Subgroups — #33077892

Order 18688.

554 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622861	C₂	1	2	yes	maximal
2	#22622610	C₂	1	2	yes	maximal
3	#22609528	C₂	1	1	yes	maximal, characteristic
4	#22609438	C₂	1	2	yes	maximal
5	#22609389	C₂	1	2	yes	maximal
6	#22609176	C₂	1	2	yes	maximal
7	#22609073	C₂	1	2	yes	maximal
8	#22608829	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
10	Q₈×Dic₁₄₆	C₂²	1	2	yes	—
11	#17989316	C₂²	1	2	yes	—
12	#17988995	—	2	2	no	—
13	#17988992	C₂²	1	1	yes	characteristic
14	#17988985	C₂²	1	2	yes	—
15	#17988967	C₂²	1	1	yes	characteristic
16	#17988938	—	2	2	no	—
17	#17988938	C₂²	1	2	yes	—
18	#17988892	—	2	2	no	—
19	#17988883	—	2	2	no	—
20	#17988883	C₂²	1	2	yes	—
21	#17988882	C₂²	1	2	yes	—
22	#17988872	C₂²	1	2	yes	—
23	#17988870	—	2	2	no	—
24	#17988798	—	2	2	no	—
25	#17988798	C₂²	1	2	yes	—
26	#17988717	C₂²	1	2	yes	—
27	#17988694	—	2	2	no	—
28	C₄×Dic₂₉₂	—	2	2	no	—
29	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
30	#17988340	C₂²	1	1	yes	characteristic
31	#17988335	C₂²	1	2	yes	—
32	#17988334	C₂²	1	2	yes	—
33	#17988319	C₂²	1	2	yes	—
34	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
36	#335×C₇₃	C₂²	1	1	yes	characteristic
37	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
38	C₈²×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₉₂	—	2	2	no	—
40	C₂×Dic₂₉₂	—	4	1	no	—
41	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
42	#14556591	—	2	2	no	—
43	#14556591	—	2	2	no	—
44	#14556591	C₂³	1	2	yes	—
45	#14556591	C₂³	1	2	yes	—
46	#14556590	—	2	2	no	—
47	#14556590	—	4	2	no	—
48	#14556582	—	2	2	no	—
49	#14556582	—	4	2	no	—
50	Q₈×Dic₇₃	—	2	2	no	—
51	Q₈×Dic₇₃	—	2	2	no	—
52	C₄×Dic₁₄₆	—	2	2	no	—
53	C₄×Dic₁₄₆	—	2	2	no	—
54	C₄×Dic₁₄₆	C₂³	1	2	yes	—
55	#14556525	—	2	2	no	—
56	#14556525	—	4	1	no	—
57	#14556525	D₄	1	1	yes	characteristic
58	#14556524	—	2	2	no	—
59	#14556523	—	2	2	no	—
60	#14556523	—	2	2	no	—
61	#14556522	—	2	2	no	—
62	C₄⋊#10596831	C₂³	1	2	yes	—
63	#14556520	—	2	2	no	—
64	#14556516	—	2	2	no	—
65	#14556516	—	2	2	no	—
66	#14556516	—	2	2	no	—
67	#14556516	—	2	2	no	—
68	#14556516	D₄	1	2	yes	—
69	#14556515	—	2	2	no	—
70	#14556515	—	2	2	no	—
71	#14556501	—	2	2	no	—
72	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
74	#107×C₇₃	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₇₃	—	2	2	no	—
76	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
77	#103×C₇₃	D₄	1	2	yes	—
78	C₄×C₈×C₇₃	—	2	2	no	—
79	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₄₆	—	2	2	no	—
83	C₂×Dic₁₄₆	—	2	2	no	—
84	C₂×Dic₁₄₆	—	4	2	no	—
85	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
86	Dic₂₉₂	—	4	1	no	—
87	Dic₂₉₂	—	4	4	no	—
88	Dic₂₉₂	—	8	2	no	—
89	C₂×#10596831	—	2	2	no	—
90	C₄⋊Dic₇₃	—	2	2	no	—
91	C₄⋊Dic₇₃	—	2	2	no	—
92	C₄⋊Dic₇₃	—	2	2	no	—
93	C₄⋊Dic₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	4	2	no	—
95	C₄⋊Dic₇₃	—	4	2	no	—
96	C₄⋊Dic₇₃	—	4	2	no	—
