Subgroups — #33239453

Order 18688.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989030	C₂	1	1	yes	maximal, characteristic
2	#22618849	C₂	1	1	yes	maximal, characteristic
3	#22618800	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611824	C₂	1	8	yes	maximal, complemented
5	#22611704	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988404	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611668	C₂	1	1	yes	maximal, characteristic
8	C₇₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
11	#17989353	—	2	1	no	—
12	#17989064	—	2	1	no	—
13	#17989030	C₄	1	4	yes	complemented
14	#17989030	C₂²	1	4	yes	—
15	C₂×Dic₅₈₄	C₄	1	2	yes	complemented
16	C₂×#14556536	C₄	1	2	yes	complemented
17	#17988710	C₂²	1	4	yes	—
18	#17988705	—	2	2	no	—
19	#17988695	C₂²	1	4	yes	complemented
20	C₄×Dic₂₉₂	—	2	2	no	—
21	C₂×#14556527	C₂²	1	1	yes	characteristic
22	#17988625	C₂²	1	1	yes	characteristic
23	#17988605	—	2	1	no	—
24	#17988597	—	2	1	no	—
25	#17988410	C₂²	1	4	yes	complemented
26	#17988404	C₂²	1	4	yes	complemented
27	#17988398	C₂²	1	4	yes	—
28	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
30	#405×C₇₃	C₂²	1	1	yes	characteristic
31	#347×C₇₃	C₂²	1	4	yes	complemented
32	C₂×#11924028	—	2	1	no	—
33	C₂²×Dic₁₄₆	—	2	1	no	—
34	#14556613	—	2	1	no	—
35	#14556613	—	2	1	no	—
36	#14556613	C₂×C₄	1	2	yes	—
37	#14556613	C₂³	1	2	yes	—
38	C₂×Dic₂₉₂	—	2	1	no	—
39	C₂×Dic₂₉₂	—	2	1	no	—
40	C₂×Dic₂₉₂	—	2	1	no	—
41	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₉₂	C₂³	1	2	yes	—
45	C₂×#11924017	—	2	1	no	—
46	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
47	#14556599	—	4	2	no	—
48	#14556597	—	2	4	no	—
49	C₂×#11924012	—	2	1	no	—
50	M₁₆×D₇₃	—	4	2	no	—
51	C₄×Dic₁₄₆	—	2	4	no	—
52	#14556556	—	4	2	no	—
53	#14556550	—	2	1	no	—
54	#14556547	—	4	2	no	—
55	Dic₅₈₄	C₂×C₄	1	8	yes	—
56	#14556536	C₂×C₄	1	8	yes	—
57	#14556527	—	2	1	no	—
58	#14556527	C₂³	1	2	yes	—
59	#14556525	—	2	1	no	—
60	#14556523	C₂³	1	2	yes	—
61	#14556519	—	2	4	no	—
62	#14556516	—	2	4	no	—
63	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	#117×C₇₃	D₄	1	1	yes	characteristic
66	M₃₂×C₇₃	—	2	1	no	—
67	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
68	M₃₂×C₇₃	C₂³	1	2	yes	—
69	M₃₂×C₇₃	C₂³	1	2	yes	—
70	C₂×C₁₆×C₇₃	—	2	1	no	—
71	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
72	#97×C₇₃	C₂³	1	2	yes	—
73	C₄×C₈×C₇₃	D₄	1	2	yes	—
74	#11924028	—	2	2	no	—
75	#11924028	—	2	2	no	—
76	#11924028	—	4	1	no	—
77	#11924028	—	4	1	no	—
78	C₂×Dic₁₄₆	—	2	1	no	—
79	C₂×Dic₁₄₆	—	2	2	no	—
80	C₂×Dic₁₄₆	—	2	2	no	—
81	C₂×Dic₁₄₆	—	2	2	no	—
82	C₂×Dic₁₄₆	—	4	1	no	—
83	C₂×Dic₁₄₆	—	4	1	no	—
