Subgroups — #33243656

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622944	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622799	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612163	C₂	1	2	yes	maximal, complemented
4	#22612155	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612089	C₂	1	2	yes	maximal, complemented
6	#22612084	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611987	C₂	1	2	yes	maximal, complemented
8	#22611967	C₂	1	1	yes	maximal, characteristic
9	C₄×#14556536	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₁₆₈	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989322	—	2	1	no	—
13	#17989297	—	2	1	no	—
14	C₂×#14556536	C₂²	1	1	yes	characteristic
15	C₂×D₁₁₆₈	C₂²	1	2	yes	—
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988989	C₂²	1	2	yes	—
18	#17988971	C₂²	1	2	yes	—
19	#17988964	C₂²	1	1	yes	characteristic
20	#17988888	—	2	2	no	—
21	#17988878	—	2	1	no	—
22	#17988803	—	2	2	no	—
23	#17988788	—	2	1	no	—
24	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
25	C₄×D₅₈₄	C₂²	1	2	yes	—
26	C₄×D₅₈₄	C₂²	1	2	yes	—
27	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
28	#17988415	C₂²	1	1	yes	characteristic
29	#17988413	C₂²	1	2	yes	—
30	#17988392	C₂²	1	2	yes	—
31	#17988392	C₂²	1	1	yes	characteristic
32	#17988389	C₂²	1	2	yes	—
33	#17988389	C₂²	1	1	yes	characteristic
34	#17988389	C₂²	1	2	yes	—
35	#17988389	C₂²	1	2	yes	—
36	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₉₂	C₂³	1	1	yes	characteristic
42	C₂×#11924017	—	2	2	no	—
43	C₂×D₅₈₄	—	2	1	no	—
44	C₂×D₅₈₄	C₂³	1	2	yes	—
45	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
46	#14556591	—	2	1	no	—
47	#14556588	—	2	1	no	—
48	#14556583	—	2	2	no	—
49	#14556580	—	4	1	no	—
50	#14556579	—	2	2	no	—
51	#14556576	—	4	2	no	—
52	#14556569	—	4	2	no	—
53	#14556567	—	4	1	no	—
54	C₄×Dic₁₄₆	—	2	1	no	—
55	C₄×D₂₉₂	—	2	1	no	—
56	C₄×D₂₉₂	—	2	2	no	—
57	C₄×D₂₉₂	—	2	2	no	—
58	#14556553	—	4	1	no	—
59	C₄⋊C₄×D₇₃	—	4	1	no	—
60	#14556536	—	2	2	no	—
61	D₁₁₆₈	—	2	4	no	—
62	#14556525	—	2	1	no	—
63	#14556525	C₂³	1	2	yes	—
64	#14556525	C₂³	1	1	yes	characteristic
65	#14556525	C₂³	1	1	yes	characteristic
66	#14556523	—	2	2	no	—
67	#14556516	—	2	1	no	—
68	#14556513	—	2	2	no	—
69	#14556513	—	2	1	no	—
70	#14556513	—	2	2	no	—
71	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
75	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
77	C₄×C₈×C₇₃	C₂³	1	2	yes	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₆	—	2	1	no	—
80	C₂×Dic₁₄₆	—	4	1	no	—
81	C₂×D₂₉₂	—	2	1	no	—
82	C₂×D₂₉₂	—	2	2	no	—
83	C₂×D₂₉₂	—	4	1	no	—
84	C₂×D₂₉₂	—	4	1	no	—
85	C₂×C₄×D₇₃	—	4	1	no	—
86	C₂×C₄×D₇₃	—	4	2	no	—
87	C₂×C₄×D₇₃	—	4	2	no	—
88	Dic₂₉₂	—	2	1	no	—
89	Dic₂₉₂	—	2	1	no	—
90	#11924017	—	8	2	no	—
91	D₅₈₄	—	2	2	no	—
92	D₅₈₄	—	2	1	no	—
93	D₅₈₄	—	2	2	no	—
94	D₅₈₄	—	8	1	no	—
95	D₅₈₄	#41	1	2	yes	—
96	C₄⋊Dic₇₃	—	2	1	no	—
97	C₄⋊Dic₇₃	—	2	1	no	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	C₄⋊Dic₇₃	—	4	1	no	—
100	#11924007	—	4	2	no	—
101	#11924007	—	4	1	no	—
102	#11924007	—	4	2	no	—
103	#11924005	—	4	2	no	—
104	#11924005	—	4	2	no	—
105	#11924005	—	4	1	no	—
106	#11924005	—	4	2	no	—
107	#11924005	—	4	2	no	—
108	C₄×Dic₇₃	—	4	1	no	—
109	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
