Subgroups — #33002986

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988828	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556527	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988404	C₂	1	2	yes	maximal, complemented
4	#22608040	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂²×#11924012	—	2	1	no	—
8	C₂×#14556570	—	2	1	no	—
9	#17988828	—	2	1	no	—
10	#17988828	—	2	1	no	—
11	#17988828	C₄	1	2	yes	—
12	#17988828	C₄	1	2	yes	—
13	C₂×#14556527	—	2	4	no	—
14	C₂×#14556527	—	2	1	no	—
15	C₂×#14556527	C₂²	1	2	yes	—
16	C₂×#14556525	C₂²	1	1	yes	characteristic
17	C₂×#14556516	—	2	1	no	—
18	#17988404	—	2	4	no	—
19	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
21	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
22	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
23	C₂²×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₂₉₂	—	2	2	no	—
25	C₂×Dic₂₉₂	—	2	1	no	—
26	C₂×Dic₂₉₂	—	2	2	no	—
27	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
28	C₂×#11924012	—	2	4	no	—
29	C₂×#11924012	—	2	2	no	—
30	C₂×#11924012	—	4	2	no	—
31	C₂×#11924012	—	4	1	no	—
32	C₂²×#10596831	—	4	1	no	—
33	#14556570	—	2	2	no	—
34	#14556570	—	2	2	no	—
35	#14556570	—	4	1	no	—
36	#14556570	—	4	1	no	—
37	C₂×C₄⋊Dic₇₃	—	2	1	no	—
38	C₂×#11924007	—	4	1	no	—
39	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
40	#14556527	—	2	4	no	—
41	#14556527	—	2	2	no	—
42	#14556527	D₄	1	4	yes	—
43	#14556525	—	2	1	no	—
44	#14556525	C₂×C₄	1	2	yes	—
45	#14556516	—	2	2	no	—
46	#14556516	—	2	2	no	—
47	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	—	2	2	no	—
49	C₂²×C₈×C₇₃	—	2	2	no	—
50	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
56	M₃₂×C₇₃	—	2	2	no	—
57	M₃₂×C₇₃	—	2	2	no	—
58	C₂×C₁₆×C₇₃	—	2	2	no	—
59	C₂×Dic₁₄₆	—	2	2	no	—
60	C₂×Dic₁₄₆	—	4	1	no	—
61	C₂×Dic₁₄₆	—	4	1	no	—
62	C₂×Dic₁₄₆	—	4	1	no	—
63	C₂×Dic₁₄₆	—	4	1	no	—
64	C₂×Dic₁₄₆	—	4	1	no	—
65	C₂²×Dic₇₃	—	4	1	no	—
66	C₂²×Dic₇₃	—	4	1	no	—
67	Dic₂₉₂	—	2	4	no	—
68	Dic₂₉₂	—	4	2	no	—
69	#11924012	—	2	4	no	—
70	#11924012	—	4	4	no	—
71	#11924012	—	4	2	no	—
72	C₂×#10596831	—	4	4	no	—
73	C₂×#10596831	—	4	2	no	—
74	C₄⋊Dic₇₃	—	2	2	no	—
75	C₄⋊Dic₇₃	—	4	1	no	—
76	#11924007	—	4	2	no	—
77	#11924007	—	4	2	no	—
78	C₂²⋊Dic₇₃	—	4	2	no	—
79	C₂²⋊Dic₇₃	—	4	2	no	—
80	C₂⁴×C₇₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	—	2	1	no	—
82	C₂²×C₄×C₇₃	—	2	1	no	—
83	C₂²×C₄×C₇₃	—	2	1	no	—
84	C₂²×C₄×C₇₃	—	2	1	no	—
85	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	—	2	1	no	—
92	C₂×C₈×C₇₃	—	2	1	no	—
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	—	2	2	no	—
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	#41	1	2	yes	—
103	C₁₆×C₇₃	—	4	2	no	—
104	C₂³×C₇₃	—	2	2	no	—
105	C₂³×C₇₃	—	2	1	no	—
106	C₂³×C₇₃	—	2	1	no	—
107	C₂³×C₇₃	—	4	1	no	—
108	C₂³×C₇₃	#102	1	1	yes	characteristic
109	C₂³×C₇₃	#102	1	1	yes	characteristic
110	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	—	2	1	no	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
122	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