84	C₂×Dic₁₄₆	—	4	1	no	—
85	C₂×D₂₉₂	—	2	1	no	—
86	C₂×C₂²⋊D₇₃	—	4	1	no	—
87	C₂²×Dic₇₃	—	4	1	no	—
88	C₂×C₄×D₇₃	—	4	1	no	—
89	Dic₂₉₂	—	2	2	no	—
90	Dic₂₉₂	—	2	2	no	—
91	Dic₂₉₂	—	2	1	no	—
92	Dic₂₉₂	—	2	2	no	—
93	Dic₂₉₂	C₂²×C₄	1	4	yes	—
94	Dic₂₉₂	C₂²×C₄	1	2	yes	—
95	#11924017	—	2	2	no	—
96	#11924017	—	2	2	no	—
97	D₅₈₄	—	2	1	no	—
98	D₅₈₄	C₂²×C₄	1	2	yes	—
99	#11924012	—	2	2	no	—
100	#11924012	—	4	1	no	—
101	#11924011	—	4	4	no	—
102	C₂×#10596831	—	4	1	no	—
103	C₈×D₇₃	—	4	4	no	—
104	C₄⋊Dic₇₃	—	2	2	no	—
105	#11924007	—	4	4	no	—
106	C₂²⋊Dic₇₃	—	4	1	no	—
107	C₄×Dic₇₃	—	4	4	no	—
108	C₄×Dic₇₃	—	4	1	no	—
109	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₇₃	—	2	1	no	—
111	M₁₆×C₇₃	—	2	1	no	—
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
117	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₇₃	—	2	1	no	—
120	C₂²⋊C₄×C₇₃	—	2	1	no	—
121	C₄²×C₇₃	C₂×D₄	1	2	yes	—
122	C₁₆×C₇₃	—	2	2	no	—
123	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
124	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₇₃	—	4	1	no	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₇₃	—	4	2	no	—
134	C₂²⋊D₇₃	—	4	2	no	—
135	C₂×Dic₇₃	—	4	2	no	—
136	C₂×Dic₇₃	—	4	1	no	—
137	C₂×Dic₇₃	—	4	2	no	—
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	—	4	2	no	—
140	D₂₉₂	—	2	2	no	—
141	D₂₉₂	—	4	1	no	—
142	C₄×D₇₃	—	4	2	no	—
143	C₄×D₇₃	—	4	2	no	—
144	Dic₁₄₆	—	2	2	no	—
145	Dic₁₄₆	—	2	4	no	—
146	Dic₁₄₆	—	4	2	no	—
147	Dic₁₄₆	—	4	2	no	—
148	Dic₁₄₆	—	4	2	no	—
149	Dic₁₄₆	—	4	1	no	—
150	C₈×C₇₃	—	2	2	no	—
151	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
152	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
155	#10596831	—	4	2	no	—
156	C₂²×C₇₃	—	4	1	no	—
157	C₂²×C₇₃	#442	1	1	yes	characteristic
158	C₂²×C₇₃	#442	1	1	yes	characteristic
159	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×D₇₃	—	4	2	no	—
161	C₂×D₇₃	—	16	1	no	—
162	C₄×C₇₃	—	4	2	no	—
163	C₄×C₇₃	#442	1	1	yes	characteristic
164	C₄×C₇₃	#442	1	1	yes	characteristic
165	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	Dic₇₃	—	4	4	no	—
168	Dic₇₃	—	4	2	no	—
169	Dic₇₃	—	8	2	no	—
170	#33470	—	73	1	no	maximal, Sylow
171	C₂×C₇₃	—	2	1	no	—
172	C₂×C₇₃	—	2	1	no	—
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	C₂×#442	1	1	yes	characteristic
175	D₇₃	—	8	2	no	—
176	C₂×#510	—	73	1	no	—
177	#2957	—	73	1	no	—
178	#2950	—	73	1	no	—
179	#2180	—	73	8	no	—
180	#2155	—	73	1	no	—
181	C₂×#362	—	73	1	no	—
182	#2145	—	73	1	no	—
183	#2110	D₇₃	1	1	yes	characteristic, complemented
184	C₇₃	#33470	1	1	yes	characteristic, complemented, Sylow
185	C₂×#135	—	73	1	no	—
186	C₂²×Q₁₆	—	73	1	no	—
187	#568	—	146	1	no	—
