Subgroups — #33057742

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618675	C₂	1	2	yes	maximal, complemented
2	#22608816	C₂	1	2	yes	maximal, complemented
3	#22608761	C₂	1	4	yes	maximal, complemented
4	D₈×#10596831	C₂	1	2	yes	maximal, complemented
5	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608330	C₂	1	2	yes	maximal, complemented
7	C₈×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#3169	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556598	—	2	2	no	—
10	#17988665	C₂²	1	4	yes	—
11	#17988659	—	2	4	no	—
12	D₄×#10596831	C₂²	1	4	yes	—
13	#17988640	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
15	#17988554	—	2	2	no	—
16	C₄×#11924012	C₂²	1	1	yes	characteristic
17	#17988340	C₂²	1	1	yes	characteristic
18	#17988340	C₂²	1	2	yes	—
19	#17988340	C₂²	1	1	yes	characteristic
20	D₄⋊#10596831	C₂²	1	4	yes	—
21	D₄⋊#10596831	C₂²	1	4	yes	—
22	#17988309	C₂²	1	2	yes	—
23	C₈×#10596831	C₂²	1	2	yes	—
24	#532×C₇₃	C₂²	1	2	yes	—
25	#500×C₇₃	C₄	1	1	yes	characteristic
26	#493×C₇₃	C₄	1	1	yes	characteristic
27	#486×C₇₃	C₄	1	2	yes	—
28	#466×C₇₃	C₄	1	2	yes	—
29	#466×C₇₃	C₄	1	2	yes	—
30	C₄×D₈×C₇₃	C₂²	1	2	yes	—
31	C₄×D₈×C₇₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	#14556598	—	4	8	no	—
34	C₂×#11924012	—	2	4	no	—
35	C₂×#11924012	D₄	1	1	yes	characteristic
36	C₂×#11924012	D₄	1	1	yes	characteristic
37	C₂²×#10596831	—	2	4	no	—
38	C₂²×#10596831	—	2	2	no	—
39	C₄⋊#10596831	—	2	2	no	—
40	C₄⋊#10596831	—	2	1	no	—
41	C₄⋊#10596831	C₂³	1	2	yes	—
42	C₄⋊#10596831	C₂³	1	2	yes	—
43	C₂²⋊#10596831	—	2	4	no	—
44	C₂²⋊#10596831	—	2	4	no	—
45	C₂²⋊#10596831	—	2	1	no	—
46	#14556502	D₄	1	2	yes	—
47	C₄×#10596831	C₂³	1	2	yes	—
48	C₄×#10596831	C₂³	1	2	yes	—
49	C₂×#41×C₇₃	—	2	2	no	—
50	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
55	#127×C₇₃	C₂×C₄	1	2	yes	—
56	#124×C₇₃	—	2	2	no	—
57	#121×C₇₃	—	2	2	no	—
58	#121×C₇₃	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₇₃	—	2	2	no	—
60	C₄×D₄×C₇₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	#107×C₇₃	C₂×C₄	1	2	yes	—
63	#107×C₇₃	C₂×C₄	1	2	yes	—
64	#102×C₇₃	C₂×C₄	1	4	yes	—
65	#102×C₇₃	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
70	#11924012	—	2	2	no	—
71	#11924012	—	4	8	no	—
72	#11924012	D₈	1	4	yes	complemented
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	—	2	2	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	4	4	no	—
77	C₂×#10596831	—	4	8	no	—
78	C₂×#10596831	C₂×D₄	1	2	yes	—
79	C₂×#10596831	C₂×D₄	1	2	yes	—
80	#41×C₇₃	—	4	4	no	—
81	#41×C₇₃	—	4	4	no	—
82	C₂×Q₈×C₇₃	—	2	2	no	—
83	C₂×D₄×C₇₃	—	2	4	no	—
84	C₂×D₄×C₇₃	—	2	2	no	—
85	C₂×D₄×C₇₃	—	2	2	no	—
86	C₂×D₄×C₇₃	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₇₃	—	2	4	no	—
88	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
91	D₈×C₇₃	—	2	4	no	—
92	D₈×C₇₃	—	4	2	no	—
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	4	1	no	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₃	—	2	2	no	—