125	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
129	C₂×Dic₇₃	—	4	2	no	—
130	C₂×Dic₇₃	—	4	2	no	—
131	C₂×Dic₇₃	—	8	2	no	—
132	C₂×Dic₇₃	—	8	1	no	—
133	C₂×Dic₇₃	—	8	1	no	—
134	Dic₁₄₆	—	4	2	no	—
135	Dic₁₄₆	—	4	2	no	—
136	Dic₁₄₆	—	8	1	no	—
137	Dic₁₄₆	—	8	1	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	C₄×D₄	1	2	yes	—
141	C₈×C₇₃	#121	1	2	yes	—
142	#10596831	—	4	4	no	—
143	C₂²×C₇₃	—	2	1	no	—
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	4	2	no	—
151	C₂²×C₇₃	—	4	2	no	—
152	C₂²×C₇₃	#362	1	2	yes	complemented
153	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
154	C₂²×C₇₃	#448	1	1	yes	characteristic
155	C₂²×C₇₃	#481	1	1	yes	characteristic
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	#386	1	1	yes	characteristic
159	C₄×C₇₃	#417	1	1	yes	characteristic
160	C₄×C₇₃	#447	1	1	yes	characteristic
161	C₄×C₇₃	#480	1	1	yes	characteristic
162	Dic₇₃	—	8	2	no	—
163	Dic₇₃	—	8	2	no	—
164	#13003	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	2	2	no	—
166	C₂×C₇₃	—	2	1	no	—
167	C₂×C₇₃	—	2	1	no	—
168	C₂×C₇₃	—	4	1	no	—
169	C₂×C₇₃	#1874	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#362	1	1	yes	characteristic
171	C₂×C₇₃	#2148	1	1	yes	characteristic
172	C₂×#467	—	73	1	no	—
173	C₂²×#108	—	73	1	no	—
174	C₂×#362	—	73	2	no	—
175	#2112	D₇₃	1	1	yes	characteristic
176	#1387	—	73	2	no	—
177	C₇₃	#13003	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	73	1	no	—
179	C₂²×M₁₆	—	146	1	no	—
180	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
181	C₂×C₂²⋊Q₈	—	146	1	no	—
182	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
183	#467	—	73	2	no	—
184	#467	—	73	2	no	—
185	#467	—	146	1	no	—
186	#467	—	146	1	no	—
187	C₂×#108	—	73	2	no	—
188	C₂×#108	—	146	4	no	—
189	C₂×#108	—	146	1	no	—
190	C₂×#107	—	73	1	no	—
191	C₂×#103	—	146	1	no	—
192	#362	—	146	4	no	—
193	C₂²⋊C₁₆	—	2	1	no	—
194	C₂²⋊C₁₆	—	2	1	no	—
195	C₂²×Q₈	—	146	1	no	—
196	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
197	C₂×Q₁₆	—	73	2	no	—
198	C₂×Q₁₆	—	146	2	no	—
199	C₂×Q₁₆	—	146	1	no	—
200	C₂×Q₁₆	—	146	2	no	—
201	C₂×M₁₆	—	292	2	no	—
202	C₂×M₁₆	—	292	1	no	—
203	C₂×M₁₆	—	146	4	no	—
204	C₂×M₁₆	—	146	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	292	1	no	—
208	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
210	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
211	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
214	C₂²⋊Q₈	—	292	1	no	—
215	C₂²⋊Q₈	—	292	1	no	—
216	C₂²⋊Q₈	—	146	2	no	—
217	C₂²⋊Q₈	—	146	2	no	—
218	C₂×C₄⋊C₄	—	292	1	no	—
219	C₂×C₄⋊C₄	—	146	1	no	—
220	C₂×C₂²⋊C₄	—	292	1	no	—
221	M₃₂	—	2	2	no	—
222	M₃₂	—	2	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	#108	—	73	4	no	—
225	#108	—	146	4	no	—
226	#108	—	146	2	no	—
227	#107	—	73	2	no	—
228	#107	—	146	1	no	—
229	#103	—	146	2	no	—
230	#103	—	146	2	no	—
231	C₂⁴	#11924017	1	1	yes	characteristic
232	C₂×Q₈	—	292	1	no	—
233	C₂×Q₈	—	292	1	no	—
234	C₂×Q₈	—	292	1	no	—
235	C₂×Q₈	—	292	1	no	—
236	C₂×Q₈	—	292	1	no	—
