Subgroups — #33055761

Order 18688.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618892	C₂	1	2	yes	maximal, complemented
2	C₄×#14556613	C₂	1	1	yes	maximal, characteristic
3	#22608844	C₂	1	2	yes	maximal, complemented
4	#22608839	C₂	1	2	yes	maximal, complemented
5	#22608832	C₂	1	4	yes	maximal, complemented
6	#22608311	C₂	1	2	yes	maximal, complemented
7	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₄	1	1	yes	characteristic
10	C₄×#11924028	C₂²	1	2	yes	—
11	#17988714	C₂²	1	4	yes	—
12	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
14	C₄×#11924017	C₄	1	2	yes	—
15	C₄×#11924017	C₂²	1	2	yes	—
16	C₄×D₅₈₄	C₄	1	1	yes	characteristic
17	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
18	#17988664	C₂²	1	4	yes	—
19	#17988655	—	2	4	no	—
20	#17988638	—	2	2	no	—
21	#17988625	C₄	1	1	yes	characteristic
22	#17988584	C₄	1	2	yes	—
23	C₄×#11924011	—	2	2	no	—
24	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
26	#17988317	C₂²	1	4	yes	—
27	#17988314	C₂²	1	4	yes	—
28	#17988308	C₂²	1	2	yes	—
29	#17988308	C₂²	1	2	yes	—
30	#431×C₇₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#335×C₇₃	C₂²	1	2	yes	—
33	#334×C₇₃	C₂²	1	2	yes	—
34	C₂×#11924028	C₂×C₄	1	2	yes	—
35	#14556613	—	2	4	no	—
36	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924017	C₂×C₄	1	2	yes	—
38	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924011	—	2	4	no	—
40	C₂²×#10596831	—	2	1	no	—
41	C₂²×#10596831	D₄	1	2	yes	—
42	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₄₆	C₂³	1	2	yes	—
44	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
45	C₄×D₂₉₂	C₂³	1	2	yes	—
46	C₄×C₂²⋊D₇₃	—	2	2	no	—
47	C₄×C₂²⋊D₇₃	—	2	2	no	—
48	#14556550	C₂×C₄	1	2	yes	—
49	#14556548	—	2	2	no	—
50	C₄²×D₇₃	—	2	2	no	—
51	#14556525	C₂×C₄	1	1	yes	characteristic
52	#14556525	C₂×C₄	1	1	yes	characteristic
53	#14556523	C₂×C₄	1	2	yes	—
54	#14556520	—	2	4	no	—
55	#14556516	C₂×C₄	1	2	yes	—
56	#14556516	C₂×C₄	1	2	yes	—
57	#14556513	C₂×C₄	1	2	yes	—
58	#14556513	C₂×C₄	1	2	yes	—
59	#14556503	—	2	4	no	—
60	#14556501	—	2	4	no	—
61	C₄×#10596831	—	2	1	no	—
62	C₄×#10596831	D₄	1	2	yes	—
63	C₄×#10596831	C₂³	1	2	yes	—
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
69	C₂²⋊C₈×C₇₃	—	2	2	no	—
70	C₄×C₈×C₇₃	—	2	2	no	—
71	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	#11924028	—	2	4	no	—
76	#11924028	—	2	4	no	—
77	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
78	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
79	C₂×C₂²⋊D₇₃	—	2	2	no	—
80	C₂×C₄×D₇₃	—	2	2	no	—
81	C₂×C₄×D₇₃	—	2	2	no	—
82	C₂×C₄×D₇₃	—	2	2	no	—
83	Dic₂₉₂	—	2	2	no	—
84	#11924017	—	2	4	no	—
85	D₅₈₄	—	2	2	no	—
86	#11924011	—	4	8	no	—
87	C₂×#10596831	—	2	2	no	—
88	C₂×#10596831	—	4	2	no	—
89	C₂×#10596831	—	4	2	no	—
90	C₂×#10596831	C₂×D₄	1	4	yes	—
91	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
92	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
93	#11924007	—	2	2	no	—
94	#11924007	—	2	2	no	—
95	C₂²⋊Dic₇₃	—	2	2	no	—
96	#11924005	—	2	2	no	—
97	#11924005	—	2	2	no	—
98	C₄×Dic₇₃	—	2	2	no	—
99	C₄×Dic₇₃	—	2	2	no	—
100	C₄×Dic₇₃	—	2	2	no	—
101	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	—	2	4	no	—
106	C₂×C₈×C₇₃	M₁₆	1	4	yes	complemented
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇₃	#41	1	1	yes	characteristic
