Subgroups — #32979107

Order 18688.

482 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988652	C₂	1	1	yes	maximal, characteristic
2	#22610102	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608677	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608357	C₂	1	1	yes	maximal, characteristic
5	#22608355	C₂	1	1	yes	maximal, characteristic
6	#22607828	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	—	2	1	no	—
9	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
10	#17988652	—	2	1	no	—
11	#17988652	—	2	1	no	—
12	#17988652	C₄	1	2	yes	—
13	#17988652	C₄	1	2	yes	—
14	#17988640	C₂²	1	1	yes	characteristic
15	C₂×#14556526	—	2	1	no	—
16	C₂×#14556526	—	2	1	no	—
17	#17988611	C₂²	1	1	yes	characteristic
18	C₂×#14556503	—	2	1	no	—
19	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
20	#17988339	—	2	1	no	—
21	#17988337	—	2	1	no	—
22	#17988316	—	2	2	no	—
23	#17988314	—	2	2	no	—
24	#431×C₇₃	C₂²	1	1	yes	characteristic
25	#423×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₂²×D₂₉₂	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₇₃	—	2	1	no	—
29	C₂×#11924012	—	2	1	no	—
30	C₂²×#10596831	D₄	1	1	yes	characteristic
31	C₂²×#10596831	D₄	1	1	yes	characteristic
32	C₂×C₈×D₇₃	—	2	2	no	—
33	C₂×C₈×D₇₃	—	2	2	no	—
34	C₂×C₈×D₇₃	—	4	2	no	—
35	C₂×C₈×D₇₃	—	4	2	no	—
36	C₄×D₂₉₂	—	2	1	no	—
37	C₄×D₂₉₂	—	2	1	no	—
38	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
39	C₄×D₂₉₂	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₇₃	Q₈	1	1	yes	characteristic
41	C₂×#11924005	—	2	1	no	—
42	#14556526	—	4	2	no	—
43	#14556526	—	4	2	no	—
44	C₄⋊#10596831	—	2	1	no	—
45	C₄⋊#10596831	—	2	1	no	—
46	C₄⋊#10596831	—	2	1	no	—
47	C₂²⋊#10596831	—	2	1	no	—
48	#14556503	—	2	2	no	—
49	#14556503	—	2	2	no	—
50	C₄×#10596831	—	2	1	no	—
51	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
52	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
53	C₄×#10596831	D₄	1	1	yes	characteristic
54	C₄×#10596831	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₇₃	—	2	1	no	—
56	C₂²×C₈×C₇₃	—	2	1	no	—
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₃	—	2	1	no	—
59	C₄⋊C₈×C₇₃	—	2	1	no	—
60	C₄⋊C₈×C₇₃	—	2	1	no	—
61	C₄⋊C₈×C₇₃	—	2	1	no	—
62	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₇₃	—	2	1	no	—
64	C₄×C₈×C₇₃	—	2	1	no	—
65	C₂³×D₇₃	—	2	1	no	—
66	C₂×D₂₉₂	—	2	1	no	—
67	C₂×D₂₉₂	—	2	4	no	—
68	C₂×D₂₉₂	C₄⋊C₄	1	2	yes	—
69	C₂²×Dic₇₃	—	2	1	no	—
70	C₂×C₄×D₇₃	—	2	2	no	—
71	C₂×C₄×D₇₃	—	2	2	no	—
72	C₂×C₄×D₇₃	—	4	1	no	—
73	C₂×C₄×D₇₃	—	4	1	no	—
74	C₂×C₄×D₇₃	—	4	1	no	—
75	C₂×C₄×D₇₃	—	4	1	no	—
76	#11924012	—	4	1	no	—
77	#11924012	—	4	1	no	—
78	C₂×#10596831	—	2	1	no	—
79	C₂×#10596831	—	2	1	no	—
80	C₂×#10596831	—	4	1	no	—
81	C₂×#10596831	—	4	1	no	—
82	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#10596831	D₈	1	2	yes	—
87	C₂×#10596831	SD₁₆	1	2	yes	—
88	C₈×D₇₃	—	4	4	no	—
89	C₈×D₇₃	—	4	4	no	—
90	C₄⋊Dic₇₃	—	2	1	no	—
91	C₄⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
92	#11924005	—	2	2	no	—
93	#11924005	—	2	2	no	—
94	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
97	M₁₆×C₇₃	—	4	1	no	—
98	M₁₆×C₇₃	—	4	1	no	—
99	C₂×C₈×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	—	4	1	no	—
101	C₂×C₈×C₇₃	—	4	1	no	—
102	C₂×C₈×C₇₃	—	4	1	no	—
103	C₂×C₈×C₇₃	—	4	1	no	—
