Subgroups — #33077846

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622828	C₂	1	2	yes	maximal
2	#22621808	C₂	1	2	yes	maximal, complemented
3	#22609528	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609438	C₂	1	2	yes	maximal, complemented
5	#22609382	C₂	1	2	yes	maximal, complemented
6	#22609183	C₂	1	2	yes	maximal, complemented
7	#22609067	C₂	1	2	yes	maximal, complemented
8	#22608829	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
10	#17989294	C₂²	1	2	yes	—
11	D₄×Dic₁₄₆	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	C₂²	1	2	yes	—
17	#17988923	—	2	2	no	—
18	#17988889	—	2	2	no	—
19	#17988883	C₂²	1	2	yes	—
20	#17988877	C₂²	1	2	yes	—
21	#17988875	—	2	2	no	—
22	#17988867	C₂²	1	2	yes	—
23	#17988828	—	2	2	no	—
24	#17988798	C₂²	1	2	yes	—
25	#17988745	—	2	2	no	—
26	#17988717	C₂²	1	2	yes	—
27	#17988699	—	2	2	no	—
28	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
29	C₂×#14556516	—	2	2	no	—
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	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₇₃	C₂²	1	2	yes	—
36	#335×C₇₃	C₂²	1	1	yes	characteristic
37	D₄⋊C₈×C₇₃	C₂²	1	2	yes	—
38	C₈²×C₇₃	C₂²	1	1	yes	characteristic
39	C₂²×Dic₁₄₆	—	2	2	no	—
40	C₂×Dic₂₉₂	—	2	2	no	—
41	C₂×Dic₂₉₂	—	4	1	no	—
42	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
43	#14556591	C₂³	1	2	yes	—
44	#14556591	C₂³	1	2	yes	—
45	#14556590	—	2	2	no	—
46	#14556582	—	2	2	no	—
47	#14556570	—	2	2	no	—
48	#14556570	—	2	2	no	—
49	#14556568	—	4	2	no	—
50	#14556568	—	4	2	no	—
51	C₄×Dic₁₄₆	C₂³	1	2	yes	—
52	D₄×Dic₇₃	—	2	2	no	—
53	D₄×Dic₇₃	—	2	2	no	—
54	C₂×C₄⋊Dic₇₃	—	2	2	no	—
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	#14556512	—	2	2	no	—
70	#14556512	—	2	2	no	—
71	#14556501	—	2	2	no	—
72	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₇₃	—	2	2	no	—
74	C₄×D₄×C₇₃	C₂³	1	2	yes	—
75	#107×C₇₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
77	#102×C₇₃	D₄	1	2	yes	—
78	C₂²⋊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	4	no	—
85	C₂×Dic₁₄₆	—	4	2	no	—
86	C₂×Dic₁₄₆	—	8	2	no	—
87	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
88	C₂²×Dic₇₃	—	4	2	no	—
89	C₂²×Dic₇₃	—	4	2	no	—
90	Dic₂₉₂	—	4	1	no	—
91	Dic₂₉₂	—	8	2	no	—
92	Dic₂₉₂	—	8	2	no	—
93	C₂×#10596831	—	2	2	no	—
94	C₄⋊Dic₇₃	—	2	2	no	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	—	2	2	no	—
97	C₄⋊Dic₇₃	—	2	2	no	—
98	C₄⋊Dic₇₃	—	4	2	no	—
99	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
100	#11924007	—	2	2	no	—
101	#11924007	—	4	2	no	—
102	#11924007	—	4	2	no	—
103	C₂²⋊Dic₇₃	—	4	2	no	—
104	C₂²⋊Dic₇₃	—	4	2	no	—
105	C₄×Dic₇₃	—	2	2	no	—
106	C₂×D₄×C₇₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₇₃	—	2	2	no	—
108	D₈×C₇₃	Q₁₆	1	4	yes	complemented
109	C₂×C₈×C₇₃	—	2	1	no	—
110	C₂×C₈×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	—	8	2	no	—
112	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄×C₇₃	—	2	2	no	—