188	#519	—	146	1	no	—
189	#510	—	73	4	no	—
190	#510	—	73	4	no	—
191	C₂×Q₃₂	—	73	2	no	—
192	C₂×SD₃₂	—	73	2	no	—
193	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
194	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
195	#444	—	73	4	no	—
196	#443	—	146	2	no	—
197	#441	—	73	4	no	—
198	C₄×Q₁₆	—	146	2	no	—
199	C₂×#108	—	73	1	no	—
200	#427	—	73	1	no	—
201	#421	—	146	1	no	—
202	#419	—	146	1	no	—
203	#405	C₂×D₇₃	1	1	yes	characteristic
204	#365	—	73	4	no	—
205	#362	—	73	4	no	—
206	#360	—	73	4	no	—
207	#347	C₂×D₇₃	1	4	yes	complemented
208	C₂×#41	—	146	1	no	—
209	C₂²×Q₈	—	146	1	no	—
210	#135	—	73	2	no	—
211	#135	—	73	2	no	—
212	#135	—	146	1	no	—
213	#135	—	146	1	no	—
214	C₂×Q₁₆	—	73	2	no	—
215	C₂×Q₁₆	—	73	2	no	—
216	C₂×Q₁₆	—	73	1	no	—
217	C₂×Q₁₆	—	73	2	no	—
218	C₂×Q₁₆	—	146	1	no	—
219	C₂×Q₁₆	—	146	1	no	—
220	C₂×Q₁₆	—	146	1	no	—
221	C₂×SD₁₆	—	146	1	no	—
222	C₂×D₈	—	73	1	no	—
223	#131	—	292	2	no	—
224	#131	—	146	4	no	—
225	C₂×M₁₆	—	292	2	no	—
226	C₂×M₁₆	—	146	1	no	—
227	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
229	C₄×Q₈	—	292	2	no	—
230	C₄×Q₈	—	146	4	no	—
231	#117	—	292	2	no	—
232	#117	—	146	1	no	—
233	#117	D₂₉₂	1	1	yes	characteristic
234	Q₃₂	—	73	8	no	—
235	SD₃₂	—	73	8	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	C₄×D₇₃	1	2	yes	—
238	M₃₂	C₂²×D₇₃	1	2	yes	—
239	M₃₂	C₂²×D₇₃	1	2	yes	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
242	#108	—	73	2	no	—
243	#108	—	146	1	no	—
244	#107	—	146	1	no	—
245	#106	—	73	2	no	—
246	#104	—	146	4	no	—
247	#103	—	146	4	no	—
248	#97	C₂²×D₇₃	1	2	yes	—
249	C₄×C₈	D₂₉₂	1	2	yes	—
250	#41	—	292	1	no	—
251	#41	—	292	1	no	—
252	#41	—	146	2	no	—
253	#41	—	146	2	no	—
254	C₂×Q₈	—	292	1	no	—
255	C₂×Q₈	—	292	1	no	—
256	C₂×Q₈	—	292	1	no	—
257	C₂×Q₈	—	146	1	no	—
258	C₂×Q₈	—	146	2	no	—
259	C₂×Q₈	—	146	2	no	—
260	C₂×Q₈	—	146	2	no	—
261	C₂×D₄	—	292	1	no	—
262	C₂×D₄	—	146	1	no	—
263	C₂²×C₄	—	292	1	no	—
264	C₂²×C₄	—	292	1	no	—
265	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
266	Q₁₆	—	73	4	no	—
267	Q₁₆	—	73	2	no	—
268	Q₁₆	—	146	2	no	—
269	Q₁₆	—	146	2	no	—
270	Q₁₆	—	146	1	no	—
271	Q₁₆	—	146	2	no	—
272	SD₁₆	—	146	2	no	—
273	SD₁₆	—	146	2	no	—
274	D₈	—	73	2	no	—
275	D₈	—	146	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	2	1	no	—
278	M₁₆	—	292	4	no	—
279	M₁₆	—	292	1	no	—
280	M₁₆	—	146	2	no	—
281	C₂×C₈	—	292	4	no	—
282	C₂×C₈	—	292	1	no	—
283	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
285	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
286	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
287	C₂×C₈	#11924028	1	1	yes	characteristic
