Subgroups — #33243660

Order 18688.

528 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622904	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612154	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612153	C₂	1	2	yes	maximal, complemented
5	#22612079	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612076	C₂	1	2	yes	maximal, complemented
7	#22611969	C₂	1	2	yes	maximal, complemented
8	#22611833	C₂	1	1	yes	maximal, characteristic, complemented
9	SD₃₂×Dic₇₃	C₂	1	2	yes	maximal, complemented
10	C₄×#14556534	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2207	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₄₆	—	2	1	no	—
13	#17989257	—	2	1	no	—
14	C₂×#14556534	C₂²	1	1	yes	characteristic
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988991	C₂²	1	2	yes	—
17	#17988962	C₂²	1	2	yes	—
18	#17988882	—	2	1	no	—
19	#17988880	—	2	2	no	—
20	#17988780	—	2	2	no	—
21	#17988779	—	2	1	no	—
22	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
24	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
25	D₈×Dic₇₃	C₂²	1	2	yes	—
26	#17988415	C₂²	1	2	yes	—
27	#17988414	C₂²	1	1	yes	characteristic
28	#17988405	C₂²	1	2	yes	—
29	#17988391	C₂²	1	1	yes	characteristic
30	#17988389	C₂²	1	2	yes	—
31	#17988388	C₂²	1	2	yes	—
32	#17988388	C₂²	1	2	yes	—
33	#17988387	C₂²	1	1	yes	characteristic
34	C₄×#11924003	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₃	C₂²	1	2	yes	—
36	C₂×SD₃₂×C₇₃	C₂²	1	2	yes	—
37	#493×C₇₃	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	#357×C₇₃	C₂²	1	2	yes	—
41	C₂×#11924027	—	4	1	no	—
42	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
43	C₂×#11924013	—	2	2	no	—
44	#14556591	—	2	1	no	—
45	#14556591	—	4	1	no	—
46	#14556590	—	2	2	no	—
47	#14556590	—	4	2	no	—
48	#14556587	—	2	2	no	—
49	#14556563	—	4	2	no	—
50	#14556562	—	4	1	no	—
51	Q₈×Dic₇₃	—	2	2	no	—
52	C₄×Dic₁₄₆	—	4	1	no	—
53	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
54	C₄×D₂₉₂	—	2	1	no	—
55	D₄×Dic₇₃	—	2	2	no	—
56	#14556534	—	2	2	no	—
57	C₂×#11924003	C₂³	1	1	yes	characteristic
58	C₂×#11924003	C₂³	1	1	yes	characteristic
59	#14556525	—	2	1	no	—
60	#14556525	C₂³	1	2	yes	—
61	#14556525	C₂³	1	1	yes	characteristic
62	#14556524	—	2	2	no	—
63	C₄⋊#10596831	D₄	1	1	yes	characteristic
64	#14556520	D₄	1	2	yes	—
65	#14556515	—	2	2	no	—
66	#14556513	—	2	1	no	—
67	#14556512	—	2	2	no	—
68	C₈×Dic₇₃	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₇₃	—	2	1	no	—
71	C₂×D₈×C₇₃	C₂³	1	2	yes	—
72	#127×C₇₃	—	2	1	no	—
73	C₄×Q₈×C₇₃	—	2	1	no	—
74	SD₃₂×C₇₃	—	2	4	no	—
75	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
76	#107×C₇₃	C₂³	1	1	yes	characteristic