97	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
98	#11924007	—	2	2	no	—
99	#11924007	—	4	2	no	—
100	#11924007	—	4	2	no	—
101	C₄×Dic₇₃	—	2	2	no	—
102	C₄×Dic₇₃	—	4	2	no	—
103	C₄×Dic₇₃	—	4	2	no	—
104	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
105	Q₁₆×C₇₃	Q₁₆	1	4	yes	complemented
106	C₂×C₈×C₇₃	—	2	1	no	—
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
114	C₄²×C₇₃	—	2	2	no	—
115	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
116	Q₈×C₇₃	—	2	2	no	—
117	Q₈×C₇₃	C₂×Q₁₆	1	4	yes	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×Dic₇₃	—	2	2	no	—
124	C₂×Dic₇₃	—	2	2	no	—
125	C₂×Dic₇₃	—	4	2	no	—
126	C₂×Dic₇₃	—	4	2	no	—
127	Dic₁₄₆	—	2	2	no	—
128	Dic₁₄₆	—	2	2	no	—
129	Dic₁₄₆	—	4	2	no	—
130	Dic₁₄₆	—	4	2	no	—
131	Dic₁₄₆	—	8	2	no	—
132	Dic₁₄₆	—	16	2	no	—
133	C₈×C₇₃	—	2	2	no	—
134	C₈×C₇₃	—	2	1	no	—
135	C₈×C₇₃	—	2	1	no	—
136	C₈×C₇₃	—	2	1	no	—
137	C₈×C₇₃	—	4	2	no	—
138	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
139	#10596831	—	4	2	no	—
140	C₂²×C₇₃	D₄²	1	1	yes	characteristic
141	C₄×C₇₃	—	2	4	no	—
142	C₄×C₇₃	—	2	1	no	—
143	C₄×C₇₃	—	4	2	no	—
144	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
145	C₄×C₇₃	#575	1	1	yes	characteristic
146	C₄×C₇₃	#578	1	1	yes	characteristic
147	C₄×C₇₃	#578	1	1	yes	characteristic
148	Dic₇₃	—	4	2	no	—
149	Dic₇₃	—	4	2	no	—
150	Dic₇₃	—	8	2	no	—
151	Dic₇₃	—	8	2	no	—
152	#19931	—	73	1	no	maximal, Sylow
153	C₂×C₇₃	#3277	1	1	yes	characteristic
154	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
155	C₂×C₇₃	#3291	1	1	yes	characteristic
156	#3387	—	73	2	no	—
157	#3363	—	73	2	no	—
158	#1707	—	73	1	no	—
159	#1692	—	73	2	no	—
160	#1684	—	73	2	no	—
161	#1648	—	73	2	no	—
162	#1633	—	73	2	no	—
163	#1591	—	73	1	no	—
164	C₈×Q₁₆	D₇₃	1	1	yes	characteristic
165	C₇₃	#19931	1	1	yes	characteristic, complemented, Sylow
166	Q₈²	—	73	2	no	—
167	#557	—	73	2	no	—
168	#501	—	146	2	no	—
169	#500	—	73	1	no	—
170	#498	—	73	2	no	—
171	#494	—	73	1	no	—
172	#487	—	73	2	no	—
173	#487	—	146	2	no	—
174	#479	—	146	2	no	—
175	#477	—	73	2	no	—
176	#477	—	73	2	no	—
177	#477	—	146	2	no	—
178	#475	—	73	2	no	—
179	#475	—	146	2	no	—
180	#462	—	73	2	no	—
181	#462	—	146	2	no	—
182	#446	—	73	2	no	—
183	C₈×Q₈	C₂×D₇₃	1	2	yes	—
184	#441	—	146	2	no	—
185	C₄×Q₁₆	—	73	1	no	—
186	C₄×Q₁₆	—	146	2	no	—
187	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
188	#335	—	73	1	no	—
189	#335	C₂×D₇₃	1	1	yes	characteristic
190	#333	—	73	2	no	—
191	#332	—	73	2	no	—
192	Q₈⋊C₈	—	73	2	no	—
193	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
194	C₈²	C₂×D₇₃	1	1	yes	characteristic
195	C₂×Q₁₆	—	292	1	no	—
196	C₂×Q₁₆	—	73	1	no	—
197	C₂×Q₁₆	—	146	2	no	—
198	C₂×Q₁₆	D₂₉₂	1	1	yes	characteristic
199	C₄⋊Q₈	—	292	2	no	—
200	C₄⋊Q₈	—	73	2	no	—
201	C₄⋊Q₈	—	73	2	no	—
202	C₄⋊Q₈	—	146	2	no	—
203	C₄⋊Q₈	—	146	2	no	—
204	C₄⋊Q₈	—	146	2	no	—
205	#125	—	292	2	no	—
206	#125	—	146	2	no	—
207	C₄×Q₈	—	73	2	no	—
208	C₄×Q₈	—	146	2	no	—
209	C₄×Q₈	—	146	2	no	—