118	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
119	C₂³×C₇₃	—	2	2	no	—
120	D₄×C₇₃	—	2	2	no	—
121	D₄×C₇₃	C₂×Q₁₆	1	4	yes	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	—	4	2	no	—
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×Dic₇₃	—	2	2	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	4	2	no	—
131	C₂×Dic₇₃	—	4	2	no	—
132	C₂×Dic₇₃	—	8	4	no	—
133	C₂×Dic₇₃	—	16	2	no	—
134	Dic₁₄₆	—	2	2	no	—
135	Dic₁₄₆	—	2	2	no	—
136	Dic₁₄₆	—	4	2	no	—
137	Dic₁₄₆	—	4	2	no	—
138	Dic₁₄₆	—	8	4	no	—
139	Dic₁₄₆	—	8	2	no	—
140	C₈×C₇₃	—	2	2	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	—	2	1	no	—
144	C₈×C₇₃	—	4	2	no	—
145	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
146	#10596831	—	4	2	no	—
147	C₂²×C₇₃	—	2	4	no	—
148	C₂²×C₇₃	—	4	2	no	—
149	C₂²×C₇₃	—	4	2	no	—
150	C₂²×C₇₃	D₄²	1	1	yes	characteristic
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	4	2	no	—
153	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
154	C₄×C₇₃	#575	1	1	yes	characteristic
155	C₄×C₇₃	#576	1	1	yes	characteristic
156	C₄×C₇₃	#578	1	1	yes	characteristic
157	Dic₇₃	—	4	2	no	—
158	Dic₇₃	—	4	2	no	—
159	Dic₇₃	—	8	2	no	—
160	Dic₇₃	—	8	2	no	—
161	#19927	—	73	1	no	maximal, Sylow
162	C₂×C₇₃	—	4	4	no	—
163	C₂×C₇₃	#3273	1	1	yes	characteristic
164	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
165	C₂×C₇₃	#3291	1	1	yes	characteristic
166	#3384	—	73	2	no	—
167	#3299	—	73	2	no	—
168	#1707	—	73	1	no	—
169	#1692	—	73	2	no	—
170	#1683	—	73	2	no	—
171	#1649	—	73	2	no	—
172	#1632	—	73	2	no	—
173	#1591	—	73	1	no	—
174	C₈×D₈	D₇₃	1	1	yes	characteristic
175	C₇₃	#19927	1	1	yes	characteristic, complemented, Sylow
176	#554	—	73	2	no	—
177	D₄×Q₈	—	73	2	no	—
178	#501	—	146	2	no	—
179	#500	—	73	1	no	—
180	#498	—	73	2	no	—
181	#494	—	73	1	no	—
182	#487	—	73	2	no	—
183	#484	—	146	2	no	—
184	#478	—	146	2	no	—
185	#477	—	73	2	no	—
186	#476	—	73	2	no	—
187	#476	—	146	2	no	—
188	#474	—	73	2	no	—
189	#467	—	146	2	no	—
190	#462	—	73	2	no	—
191	#451	—	146	2	no	—
192	#446	—	73	2	no	—
193	#442	—	146	2	no	—
194	C₄×Q₁₆	—	73	1	no	—
195	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
196	C₈×D₄	C₂×D₇₃	1	2	yes	—
197	C₂×#103	—	146	2	no	—
198	#335	—	73	1	no	—
199	#335	C₂×D₇₃	1	1	yes	characteristic
200	#333	—	73	2	no	—
201	#332	—	73	2	no	—
202	Q₈⋊C₈	—	73	2	no	—
203	D₄⋊C₈	C₂×D₇₃	1	2	yes	—
204	C₈²	C₂×D₇₃	1	1	yes	characteristic
205	C₂²×Q₈	—	146	2	no	—
206	C₂×Q₁₆	—	292	1	no	—
207	C₂×Q₁₆	—	73	1	no	—
208	C₂×Q₁₆	—	146	2	no	—
209	C₂×D₈	D₂₉₂	1	1	yes	characteristic
210	C₂²×C₈	—	2	2	no	—
211	C₄⋊Q₈	—	73	2	no	—
212	C₄⋊Q₈	—	73	2	no	—
213	C₄⋊Q₈	—	146	2	no	—
214	#125	—	146	2	no	—
215	C₂²⋊Q₈	—	292	2	no	—
216	C₂²⋊Q₈	—	292	2	no	—
217	C₂²⋊Q₈	—	146	2	no	—
218	C₂²⋊Q₈	—	146	2	no	—
219	C₄×Q₈	—	73	2	no	—
220	C₄×D₄	—	146	2	no	—
221	C₄×D₄	—	146	2	no	—
222	C₄×D₄	C₂²×D₇₃	1	2	yes	—
223	C₂×C₄⋊C₄	—	146	2	no	—
224	#107	—	292	1	no	—
225	#107	—	73	1	no	—
226	#107	—	146	2	no	—