77	#103×C₇₃	—	2	1	no	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	#11924027	—	8	4	no	—
80	C₂×Dic₁₄₆	—	4	1	no	—
81	C₂×Dic₁₄₆	—	4	2	no	—
82	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
83	C₂×D₂₉₂	—	2	1	no	—
84	C₂×C₂²⋊D₇₃	—	4	2	no	—
85	C₂×C₂²⋊D₇₃	—	4	2	no	—
86	C₂²×Dic₇₃	—	4	2	no	—
87	C₂×C₄×D₇₃	—	4	1	no	—
88	D₅₈₄	—	2	1	no	—
89	D₅₈₄	—	2	1	no	—
90	#11924013	—	8	2	no	—
91	C₂×#10596831	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₇₃	—	2	1	no	—
93	C₄⋊Dic₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	4	1	no	—
95	C₄⋊Dic₇₃	—	4	2	no	—
96	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
97	#11924007	—	4	2	no	—
98	#11924007	D₈	1	2	yes	—
99	C₂²⋊Dic₇₃	—	4	2	no	—
100	#11924005	—	4	1	no	—
101	C₄×Dic₇₃	—	4	2	no	—
102	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
103	#11924003	—	2	1	no	—
104	#11924003	—	2	1	no	—
105	C₂×Q₈×C₇₃	—	2	1	no	—
106	C₂×D₄×C₇₃	—	2	2	no	—
107	C₂×D₄×C₇₃	—	4	1	no	—
108	C₂×D₄×C₇₃	—	4	1	no	—
109	Q₁₆×C₇₃	—	2	1	no	—
110	Q₁₆×C₇₃	#41	1	2	yes	complemented
111	D₈×C₇₃	—	2	2	no	—
112	D₈×C₇₃	—	2	2	no	—
113	D₈×C₇₃	—	8	1	no	—
114	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₇₃	—	2	1	no	—
117	C₄⋊C₄×C₇₃	—	4	1	no	—
118	C₄²×C₇₃	—	4	1	no	—
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₇₃	—	2	2	no	—
121	C₂³×C₇₃	—	4	2	no	—
122	C₂²×D₇₃	—	4	1	no	—
123	Q₈×C₇₃	—	2	2	no	—
124	Q₈×C₇₃	—	4	1	no	—
125	D₄×C₇₃	—	4	2	no	—
126	D₄×C₇₃	—	4	2	no	—
127	D₄×C₇₃	—	8	2	no	—
128	D₄×C₇₃	—	16	1	no	—
129	C₂×C₄×C₇₃	—	4	1	no	—
130	C₂×C₄×C₇₃	—	4	1	no	—
131	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₇₃	—	8	4	no	—
135	C₂²⋊D₇₃	—	16	2	no	—
136	C₂×Dic₇₃	—	4	2	no	—
137	C₂×Dic₇₃	—	4	1	no	—
138	C₂×Dic₇₃	—	8	4	no	—
139	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
140	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
141	D₂₉₂	—	4	1	no	—
142	D₂₉₂	—	4	1	no	—
143	C₄×D₇₃	—	8	2	no	—
144	Dic₁₄₆	—	16	2	no	—
145	Dic₁₄₆	—	16	1	no	—
146	Dic₁₄₆	SD₃₂	1	4	yes	complemented
147	C₈×C₇₃	—	2	1	no	—
148	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
150	#10596831	—	2	2	no	—
151	C₂²×C₇₃	—	8	2	no	—
152	C₂²×C₇₃	—	8	2	no	—
153	C₂²×C₇₃	—	8	2	no	—
154	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₇₃	—	8	1	no	—
156	C₂×D₇₃	—	8	1	no	—
157	C₂×D₇₃	—	8	1	no	—
158	C₄×C₇₃	—	2	1	no	—
159	C₄×C₇₃	—	4	2	no	—
160	C₄×C₇₃	—	8	1	no	—
161	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
162	C₄×C₇₃	#548	1	1	yes	characteristic
163	C₄×C₇₃	#577	1	1	yes	characteristic
164	Dic₇₃	—	2	2	no	—
165	Dic₇₃	—	8	1	no	—
166	Dic₇₃	—	8	2	no	—
167	Dic₇₃	C₂×SD₃₂	1	4	yes	—
168	#33929	—	73	1	no	maximal, Sylow
169	C₂×C₇₃	—	16	2	no	—
170	C₂×C₇₃	#3358	1	1	yes	characteristic
