—	146	2	no	—
168	C₂×#97	—	146	1	no	—
169	C₂×C₄×C₈	—	73	2	no	—
170	C₄×C₄⋊C₄	—	73	2	no	—
171	C₄×C₄⋊C₄	—	73	2	no	—
172	C₄×C₂²⋊C₄	—	146	1	no	—
173	C₄×C₂²⋊C₄	—	146	1	no	—
174	#376	—	146	1	no	—
175	C₄³	—	146	1	no	—
176	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
177	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
178	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	4	4	no	—
181	C₂×M₁₆	—	146	8	no	—
182	C₂×M₁₆	—	146	4	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	146	2	no	—
185	#117	—	146	8	no	—
186	C₂×C₄⋊C₄	—	73	4	no	—
187	C₂×C₂²⋊C₄	—	146	2	no	—
188	C₂×C₄²	—	146	2	no	—
189	C₂×C₄²	—	146	2	no	—
190	C₂×C₄²	—	146	2	no	—
191	C₂×C₄²	Dic₁₄₆	1	2	yes	—
192	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₇₃	1	2	yes	—
194	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
202	#97	—	146	4	no	—
203	C₄×C₈	—	73	4	no	—
204	C₄×C₈	—	146	2	no	—
205	#95	—	146	2	no	—
206	#95	—	146	2	no	—
207	C₂⁴	#11924005	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	146	4	no	—
213	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
214	C₂²×C₄	#11924005	1	1	yes	characteristic
215	C₂²×C₄	#11924005	1	1	yes	characteristic
216	C₂²×C₄	#11924005	1	2	yes	—
217	C₂²×C₄	#11924005	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
219	C₂²×C₄	#11924007	1	2	yes	—
220	C₂²×C₄	#11924007	1	2	yes	—
221	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
225	M₁₆	—	4	8	no	—
226	M₁₆	—	146	16	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	146	8	no	—
230	C₂×C₈	—	146	4	no	—
231	C₄⋊C₄	—	73	8	no	—
232	C₄⋊C₄	—	146	4	no	—
233	C₂²⋊C₄	—	146	8	no	—
234	C₄²	—	292	4	no	—
235	C₄²	—	292	2	no	—
236	C₄²	—	292	2	no	—
237	C₄²	—	146	8	no	—
238	C₄²	C₄×Dic₇₃	1	2	yes	—
239	C₄²	#11924005	1	2	yes	—
240	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
241	C₄²	#11924007	1	2	yes	—
242	C₄²	#11924007	1	2	yes	—
243	C₄²	C₄⋊Dic₇₃	1	2	yes	—
244	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
245	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
247	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	#14556498	1	2	yes	—
252	C₂³	C₂×#11924005	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	292	4	no	—
264	C₂×C₄	—	292	4	no	—
265	C₂×C₄	—	146	8	no	—
266	C₂×C₄	#14556498	1	2	yes	—
267	C₂×C₄	#14556498	1	2	yes	—
268	C₂×C₄	#14556498	1	2	yes	—
269	C₂×C₄	#14556517	1	4	yes	—
270	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
274	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
276	C₂×C₄	M₁₆×D₇₃	1	4	yes	—
277	C₂×C₄	#14556617	1	2	yes	—
278	C₂×C₄	#14556622	1	2	yes	—
279	C₈	—	4	4	no	—
280	C₈	—	146	8	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×#14556498	1	1	yes	characteristic
286	C₂²	#17988550	1	2	yes	—
287	C₂²	#17988586	1	2	yes	—
288	C₂²	C₂×#14556517	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	292	8	no	—
294	C₄	#17988348	1	4	yes	—
295	C₄	M₁₆×Dic₇₃	1	2	yes	—
296	C₄	#17988592	1	1	yes	characteristic
297	C₄	#17988596	1	1	yes	characteristic
298	C₄	#17988610	1	2	yes	—
299	C₄	#17988627	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	#22608368	1	4	yes	—
302	C₂	#22609624	1	1	yes	characteristic
303	C₂	C₂×#17988348	1	1	yes	characteristic
304	C₂	#22609650	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	#7903
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #22607770	C₂	semidirect
8. C₄×#11924012	C₄	semidirect
20. #423×C₇₃	C₄	semidirect
59. C₄⋊Dic₇₃	M₁₆	semidirect
157. C₇₃	#7903	semidirect
163. #423	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924012	2
C₄×#14556550	2
#22607770	2
C₂×#423×C₇₃	2
#7903	73