115	C₄²×C₇₃	C₂×D₄	1	2	yes	—
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₃	—	2	2	no	—
118	C₁₆×C₇₃	—	2	1	no	—
119	C₁₆×C₇₃	—	2	1	no	—
120	C₂²×D₇₃	—	4	1	no	—
121	C₂²×D₇₃	—	4	2	no	—
122	Q₈×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₇₃	—	4	1	no	—
129	C₂×Dic₇₃	—	4	1	no	—
130	C₂×Dic₇₃	—	4	1	no	—
131	C₂×Dic₇₃	—	4	2	no	—
132	D₂₉₂	—	2	2	no	—
133	D₂₉₂	—	4	2	no	—
134	D₂₉₂	—	4	1	no	—
135	D₂₉₂	—	4	2	no	—
136	D₂₉₂	—	8	2	no	—
137	D₂₉₂	—	16	1	no	—
138	C₄×D₇₃	—	8	2	no	—
139	C₄×D₇₃	—	16	2	no	—
140	C₄×D₇₃	—	16	2	no	—
141	Dic₁₄₆	—	4	1	no	—
142	Dic₁₄₆	—	4	1	no	—
143	Dic₁₄₆	—	16	1	no	—
144	C₈×C₇₃	—	2	1	no	—
145	C₈×C₇₃	—	2	2	no	—
146	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₃	—	4	2	no	—
150	C₂×D₇₃	—	8	1	no	—
151	C₂×D₇₃	—	8	1	no	—
152	C₂×D₇₃	—	8	2	no	—
153	C₂×D₇₃	—	8	2	no	—
154	C₂×D₇₃	—	8	2	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	#508	1	2	yes	—
159	C₄×C₇₃	#548	1	1	yes	characteristic
160	C₄×C₇₃	#577	1	1	yes	characteristic
161	Dic₇₃	—	8	1	no	—
162	Dic₇₃	—	8	1	no	—
163	Dic₇₃	—	8	2	no	—
164	Dic₇₃	—	8	1	no	—
165	#33928	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	#3358	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
168	C₂×C₇₃	#3415	1	1	yes	characteristic
169	D₇₃	—	8	2	no	—
170	D₇₃	—	16	2	no	—
171	#3396	—	73	1	no	—
172	#3380	—	73	1	no	—
173	#2242	—	73	2	no	—
174	#2240	—	73	1	no	—
175	#2228	—	73	2	no	—
176	#2227	—	73	1	no	—
177	#2210	—	73	2	no	—
178	#2206	—	73	1	no	—
179	Q₈×C₁₆	D₇₃	1	1	yes	characteristic, complemented
180	C₄×SD₃₂	—	73	1	no	—
181	C₄×D₁₆	—	73	2	no	—
182	C₇₃	#33928	1	1	yes	characteristic, complemented, Sylow
183	#559	—	146	1	no	—
184	#554	—	146	1	no	—
185	C₂×SD₃₂	—	73	1	no	—
186	C₂×D₁₆	—	73	2	no	—
187	#500	—	73	1	no	—
188	#499	—	73	2	no	—
189	#495	—	73	2	no	—
190	#493	—	73	1	no	—
191	#478	—	146	2	no	—
192	#476	—	146	1	no	—
193	#463	—	146	2	no	—
194	#460	—	146	1	no	—
195	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
196	C₄×Q₁₆	—	73	1	no	—
197	C₄×D₈	—	73	2	no	—
198	C₄×D₈	—	73	1	no	—
199	C₄×D₈	—	73	2	no	—
200	#367	—	73	1	no	—
201	#366	—	73	2	no	—
202	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
204	#358	—	73	1	no	—
205	#358	—	73	2	no	—
206	#357	—	73	1	no	—
207	#357	—	73	2	no	—
208	#357	—	73	2	no	—
209	#357	—	73	2	no	—
210	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
211	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
212	C₂×Q₁₆	—	73	1	no	—
213	C₂×SD₁₆	—	146	2	no	—
214	C₂×D₈	—	73	1	no	—
215	C₂×D₈	—	73	2	no	—
216	C₂×D₈	—	146	1	no	—
217	C₄⋊Q₈	—	146	1	no	—
218	#127	—	146	1	no	—
219	#125	—	146	2	no	—
220	#124	—	292	1	no	—
221	#124	—	146	2	no	—
222	#123	—	292	2	no	—
223	C₂²⋊Q₈	—	292	1	no	—
224	C₂²⋊Q₈	—	292	2	no	—
225	C₄×Q₈	—	146	1	no	—
226	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
227	C₄×D₄	—	292	1	no	—
228	C₄×D₄	—	146	1	no	—
229	C₄×D₄	—	146	2	no	—
230	C₄×D₄	—	146	2	no	—
231	C₂×C₄⋊C₄	—	292	1	no	—
232	SD₃₂	—	146	2	no	—
233	D₁₆	—	146	4	no	—
234	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
237	#107	—	73	2	no	—
238	#107	—	73	1	no	—
239	#107	—	73	1	no	—
240	#107	—	146	1	no	—
241	#106	—	146	2	no	—
242	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
243	C₄⋊C₈	D₂₉₂	1	2	yes	—