114	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
115	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×D₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₇₃	#131	1	2	yes	—
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
127	C₂²⋊D₇₃	—	4	4	no	—
128	C₂×Dic₇₃	—	2	2	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	2	no	—
131	D₂₉₂	—	2	2	no	—
132	D₂₉₂	—	2	2	no	—
133	C₄×D₇₃	—	4	4	no	—
134	C₄×D₇₃	—	4	4	no	—
135	C₄×D₇₃	—	4	4	no	—
136	Dic₁₄₆	—	2	2	no	—
137	Dic₁₄₆	—	2	2	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	—	4	4	no	—
141	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
142	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
143	#10596831	—	2	4	no	—
144	#10596831	—	4	2	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	#436	1	2	yes	—
148	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×D₇₃	—	4	2	no	—
150	C₂×D₇₃	—	4	2	no	—
151	C₂×D₇₃	—	4	2	no	—
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	#436	1	2	yes	—
155	C₄×C₇₃	#444	1	2	yes	—
156	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
158	C₄×C₇₃	#575	1	2	yes	—
159	Dic₇₃	—	4	2	no	—
160	Dic₇₃	—	4	2	no	—
161	Dic₇₃	—	4	2	no	—
162	#17890	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	—	2	2	no	—
164	C₂×C₇₃	C₂×#436	1	1	yes	characteristic
165	C₂×C₇₃	#2943	1	1	yes	characteristic
166	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
167	D₇₃	—	8	2	no	—
168	#2965	—	73	2	no	—
169	C₄×#135	—	73	1	no	—
170	#1594	—	73	2	no	—
171	#1593	—	73	2	no	—
172	#1592	—	73	4	no	—
173	#1564	D₇₃	1	1	yes	characteristic, complemented
174	#1476	—	73	2	no	—
175	C₂×#322	—	73	1	no	—
176	C₇₃	#17890	1	1	yes	characteristic, complemented, Sylow
177	C₂×#135	—	73	1	no	—
178	C₄×#41	—	73	2	no	—
179	#446	—	73	4	no	—
180	C₄×Q₁₆	—	73	1	no	—
181	C₄×Q₁₆	—	73	1	no	—
182	C₄×SD₁₆	—	73	2	no	—
183	C₄×SD₁₆	—	73	2	no	—
184	C₄×D₈	—	73	1	no	—
185	C₄×D₈	—	73	1	no	—
186	#436	—	73	4	no	—
187	#435	—	146	4	no	—
188	#431	—	146	2	no	—
189	#431	C₂×D₇₃	1	2	yes	—
190	#427	—	73	1	no	—
191	#416	—	73	2	no	—
192	C₄×M₁₆	—	146	2	no	—
193	C₂×C₄×C₈	—	73	1	no	—
194	C₂×C₄×C₈	—	73	1	no	—
195	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
196	#335	C₂×D₇₃	1	2	yes	—
197	#334	C₂×D₇₃	1	2	yes	—
198	Q₈⋊C₈	—	73	4	no	—
199	D₄⋊C₈	—	73	4	no	—
200	#322	—	73	2	no	—
201	#322	—	73	2	no	—
202	C₂×#41	—	73	2	no	—
203	#135	—	146	4	no	—
204	C₂×Q₁₆	—	73	1	no	—
205	C₂×SD₁₆	—	73	2	no	—
206	C₂×D₈	—	73	1	no	—
207	C₂×M₁₆	—	146	4	no	—
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	146	1	no	—
210	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
212	C₄×Q₈	—	73	2	no	—
213	C₄×Q₈	—	73	2	no	—
214	C₄×D₄	—	73	2	no	—
215	C₄×D₄	—	73	2	no	—
216	C₄×D₄	—	146	2	no	—
217	C₄×D₄	—	146	2	no	—
218	#117	—	73	2	no	—
219	#117	—	146	2	no	—
220	C₂×C₄²	—	146	2	no	—
221	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
222	#107	—	73	1	no	—
223	#107	—	73	1	no	—
224	#106	—	73	2	no	—
225	C₄⋊C₈	—	146	4	no	—
226	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
227	#103	—	73	2	no	—
228	#103	—	73	2	no	—
229	#102	—	73	2	no	—
230	#102	—	73	2	no	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₂²⋊C₈	—	146	4	no	—
233	#97	—	146	4	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	—	73	2	no	—
236	C₄×C₈	—	73	2	no	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	146	1	no	—