100	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄×C₇₃	—	2	4	no	—
102	C₂²⋊C₄×C₇₃	—	2	4	no	—
103	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₃	#41	1	1	yes	characteristic
106	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
107	C₂³×C₇₃	—	2	4	no	—
108	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
109	Q₈×C₇₃	—	4	4	no	—
110	D₄×C₇₃	—	2	4	no	—
111	D₄×C₇₃	—	2	4	no	—
112	D₄×C₇₃	—	4	8	no	—
113	D₄×C₇₃	—	4	4	no	—
114	D₄×C₇₃	—	8	2	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	4	1	no	—
119	C₂×C₄×C₇₃	—	4	4	no	—
120	C₂×C₄×C₇₃	—	4	4	no	—
121	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₇₃	#135	1	2	yes	—
126	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
129	C₈×C₇₃	—	2	1	no	—
130	C₈×C₇₃	—	2	1	no	—
131	C₈×C₇₃	—	2	1	no	—
132	C₈×C₇₃	#131	1	2	yes	complemented
133	#10596831	—	2	2	no	—
134	#10596831	—	4	2	no	—
135	#10596831	—	4	2	no	—
136	#10596831	—	4	2	no	—
137	#10596831	C₂×D₈	1	4	yes	—
138	C₂²×C₇₃	—	2	1	no	—
139	C₂²×C₇₃	—	2	1	no	—
140	C₂²×C₇₃	—	4	4	no	—
141	C₂²×C₇₃	—	4	4	no	—
142	C₂²×C₇₃	—	4	4	no	—
143	C₂²×C₇₃	C₄×D₈	1	2	yes	—
144	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	4	4	no	—
149	C₄×C₇₃	C₄×D₈	1	2	yes	—
150	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
151	C₄×C₇₃	#568	1	1	yes	characteristic
152	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
154	#18066	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	2	no	—
156	C₂×C₇₃	—	8	4	no	—
157	C₂×C₇₃	#2935	1	1	yes	characteristic
158	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
159	C₂×C₇₃	#2957	1	1	yes	characteristic
160	#3169	D₇₃	1	1	yes	characteristic
161	#2935	—	73	2	no	—
162	#1589	—	73	2	no	—
163	#1581	—	73	4	no	—
164	C₈×D₈	—	73	2	no	—
165	C₂×#335	—	73	1	no	—
166	#1480	—	73	2	no	—
167	C₈×M₁₆	—	73	1	no	—
168	C₇₃	#18066	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	—	146	2	no	—
170	#532	C₂×D₇₃	1	2	yes	—
171	#500	Dic₇₃	1	1	yes	characteristic
172	#493	Dic₇₃	1	1	yes	characteristic
173	#486	Dic₇₃	1	2	yes	—
174	#466	Dic₇₃	1	2	yes	—
175	#466	Dic₇₃	1	2	yes	—
176	C₄×D₈	C₂×D₇₃	1	2	yes	—
177	C₄×D₈	C₂×D₇₃	1	2	yes	—
178	#436	—	73	4	no	—
179	#435	—	146	4	no	—
180	C₈×D₄	—	73	4	no	—
181	#431	—	73	1	no	—
182	C₂×C₄⋊C₈	—	73	2	no	—
183	#408	—	146	2	no	—
184	C₄×M₁₆	—	73	1	no	—
185	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
186	#335	—	73	1	no	—
187	#335	—	73	2	no	—
188	#335	—	73	1	no	—
189	D₄⋊C₈	—	73	4	no	—
190	D₄⋊C₈	—	73	4	no	—
191	#322	—	73	2	no	—
192	C₈²	—	73	2	no	—
193	C₂×#41	—	2	2	no	—
194	C₂×D₈	C₂×Dic₇₃	1	2	yes	—
195	C₂×D₈	C₂×Dic₇₃	1	2	yes	—
196	#131	—	292	8	no	—
197	C₂×M₁₆	—	73	1	no	—
198	C₂×M₁₆	—	73	1	no	—
199	C₂×M₁₆	—	146	4	no	—
200	C₂²×C₈	—	146	4	no	—
201	C₂²×C₈	—	146	2	no	—
202	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
204	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
205	#127	C₂×Dic₇₃	1	2	yes	—
206	#124	—	2	2	no	—
207	#121	—	2	2	no	—
208	#121	C₂×Dic₇₃	1	4	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	C₂²×D₇₃	1	4	yes	—