244	#103	—	146	1	no	—
245	#102	—	146	2	no	—
246	#102	—	146	2	no	—
247	#102	—	146	1	no	—
248	C₄×C₈	C₂²×D₇₃	1	2	yes	—
249	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
250	C₂×Q₈	—	292	1	no	—
251	C₂×Q₈	—	146	1	no	—
252	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
253	C₂×D₄	—	292	1	no	—
254	C₂×D₄	—	292	1	no	—
255	C₂×D₄	—	146	2	no	—
256	C₂×D₄	—	146	1	no	—
257	C₂²×C₄	—	292	1	no	—
258	C₂²×C₄	—	292	2	no	—
259	C₂²×C₄	—	292	2	no	—
260	Q₁₆	—	146	1	no	—
261	Q₁₆	—	146	1	no	—
262	SD₁₆	—	584	2	no	—
263	D₈	—	584	1	no	—
264	D₈	—	73	2	no	—
265	D₈	—	146	2	no	—
266	D₈	—	146	2	no	—
267	D₈	—	146	1	no	—
268	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
269	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
271	C₄⋊C₄	—	292	2	no	—
272	C₄⋊C₄	—	292	1	no	—
273	C₄⋊C₄	—	292	2	no	—
274	C₄⋊C₄	—	292	1	no	—
275	C₄⋊C₄	—	146	1	no	—
276	C₄⋊C₄	—	146	1	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
279	C₄⋊C₄	D₅₈₄	1	2	yes	—
280	C₂²⋊C₄	—	292	2	no	—
281	C₂²⋊C₄	—	292	1	no	—
282	C₂²⋊C₄	—	292	2	no	—
283	C₂²⋊C₄	—	292	2	no	—
284	C₂²⋊C₄	—	292	2	no	—
285	C₄²	—	292	1	no	—
286	C₄²	C₂×D₂₉₂	1	2	yes	—
287	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₂³	—	292	1	no	—
292	C₂³	—	292	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	292	1	no	—
295	Q₈	—	292	1	no	—
296	Q₈	—	1168	1	no	—
297	D₄	—	292	1	no	—
298	D₄	—	292	2	no	—
299	D₄	—	292	2	no	—
300	D₄	—	584	2	no	—
301	D₄	—	1168	1	no	—
302	D₄	—	146	2	no	—
303	C₂×C₄	—	292	1	no	—
304	C₂×C₄	—	292	1	no	—
305	C₂×C₄	—	292	1	no	—
306	C₂×C₄	—	292	2	no	—
307	C₂×C₄	—	584	2	no	—
308	C₂×C₄	—	1168	2	no	—
309	C₂×C₄	—	1168	2	no	—
310	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
311	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
312	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
313	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	#41×D₇₃	1	1	yes	characteristic
318	C₈	#14556639	1	1	yes	characteristic
319	C₂²	—	292	2	no	—
320	C₂²	—	584	1	no	—
321	C₂²	—	584	1	no	—
322	C₂²	—	584	2	no	—
323	C₂²	—	584	2	no	—
324	C₂²	—	584	2	no	—
325	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	584	1	no	—
330	C₄	—	584	1	no	—
331	C₄	—	584	2	no	—
332	C₄	—	584	1	no	—
333	C₄	#17989019	1	2	yes	—
334	C₄	#17989249	1	1	yes	characteristic
335	C₄	#17989399	1	1	yes	characteristic
336	C₂	—	584	2	no	—
337	C₂	—	1168	2	no	—
338	C₂	#22622567	1	1	yes	characteristic
339	C₂	C₂×#17989019	1	1	yes	characteristic
340	C₂	#22623072	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	Q₈×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22611967	C₂
hypercenter	Q₈×C₁₆	D₇₃
hypocenter	C₇₃	#33928
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #22622944	C₂	semidirect
2. #22622799	C₂	semidirect
3. #22612163	C₂	semidirect
4. #22612155	C₂	semidirect
5. #22612089	C₂	semidirect
6. #22612084	C₂	semidirect
7. #22611987	C₂	semidirect
9. C₄×#14556536	C₂	semidirect
10. C₄×D₁₁₆₈	C₂	semidirect
11. Q₈×C₁₆×C₇₃	C₂	semidirect
179. Q₈×C₁₆	D₇₃	semidirect
182. C₇₃	#33928	semidirect

Maximal subgroups

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

subgroup	index
#22622944	2
#22622799	2
#22612163	2
#22612155	2
#22612089	2
#22612084	2
#22611987	2
#22611967	2
C₄×#14556536	2
C₄×D₁₁₆₈	2
Q₈×C₁₆×C₇₃	2
#33928	73