239	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
240	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
243	#41	—	146	4	no	—
244	#41	—	146	4	no	—
245	C₂×Q₈	—	73	2	no	—
246	C₂×D₄	—	73	2	no	—
247	C₂×D₄	—	146	2	no	—
248	C₂²×C₄	—	146	2	no	—
249	C₂²×C₄	—	146	2	no	—
250	C₂²×C₄	—	146	2	no	—
251	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
253	C₂²×C₄	#11924029	1	1	yes	characteristic
254	Q₁₆	—	146	2	no	—
255	SD₁₆	—	146	4	no	—
256	D₈	—	146	2	no	—
257	M₁₆	—	292	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	292	2	no	—
261	C₂×C₈	—	292	2	no	—
262	C₂×C₈	—	73	4	no	—
263	C₂×C₈	—	146	2	no	—
264	C₂×C₈	#11924011	1	4	yes	complemented
265	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
269	C₄⋊C₄	—	73	2	no	—
270	C₄⋊C₄	—	73	2	no	—
271	C₄⋊C₄	—	146	2	no	—
272	C₄⋊C₄	—	146	2	no	—
273	C₂²⋊C₄	—	146	2	no	—
274	C₂²⋊C₄	—	146	2	no	—
275	C₂²⋊C₄	—	146	2	no	—
276	C₄²	—	146	2	no	—
277	C₄²	—	146	2	no	—
278	C₄²	—	146	2	no	—
279	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
280	C₄²	D₄×D₇₃	1	1	yes	characteristic
281	C₄²	#11924029	1	1	yes	characteristic
282	C₄²	C₂³×D₇₃	1	1	yes	characteristic
283	C₂³	—	146	2	no	—
284	C₂³	#14556553	1	1	yes	characteristic
285	Q₈	—	146	2	no	—
286	Q₈	—	146	2	no	—
287	D₄	—	292	4	no	—
288	D₄	—	146	2	no	—
289	D₄	—	146	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	292	4	no	—
293	C₂×C₄	—	292	4	no	—
294	C₂×C₄	—	292	4	no	—
295	C₂×C₄	—	146	2	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	—	146	2	no	—
298	C₂×C₄	#14556553	1	1	yes	characteristic
299	C₂×C₄	#14556553	1	1	yes	characteristic
300	C₂×C₄	#14556553	1	1	yes	characteristic
301	C₂×C₄	C₂×#11924011	1	2	yes	—
302	C₂×C₄	#14556599	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
305	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	4	no	—
309	C₈	—	292	2	no	—
310	C₈	—	146	4	no	—
311	C₈	C₂×#11924011	1	2	yes	—
312	C₈	C₂×#11924011	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	292	2	no	—
316	C₂²	—	292	2	no	—
317	C₂²	—	292	2	no	—
318	C₂²	#17988664	1	2	yes	—
319	C₂²	C₂×#14556553	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	292	2	no	—
323	C₄	—	292	2	no	—
324	C₄	—	292	2	no	—
325	C₄	#17988664	1	2	yes	—
326	C₄	#17988708	1	2	yes	—
327	C₄	C₂²×#11924011	1	1	yes	characteristic
328	C₄	C₂×#14556599	1	1	yes	characteristic
329	C₄	#17989384	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	584	2	no	—
332	C₂	C₂×#17988664	1	1	yes	characteristic
333	C₂	#22618731	1	1	yes	characteristic
334	C₂	C₂×#17988708	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1564	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556613	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#17890
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #22618892	C₂	semidirect
3. #22608844	C₂	semidirect
4. #22608839	C₂	semidirect
5. #22608832	C₂	semidirect
6. #22608311	C₂	semidirect
7. C₂×#17988308	C₂	semidirect
8. C₇₃×#1564	C₂	semidirect
106. C₂×C₈×C₇₃	M₁₆	semidirect
173. #1564	D₇₃	semidirect
176. C₇₃	#17890	semidirect
264. C₂×C₈	#11924011	semidirect

Maximal subgroups

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

subgroup	index
#22618892	2
C₄×#14556613	2
#22608844	2
#22608839	2
#22608832	2
#22608311	2
C₂×#17988308	2
C₇₃×#1564	2
#17890	73