211	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
212	#107	C₂×Dic₇₃	1	2	yes	—
213	#107	C₂×Dic₇₃	1	2	yes	—
214	C₄⋊C₈	—	73	2	no	—
215	C₄⋊C₈	—	73	2	no	—
216	C₄⋊C₈	—	146	2	no	—
217	C₄⋊C₈	—	146	1	no	—
218	#102	C₂×Dic₇₃	1	4	yes	—
219	#102	C₂×Dic₇₃	1	4	yes	—
220	C₂²⋊C₈	—	146	4	no	—
221	C₂²⋊C₈	—	146	4	no	—
222	C₂²⋊C₈	—	146	1	no	—
223	#97	—	73	2	no	—
224	C₄×C₈	—	73	2	no	—
225	C₄×C₈	—	73	2	no	—
226	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
227	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
230	#41	—	4	4	no	—
231	#41	—	4	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂²×Dic₇₃	1	4	yes	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
239	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
240	C₂²×C₄	#11924027	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	M₁₆	—	292	8	no	—
244	M₁₆	—	73	4	no	—
245	M₁₆	—	146	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	292	4	no	—
249	C₂×C₈	—	292	8	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	146	2	no	—
253	C₂×C₈	—	146	2	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
264	C₄²	D₄×D₇₃	1	1	yes	characteristic
265	C₄²	#11924027	1	1	yes	characteristic
266	C₄²	C₂³×D₇₃	1	1	yes	characteristic
267	C₂³	—	2	4	no	—
268	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	8	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
282	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
284	C₂×C₄	D₈×D₇₃	1	2	yes	—
285	C₂×C₄	#14556610	1	2	yes	—
286	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	292	2	no	—
293	C₈	—	292	2	no	—
294	C₈	—	292	2	no	—
295	C₈	—	73	4	no	—
296	C₈	—	146	2	no	—
297	C₈	#14556598	1	2	yes	complemented
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	D₈×Dic₇₃	1	2	yes	—
304	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	4	no	—
309	C₄	D₈×Dic₇₃	1	2	yes	—
310	C₄	C₂×#14556598	1	1	yes	characteristic
311	C₄	#17989354	1	1	yes	characteristic
312	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
313	C₄	C₂×#14556610	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	8	4	no	—
316	C₂	#22618675	1	1	yes	characteristic
317	C₂	C₂×D₈×Dic₇₃	1	1	yes	characteristic
318	C₂	#22618848	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting, prime radical	C₇₃×#3169	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#3169	D₇₃
hypocenter	C₇₃	#18066
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22618675	C₂	semidirect
2. #22608816	C₂	semidirect
3. #22608761	C₂	semidirect
4. D₈×#10596831	C₂	semidirect
5. C₂×#17988340	C₂	semidirect
6. #22608330	C₂	semidirect
7. C₈×#11924012	C₂	semidirect
72. #11924012	D₈	semidirect
132. C₈×C₇₃	#131	semidirect
168. C₇₃	#18066	semidirect
297. C₈	#14556598	semidirect

Maximal subgroups

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

subgroup	index
#22618675	2
#22608816	2
#22608761	2
D₈×#10596831	2
C₂×#17988340	2
#22608330	2
C₈×#11924012	2
C₇₃×#3169	2
#18066	73