210	C₄×Q₈	—	146	2	no	—
211	C₄×Q₈	—	146	2	no	—
212	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
213	#107	—	292	1	no	—
214	#107	—	73	1	no	—
215	#107	—	146	2	no	—
216	#107	—	146	2	no	—
217	#107	D₂₉₂	1	1	yes	characteristic
218	#106	—	146	2	no	—
219	#106	—	146	2	no	—
220	#106	—	146	2	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	73	2	no	—
223	C₄⋊C₈	—	146	2	no	—
224	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
225	#103	—	73	2	no	—
226	#103	—	146	2	no	—
227	#103	—	146	2	no	—
228	#103	—	146	2	no	—
229	#103	—	146	2	no	—
230	#103	—	146	2	no	—
231	#103	—	146	2	no	—
232	#103	D₂₉₂	1	2	yes	—
233	#97	—	146	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
238	C₂×Q₈	—	292	2	no	—
239	C₂×Q₈	—	73	2	no	—
240	C₂×Q₈	—	146	2	no	—
241	C₂×Q₈	—	146	2	no	—
242	C₂×Q₈	C₂×D₂₉₂	1	2	yes	—
243	Q₁₆	—	292	1	no	—
244	Q₁₆	—	292	4	no	—
245	Q₁₆	—	584	2	no	—
246	Q₁₆	Dic₂₉₂	1	4	yes	complemented
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	146	2	no	—
250	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
251	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
253	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	292	2	no	—
256	C₄⋊C₄	—	292	2	no	—
257	C₄⋊C₄	—	292	2	no	—
258	C₄⋊C₄	—	292	2	no	—
259	C₄⋊C₄	—	292	2	no	—
260	C₄⋊C₄	—	73	2	no	—
261	C₄⋊C₄	—	146	2	no	—
262	C₄⋊C₄	—	146	2	no	—
263	C₄⋊C₄	—	146	2	no	—
264	C₄⋊C₄	—	146	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	292	2	no	—
269	C₄²	—	292	2	no	—
270	C₄²	—	146	2	no	—
271	C₄²	C₂³×D₇₃	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	292	2	no	—
274	Q₈	—	292	2	no	—
275	Q₈	—	584	2	no	—
276	Q₈	—	1168	2	no	—
277	Q₈	—	146	2	no	—
278	Q₈	—	146	2	no	—
279	Q₈	C₂×Dic₂₉₂	1	4	yes	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	292	2	no	—
283	C₂×C₄	—	292	2	no	—
284	C₂×C₄	—	146	2	no	—
285	C₂×C₄	—	146	2	no	—
286	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
288	C₂×C₄	#14556637	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	292	2	no	—
295	C₈	C₂×Dic₂₉₂	1	2	yes	—
296	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	4	no	—
299	C₄	—	4	2	no	—
300	C₄	—	292	2	no	—
301	C₄	—	292	2	no	—
302	C₄	—	584	2	no	—
303	C₄	—	584	2	no	—
304	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
305	C₄	#17989384	1	1	yes	characteristic
306	C₄	#17989400	1	1	yes	characteristic
307	C₄	#17989404	1	1	yes	characteristic
308	C₂	#22621587	1	1	yes	characteristic
309	C₂	D₄×Dic₂₉₂	1	1	yes	characteristic
310	C₂	#22621725	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×Q₁₆×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	D₄²
hypercenter	C₈×Q₁₆	D₇₃
hypocenter	C₇₃	#19931
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
105. Q₁₆×C₇₃	Q₁₆	semidirect
165. C₇₃	#19931	semidirect
246. Q₁₆	Dic₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
#22622861	2
#22622610	2
#22609528	2
#22609438	2
#22609389	2
#22609176	2
#22609073	2
#22608829	2
C₈×Q₁₆×C₇₃	2
#19931	73