104	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₃	#41	1	1	yes	characteristic
108	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×D₇₃	—	2	4	no	—
110	C₂²×D₇₃	—	2	2	no	—
111	C₂²×D₇₃	—	4	2	no	—
112	C₂²×D₇₃	—	4	1	no	—
113	C₂²×D₇₃	—	4	1	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	4	1	no	—
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	#117	1	1	yes	characteristic
119	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	—	4	1	no	—
128	C₂×Dic₇₃	—	4	1	no	—
129	D₂₉₂	—	2	4	no	—
130	D₂₉₂	—	2	2	no	—
131	D₂₉₂	#108	1	4	yes	complemented
132	C₄×D₇₃	—	4	2	no	—
133	C₄×D₇₃	—	4	2	no	—
134	C₄×D₇₃	—	4	2	no	—
135	C₄×D₇₃	—	4	2	no	—
136	C₈×C₇₃	—	4	2	no	—
137	C₈×C₇₃	—	8	1	no	—
138	C₈×C₇₃	—	8	1	no	—
139	#10596831	—	2	1	no	—
140	#10596831	—	2	1	no	—
141	#10596831	—	8	1	no	—
142	#10596831	—	8	1	no	—
143	#10596831	#102	1	2	yes	—
144	#10596831	#102	1	2	yes	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	#380	1	1	yes	characteristic
148	C₂²×C₇₃	#474	1	1	yes	characteristic
149	C₂²×C₇₃	#476	1	1	yes	characteristic
150	C₂×D₇₃	—	2	4	no	—
151	C₂×D₇₃	—	4	4	no	—
152	C₂×D₇₃	—	4	4	no	—
153	C₂×D₇₃	—	4	2	no	—
154	C₂×D₇₃	—	4	2	no	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
159	C₄×C₇₃	#417	1	1	yes	characteristic
160	C₄×C₇₃	C₂×#108	1	2	yes	—
161	C₄×C₇₃	#459	1	1	yes	characteristic
162	C₄×C₇₃	#461	1	1	yes	characteristic
163	Dic₇₃	—	4	2	no	—
164	#8658	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	2	1	no	—
166	C₂×C₇₃	—	2	1	no	—
167	C₂×C₇₃	#1802	1	1	yes	characteristic
168	C₂×C₇₃	#1810	1	1	yes	characteristic
169	C₂×C₇₃	#1814	1	1	yes	characteristic
170	D₇₃	—	4	4	no	—
171	C₂×C₈×D₄	—	73	1	no	—
172	#1848	—	73	1	no	—
173	#1564	—	73	1	no	—
174	#1486	—	73	1	no	—
175	#1486	—	73	1	no	—
176	#1301	—	73	1	no	—
177	#1301	D₇₃	1	1	yes	characteristic, complemented
178	C₇₃	#8658	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	146	1	no	—
180	C₂×C₄×D₄	—	73	1	no	—
181	C₈×D₄	—	73	2	no	—
182	C₈×D₄	—	73	2	no	—
183	C₈×D₄	—	146	1	no	—
184	C₈×D₄	—	146	1	no	—
185	#431	—	73	1	no	—
186	#431	C₂×D₇₃	1	1	yes	characteristic
187	C₂×#108	—	146	1	no	—
188	C₂×#108	—	146	1	no	—
189	#423	—	73	1	no	—
190	#423	C₂×D₇₃	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₈	—	146	1	no	—
193	C₂×C₄×C₈	—	73	1	no	—
194	#335	—	146	1	no	—
195	#334	—	146	1	no	—
196	D₄⋊C₈	—	146	2	no	—
197	D₄⋊C₈	—	146	2	no	—
198	C₂²×D₄	—	73	1	no	—
199	C₂³×C₄	—	146	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	146	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	292	2	no	—
204	C₂²×C₈	—	292	2	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	146	2	no	—
208	C₂²×C₈	—	146	2	no	—
209	C₄×D₄	—	73	2	no	—
210	C₄×D₄	—	73	2	no	—
211	C₄×D₄	—	146	1	no	—
212	C₄×D₄	—	146	1	no	—
213	C₂×C₄⋊C₄	—	73	1	no	—
214	C₂×C₂²⋊C₄	—	146	1	no	—
215	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
216	#108	—	292	2	no	—
217	#108	—	292	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	146	1	no	—
223	C₄⋊C₈	—	146	1	no	—
224	C₄⋊C₈	—	146	1	no	—
225	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	146	2	no	—
228	C₂²⋊C₈	—	146	2	no	—
229	C₂²⋊C₈	—	146	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	73	1	no	—
232	C₄×C₈	—	73	1	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	—	73	1	no	—
235	C₄×C₈	—	146	1	no	—
236	C₂⁴	—	146	1	no	—
237	C₂×D₄	—	73	2	no	—