171	C₂×C₇₃	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₇₃	#3415	1	1	yes	characteristic
173	D₇₃	—	16	1	no	—
174	#3391	—	73	1	no	—
175	Q₈×Q₁₆	—	73	1	no	—
176	#2240	—	73	1	no	—
177	#2240	—	73	2	no	—
178	#2226	—	73	1	no	—
179	#2226	—	73	2	no	—
180	#2207	—	73	2	no	—
181	#2207	D₇₃	1	1	yes	characteristic, complemented
182	Q₈×C₁₆	—	73	1	no	—
183	C₄×SD₃₂	—	73	2	no	—
184	C₄×SD₃₂	—	73	1	no	—
185	C₇₃	#33929	1	1	yes	characteristic, complemented, Sylow
186	Q₈²	—	146	1	no	—
187	#550	—	146	1	no	—
188	C₂×SD₃₂	—	73	1	no	—
189	C₂×SD₃₂	C₂×D₇₃	1	2	yes	—
190	#500	—	73	2	no	—
191	#500	—	73	1	no	—
192	#493	—	73	2	no	—
193	#493	C₂×D₇₃	1	1	yes	characteristic
194	#477	—	146	2	no	—
195	#477	—	146	1	no	—
196	#459	—	146	1	no	—
197	#459	—	146	2	no	—
198	C₈×Q₈	—	73	1	no	—
199	C₄×Q₁₆	—	73	2	no	—
200	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
201	C₄×D₈	—	73	2	no	—
202	C₄×D₈	—	73	1	no	—
203	#367	—	73	2	no	—
204	#367	—	73	1	no	—
205	C₄⋊C₁₆	—	73	2	no	—
206	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
207	#358	—	73	1	no	—
208	#357	—	73	2	no	—
209	#357	—	73	2	no	—
210	#357	—	73	2	no	—
211	#357	—	73	1	no	—
212	#357	C₂×D₇₃	1	2	yes	—
213	C₄×C₁₆	—	73	2	no	—
214	C₄×C₁₆	—	73	1	no	—
215	C₂×#41	—	292	1	no	—
216	C₂×Q₁₆	C₂²×D₇₃	1	1	yes	characteristic
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	—	73	1	no	—
219	C₂×D₈	—	146	2	no	—
220	C₂×D₈	C₂²×D₇₃	1	2	yes	—
221	C₄⋊Q₈	—	292	1	no	—
222	C₄⋊Q₈	—	292	2	no	—
223	C₄⋊Q₈	—	146	2	no	—
224	C₄⋊Q₈	—	146	1	no	—
225	#127	—	2	1	no	—
226	#127	—	146	2	no	—
227	#121	—	292	1	no	—
228	#121	—	292	2	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	292	1	no	—
231	C₄×Q₈	—	73	1	no	—
232	C₄×Q₈	—	146	2	no	—
233	C₄×D₄	—	146	2	no	—
234	C₄×D₄	—	146	1	no	—
235	SD₃₂	—	2	4	no	—
236	SD₃₂	—	146	2	no	—
237	C₂×C₁₆	—	73	1	no	—
238	C₂×C₁₆	—	73	1	no	—
239	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
240	#107	—	73	2	no	—
241	#107	—	73	1	no	—
242	#107	—	146	2	no	—
243	#107	—	146	1	no	—
244	#107	C₂²×D₇₃	1	1	yes	characteristic
245	C₄⋊C₈	—	73	2	no	—
246	C₄⋊C₈	—	73	1	no	—
247	#103	—	2	1	no	—
248	#103	—	146	2	no	—
249	#102	—	146	2	no	—
250	#102	—	146	1	no	—
251	C₄×C₈	—	73	2	no	—
252	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
253	#41	—	584	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	292	2	no	—
256	C₂×Q₈	—	292	1	no	—
257	C₂×Q₈	—	73	1	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	292	2	no	—
262	C₂×D₄	—	292	2	no	—
263	C₂×D₄	—	146	1	no	—
264	C₂²×C₄	—	292	2	no	—
265	C₂²×C₄	—	292	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	#11924027	1	2	yes	complemented
268	D₈	—	2	2	no	—
269	D₈	—	2	2	no	—
270	D₈	—	8	1	no	—