237	C₂×Q₈	—	146	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	292	1	no	—
243	C₂²×C₄	—	292	1	no	—
244	C₂²×C₄	#11924005	1	1	yes	characteristic
245	C₂²×C₄	#11924005	1	1	yes	characteristic
246	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
247	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
248	C₂²×C₄	#11924017	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
250	Q₁₆	—	292	2	no	—
251	Q₁₆	—	146	4	no	—
252	M₁₆	—	292	4	no	—
253	M₁₆	—	292	2	no	—
254	M₁₆	—	146	4	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	292	4	no	—
262	C₂×C₈	—	292	2	no	—
263	C₂×C₈	#11924005	1	1	yes	characteristic
264	C₂×C₈	#11924005	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
266	C₂×C₈	D₄×D₇₃	1	2	yes	—
267	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
268	C₂×C₈	#11924027	1	2	yes	—
269	C₄⋊C₄	—	292	2	no	—
270	C₄⋊C₄	—	292	2	no	—
271	C₄⋊C₄	—	292	1	no	—
272	C₄⋊C₄	—	146	2	no	—
273	C₂²⋊C₄	—	292	2	no	—
274	C₂²⋊C₄	—	292	2	no	—
275	C₁₆	—	4	2	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	#14556513	1	1	yes	characteristic
281	C₂³	#14556513	1	1	yes	characteristic
282	C₂³	C₂×#11924017	1	1	yes	characteristic
283	Q₈	—	292	2	no	—
284	Q₈	—	292	2	no	—
285	Q₈	—	584	1	no	—
286	Q₈	—	584	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	292	2	no	—
298	C₂×C₄	—	292	2	no	—
299	C₂×C₄	—	584	2	no	—
300	C₂×C₄	—	584	1	no	—
301	C₂×C₄	—	584	1	no	—
302	C₂×C₄	#14556513	1	1	yes	characteristic
303	C₂×C₄	#14556513	1	1	yes	characteristic
304	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
305	C₂×C₄	#14556561	1	1	yes	characteristic
306	C₂×C₄	#14556575	1	1	yes	characteristic
307	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
308	C₂×C₄	#14556619	1	1	yes	characteristic
309	C₂×C₄	#14556624	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	292	4	no	—
313	C₈	#14556551	1	2	yes	—
314	C₈	#14556563	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	#17988404	1	2	yes	complemented
325	C₂²	C₂×#14556513	1	1	yes	characteristic
326	C₂²	#17988728	1	1	yes	characteristic
327	C₂²	#17988906	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	584	2	no	—
331	C₄	—	584	2	no	—
332	C₄	#17988464	1	1	yes	characteristic
333	C₄	#17988589	1	1	yes	characteristic
334	C₄	#17988723	1	1	yes	characteristic
335	C₄	#17988901	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	4	1	no	—
340	C₂	#22610246	1	1	yes	characteristic
341	C₂	C₂×#17988404	1	1	yes	characteristic
342	C₂	#22611681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556513
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2112	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂⁴×C₇₃	SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2112	D₇₃
hypocenter	C₇₃	#13003
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
3. C₂×#17988404	C₂	semidirect
4. #22608040	C₂	semidirect
152. C₂²×C₇₃	#362	semidirect
177. C₇₃	#13003	semidirect
324. C₂²	#17988404	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988828	2
C₂²×#14556527	2
C₂×#17988404	2
#22608040	2
C₇₃×#2112	2
#13003	73