288	C₂×C₈	#11924028	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	292	4	no	—
292	C₄⋊C₄	—	146	2	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	292	1	no	—
295	C₄²	—	292	4	no	—
296	C₄²	—	292	1	no	—
297	C₄²	C₂×D₂₉₂	1	2	yes	—
298	C₁₆	—	2	2	no	—
299	C₁₆	C₂×C₄×D₇₃	1	4	yes	—
300	C₂³	—	292	1	no	—
301	C₂³	#14556619	1	1	yes	characteristic
302	Q₈	—	292	2	no	—
303	Q₈	—	292	2	no	—
304	Q₈	—	292	2	no	—
305	Q₈	—	292	1	no	—
306	Q₈	—	146	2	no	—
307	Q₈	—	146	4	no	—
308	D₄	—	292	2	no	—
309	D₄	—	292	2	no	—
310	D₄	—	292	1	no	—
311	D₄	—	146	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	292	2	no	—
314	C₂×C₄	—	292	1	no	—
315	C₂×C₄	—	292	2	no	—
316	C₂×C₄	—	292	2	no	—
317	C₂×C₄	—	292	2	no	—
318	C₂×C₄	—	292	2	no	—
319	C₂×C₄	—	292	2	no	—
320	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
321	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
322	C₂×C₄	#14556619	1	1	yes	characteristic
323	C₂×C₄	#14556619	1	1	yes	characteristic
324	C₂×C₄	#14556619	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	292	2	no	—
328	C₈	C₄×D₂₉₂	1	1	yes	characteristic
329	C₈	C₄×D₂₉₂	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
331	C₈	C₂×#11924028	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	292	2	no	—
334	C₂²	—	1168	1	no	—
335	C₂²	#17988700	1	1	yes	characteristic
336	C₂²	#17988700	1	1	yes	characteristic
337	C₂²	C₂×#14556619	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	292	2	no	—
340	C₄	—	292	4	no	—
341	C₄	—	584	2	no	—
342	C₄	#17988700	1	1	yes	characteristic
343	C₄	#17988700	1	1	yes	characteristic
344	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
345	C₄	C₂×#14556619	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	584	2	no	—
350	C₂	C₂×#17988700	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17988700
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2110	C₂
socle	C₂×C₇₃	C₂×#442
prime radical	C₂×#14556613	C₂²
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2110	D₇₃
hypocenter	C₇₃	#33470
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #22618800	C₂	semidirect
4. #22611824	C₂	semidirect
5. #22611704	C₂	semidirect
6. C₂×#17988404	C₂	semidirect
8. C₇₃×#2110	C₂	semidirect
13. #17989030	C₄	semidirect
15. C₂×Dic₅₈₄	C₄	semidirect
16. C₂×#14556536	C₄	semidirect
19. #17988695	C₂²	semidirect
25. #17988410	C₂²	semidirect
26. #17988404	C₂²	semidirect
31. #347×C₇₃	C₂²	semidirect
183. #2110	D₇₃	semidirect
184. C₇₃	#33470	semidirect
207. #347	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989030	2
#22618849	2
#22618800	2
#22611824	2
#22611704	2
C₂×#17988404	2
#22611668	2
C₇₃×#2110	2
#33470	73