271	D₈	—	584	2	no	—
272	D₈	—	146	1	no	—
273	D₈	—	146	1	no	—
274	C₂×C₈	—	73	2	no	—
275	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	292	2	no	—
280	C₄⋊C₄	—	292	2	no	—
281	C₄⋊C₄	—	292	1	no	—
282	C₄⋊C₄	—	73	2	no	—
283	C₄⋊C₄	—	73	1	no	—
284	C₄⋊C₄	—	146	1	no	—
285	C₄⋊C₄	—	146	2	no	—
286	C₂²⋊C₄	—	292	2	no	—
287	C₂²⋊C₄	—	292	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	292	2	no	—
290	C₄²	—	73	2	no	—
291	C₄²	D₄×D₇₃	1	1	yes	characteristic
292	C₁₆	—	2	2	no	—
293	C₁₆	—	146	1	no	—
294	C₁₆	—	146	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	292	1	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	73	4	no	—
300	Q₈	—	1168	1	no	—
301	Q₈	—	1168	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	16	1	no	—
306	D₄	—	292	1	no	—
307	D₄	—	292	1	no	—
308	D₄	—	584	4	no	—
309	D₄	—	1168	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	292	2	no	—
313	C₂×C₄	—	292	1	no	—
314	C₂×C₄	—	584	4	no	—
315	C₂×C₄	—	584	2	no	—
316	C₂×C₄	—	73	2	no	—
317	C₂×C₄	—	73	2	no	—
318	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
319	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	146	2	no	—
323	C₈	C₂×#11924027	1	1	yes	characteristic
324	C₈	#14556638	1	1	yes	characteristic
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	584	1	no	—
329	C₂²	—	584	1	no	—
330	C₂²	—	584	1	no	—
331	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	584	1	no	—
336	C₄	—	584	2	no	—
337	C₄	—	73	4	no	—
338	C₄	—	146	2	no	—
339	C₄	SD₃₂×D₇₃	1	2	yes	—
340	C₄	#17989243	1	1	yes	characteristic
341	C₄	#17989397	1	1	yes	characteristic
342	C₂	—	16	2	no	—
343	C₂	—	1168	1	no	—
344	C₂	#22622561	1	1	yes	characteristic
345	C₂	C₂×SD₃₂×D₇₃	1	1	yes	characteristic
346	C₂	#22623071	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2207	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22622904	C₂
hypercenter	#2207	D₇₃
hypocenter	C₇₃	#33929
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2207

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₄₆	C₂	semidirect
3. #22612154	C₂	semidirect
4. #22612153	C₂	semidirect
5. #22612079	C₂	semidirect
6. #22612076	C₂	semidirect
7. #22611969	C₂	semidirect
8. #22611833	C₂	semidirect
9. SD₃₂×Dic₇₃	C₂	semidirect
10. C₄×#14556534	C₂	semidirect
11. C₇₃×#2207	C₂	semidirect
110. Q₁₆×C₇₃	#41	semidirect
146. Dic₁₄₆	SD₃₂	semidirect
181. #2207	D₇₃	semidirect
185. C₇₃	#33929	semidirect
267. Q₁₆	#11924027	semidirect

Maximal subgroups

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

subgroup	index
#22622904	2
Q₁₆×Dic₁₄₆	2
#22612154	2
#22612153	2
#22612079	2
#22612076	2
#22611969	2
#22611833	2
SD₃₂×Dic₇₃	2
C₄×#14556534	2
C₇₃×#2207	2
#33929	73