238	C₂×D₄	—	146	4	no	—
239	C₂×D₄	—	146	1	no	—
240	C₂²×C₄	—	292	1	no	—
241	C₂²×C₄	—	292	1	no	—
242	C₂²×C₄	—	292	1	no	—
243	C₂²×C₄	—	292	1	no	—
244	C₂²×C₄	—	146	1	no	—
245	C₂²×C₄	—	146	2	no	—
246	C₂²×C₄	—	146	2	no	—
247	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
248	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
249	C₂²×C₄	#11924028	1	1	yes	characteristic
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	292	1	no	—
253	M₁₆	—	292	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	292	4	no	—
260	C₂×C₈	—	292	4	no	—
261	C₂×C₈	—	292	1	no	—
262	C₂×C₈	—	292	1	no	—
263	C₂×C₈	—	73	1	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	—	73	2	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	146	1	no	—
270	C₂×C₈	—	146	1	no	—
271	C₄⋊C₄	—	73	2	no	—
272	C₄⋊C₄	—	146	1	no	—
273	C₂²⋊C₄	—	146	2	no	—
274	C₂²⋊C₄	—	146	2	no	—
275	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
276	C₄²	D₄×D₇₃	1	1	yes	characteristic
277	C₄²	D₄×D₇₃	1	1	yes	characteristic
278	C₄²	#11924028	1	1	yes	characteristic
279	C₂³	—	292	2	no	—
280	C₂³	—	292	1	no	—
281	C₂³	—	292	1	no	—
282	C₂³	—	146	4	no	—
283	C₂³	—	146	2	no	—
284	C₂³	#14556569	1	1	yes	characteristic
285	D₄	—	73	4	no	—
286	D₄	—	146	4	no	—
287	D₄	—	146	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	292	1	no	—
291	C₂×C₄	—	292	1	no	—
292	C₂×C₄	—	292	2	no	—
293	C₂×C₄	—	292	2	no	—
294	C₂×C₄	—	292	2	no	—
295	C₂×C₄	—	292	2	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
299	C₂×C₄	#14556548	1	1	yes	characteristic
300	C₂×C₄	#14556564	1	1	yes	characteristic
301	C₂×C₄	#14556564	1	1	yes	characteristic
302	C₂×C₄	#14556569	1	1	yes	characteristic
303	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₇₃	1	1	yes	characteristic
305	C₂×C₄	#14556618	1	1	yes	characteristic
306	C₂×C₄	#14556623	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	—	8	1	no	—
309	C₈	—	8	1	no	—
310	C₈	—	584	1	no	—
311	C₈	—	584	1	no	—
312	C₈	—	73	2	no	—
313	C₈	—	73	2	no	—
314	C₈	—	146	1	no	—
315	C₈	—	146	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	292	4	no	—
319	C₂²	—	292	4	no	—
320	C₂²	—	292	2	no	—
321	C₂²	—	292	2	no	—
322	C₂²	—	146	4	no	—
323	C₂²	#17988446	1	1	yes	characteristic
324	C₂²	#17988866	1	1	yes	characteristic
325	C₂²	#17988876	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	292	2	no	—
330	C₄	#102×D₇₃	1	1	yes	characteristic
331	C₄	#17988587	1	1	yes	characteristic
332	C₄	#108×D₇₃	1	2	yes	—
333	C₄	#17988781	1	1	yes	characteristic
334	C₄	#17988791	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	292	4	no	—
338	C₂	#22609930	1	1	yes	characteristic
339	C₂	#22609968	1	1	yes	characteristic
340	C₂	#22609984	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556564
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1301	C₂
socle	C₂²×C₇₃	#380
prime radical	C₂²×D₂₉₂	Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1301	D₇₃
hypocenter	C₇₃	#8658
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #22610102	C₂	semidirect
3. #22608677	C₂	semidirect
6. #22607828	C₂	semidirect
7. C₇₃×#1301	C₂	semidirect
131. D₂₉₂	#108	semidirect
177. #1301	D₇₃	semidirect
178. C₇₃	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988652	2
#22610102	2
#22608677	2
#22608357	2
#22608355	2
#22607828	2
C₇₃×#1301	2
#8658	73