227	#107	—	146	2	no	—
228	#107	D₂₉₂	1	1	yes	characteristic
229	#106	—	146	2	no	—
230	#106	—	146	2	no	—
231	#106	—	146	2	no	—
232	C₄⋊C₈	—	73	2	no	—
233	C₄⋊C₈	—	146	2	no	—
234	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
235	#103	—	73	2	no	—
236	#103	—	146	2	no	—
237	#103	—	146	2	no	—
238	#103	—	146	2	no	—
239	#103	—	146	2	no	—
240	#102	—	146	2	no	—
241	#102	—	146	2	no	—
242	#102	D₂₉₂	1	2	yes	—
243	C₂²⋊C₈	—	2	2	no	—
244	#97	—	146	2	no	—
245	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	C₂×Q₈	—	292	2	no	—
249	C₂×Q₈	—	292	4	no	—
250	C₂×Q₈	—	584	2	no	—
251	C₂×Q₈	—	73	2	no	—
252	C₂×Q₈	—	146	2	no	—
253	C₂×Q₈	—	146	2	no	—
254	C₂×D₄	C₂×D₂₉₂	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	292	2	no	—
257	C₂²×C₄	—	292	2	no	—
258	Q₁₆	—	292	1	no	—
259	Q₁₆	—	584	2	no	—
260	Q₁₆	—	584	2	no	—
261	D₈	Dic₂₉₂	1	4	yes	complemented
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	8	2	no	—
265	C₂×C₈	—	146	2	no	—
266	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
270	C₄⋊C₄	—	292	2	no	—
271	C₄⋊C₄	—	292	2	no	—
272	C₄⋊C₄	—	292	2	no	—
273	C₄⋊C₄	—	73	2	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₄⋊C₄	—	146	2	no	—
279	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	292	2	no	—
282	C₂²⋊C₄	—	292	2	no	—
283	C₄²	—	146	2	no	—
284	C₄²	C₂³×D₇₃	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	Q₈	—	292	2	no	—
287	Q₈	—	292	2	no	—
288	Q₈	—	584	2	no	—
289	Q₈	—	584	4	no	—
290	Q₈	—	146	2	no	—
291	Q₈	—	146	2	no	—
292	D₄	—	2	2	no	—
293	D₄	C₂×Dic₂₉₂	1	4	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	292	2	no	—
298	C₂×C₄	—	292	2	no	—
299	C₂×C₄	—	584	4	no	—
300	C₂×C₄	—	1168	2	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
305	C₂×C₄	#14556637	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	292	2	no	—
312	C₈	C₂×Dic₂₉₂	1	2	yes	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	292	2	no	—
320	C₄	—	292	2	no	—
321	C₄	—	584	2	no	—
322	C₄	—	584	2	no	—
323	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
324	C₄	#17989384	1	1	yes	characteristic
325	C₄	#17989388	1	1	yes	characteristic
326	C₄	#17989404	1	1	yes	characteristic
327	C₂	—	4	4	no	—
328	C₂	#22621541	1	1	yes	characteristic
329	C₂	D₄×Dic₂₉₂	1	1	yes	characteristic
330	C₂	#22621725	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×D₈×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×D₈×C₇₃	D₄
hypercenter	C₈×D₈	D₇₃
hypocenter	C₇₃	#19927
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
2. #22621808	C₂	semidirect
3. #22609528	C₂	semidirect
4. #22609438	C₂	semidirect
5. #22609382	C₂	semidirect
6. #22609183	C₂	semidirect
7. #22609067	C₂	semidirect
8. #22608829	C₂	semidirect
108. D₈×C₇₃	Q₁₆	semidirect
175. C₇₃	#19927	semidirect
261. D₈	Dic₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
#22622828	2
#22621808	2
#22609528	2
#22609438	2
#22609382	2
#22609183	2
#22609067	2
#22608829	2
C₈×D₈×C₇₃	2
#19927	73