Subgroups — #33002469

Order 18688.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989005	C₂	1	1	yes	maximal, characteristic
2	#22611691	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988372	C₂	1	1	yes	maximal, characteristic
4	#22609768	C₂	1	1	yes	maximal, characteristic
5	#22608021	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608009	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556597	C₂²	1	1	yes	characteristic
9	#17989005	—	2	2	no	—
10	#17989005	—	2	2	no	—
11	C₂×#14556530	—	2	1	no	—
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	C₂²×#11924003	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₃	—	2	1	no	—
15	C₂×#14556519	C₄	1	1	yes	characteristic
16	#17988584	C₄	1	1	yes	characteristic
17	#17988542	C₂²	1	1	yes	characteristic
18	#17988405	—	2	1	no	—
19	#17988405	—	2	1	no	—
20	#17988372	—	2	2	no	—
21	#17988352	C₄	1	1	yes	characteristic
22	#17988347	C₄	1	1	yes	characteristic
23	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#364×C₇₃	—	2	2	no	—
27	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
28	#14556597	—	2	1	no	—
29	#14556597	—	2	1	no	—
30	#14556597	D₄	1	2	yes	—
31	#14556597	Q₈	1	2	yes	—
32	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
33	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924011	—	2	1	no	—
35	C₂×C₈×D₇₃	—	2	1	no	—
36	#14556550	C₂×C₄	1	1	yes	characteristic
37	#14556530	—	2	4	no	—
38	C₂×#11924003	—	2	1	no	—
39	C₂×#11924003	—	2	1	no	—
40	C₂×#11924003	—	2	1	no	—
41	C₂×#11924003	—	2	1	no	—
42	C₂×#11924003	—	2	1	no	—
43	C₂×#11924003	D₄	1	1	yes	characteristic
44	C₂×#11924003	D₄	1	1	yes	characteristic
45	C₁₆×D₇₃	—	2	4	no	—
46	#14556519	—	2	2	no	—
47	#14556516	C₈	1	1	yes	characteristic
48	#14556516	C₈	1	1	yes	characteristic
49	#14556513	C₈	1	1	yes	characteristic
50	#14556513	C₈	1	1	yes	characteristic
51	#14556501	—	2	1	no	—
52	C₈×Dic₇₃	—	2	1	no	—
53	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₇₃	—	2	1	no	—
57	M₃₂×C₇₃	—	2	1	no	—
58	M₃₂×C₇₃	—	2	1	no	—
59	M₃₂×C₇₃	D₄	1	2	yes	—
60	C₂×C₁₆×C₇₃	—	2	1	no	—
61	C₂×C₁₆×C₇₃	—	2	1	no	—
62	C₄×C₈×C₇₃	—	2	1	no	—
63	C₄×C₈×C₇₃	—	2	1	no	—
64	#11924028	—	2	1	no	—
65	#11924028	—	2	1	no	—
66	#11924028	C₄⋊C₄	1	2	yes	—
67	#11924028	M₁₆	1	2	yes	—
68	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
69	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
70	C₂×C₂²⋊D₇₃	—	2	1	no	—
71	C₂×C₄×D₇₃	—	2	1	no	—
72	#11924012	—	2	1	no	—
73	#11924012	—	2	1	no	—
74	#11924012	—	2	1	no	—
75	#11924012	C₄⋊C₄	1	2	yes	—
76	#11924011	—	2	2	no	—
77	#11924011	—	2	2	no	—
78	C₂×#10596831	—	2	1	no	—
79	C₂×#10596831	—	2	1	no	—
80	C₈×D₇₃	—	2	2	no	—
81	C₈×D₇₃	—	2	2	no	—
82	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
83	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊Dic₇₃	—	2	1	no	—
85	C₄×Dic₇₃	—	2	1	no	—
86	#11924003	—	2	1	no	—
87	#11924003	—	2	1	no	—
88	#11924003	—	4	1	no	—
89	C₂²×C₄×C₇₃	—	2	1	no	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	—	2	1	no	—
92	C₂×C₈×C₇₃	—	2	1	no	—
93	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
94	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
101	C₄²×C₇₃	—	2	1	no	—
102	C₄²×C₇₃	—	2	1	no	—
103	C₁₆×C₇₃	—	2	2	no	—
104	C₁₆×C₇₃	—	4	1	no	—
105	C₂³×C₇₃	#131	1	1	yes	characteristic
106	C₂²×D₇₃	—	2	1	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	4	1	no	—
109	C₂×C₄×C₇₃	—	4	1	no	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
116	C₂²⋊D₇₃	—	2	2	no	—
117	C₂²⋊D₇₃	—	2	2	no	—
118	C₂×Dic₇₃	—	2	1	no	—
119	C₂×Dic₇₃	—	2	1	no	—
120	C₂×Dic₇₃	—	2	1	no	—
121	D₂₉₂	—	2	1	no	—
122	D₂₉₂	C₄⋊C₈	1	2	yes	—
123	C₄×D₇₃	—	2	2	no	—
124	C₄×D₇₃	—	2	2	no	—
125	Dic₁₄₆	—	2	1	no	—
126	Dic₁₄₆	C₄⋊C₈	1	2	yes	—
127	C₈×C₇₃	—	2	1	no	—
128	C₈×C₇₃	—	2	1	no	—
129	C₈×C₇₃	#120	1	1	yes	characteristic
130	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₇₃	#123	1	1	yes	characteristic
133	#10596831	—	2	2	no	—
134	#10596831	—	4	1	no	—
135	C₂²×C₇₃	—	2	1	no	—
136	C₂²×C₇₃	—	2	1	no	—
137	C₂²×C₇₃	#424	1	1	yes	characteristic
138	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₇₃	#435	1	1	yes	characteristic
140	C₂×D₇₃	—	2	2	no	—
141	C₂×D₇₃	—	4	1	no	—
142	C₂×D₇₃	—	4	1	no	—
143	C₄×C₇₃	—	4	2	no	—
144	C₄×C₇₃	#385	1	1	yes	characteristic
145	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
147	C₄×C₇₃	#435	1	1	yes	characteristic
148	Dic₇₃	—	2	2	no	—
149	Dic₇₃	—	4	1	no	—
150	Dic₇₃	—	4	1	no	—
151	#12917	—	73	1	no	maximal, Sylow
152	C₂×C₇₃	—	2	1	no	—
153	C₂×C₇₃	—	2	1	no	—
154	C₂×C₇₃	#1850	1	1	yes	characteristic
155	C₂×C₇₃	#2170	1	1	yes	characteristic
156	C₂×C₇₃	#2171	1	1	yes	characteristic
157	D₇₃	—	4	2	no	—
158	C₂×#504	—	73	1	no	—
159	C₂×#364	D₇₃	1	1	yes	characteristic, complemented
160	#2151	—	73	1	no	—
161	C₂×#350	—	73	1	no	—
162	#1764	—	73	1	no	—
163	#1379	—	73	1	no	—
164	#1374	—	73	1	no	—
165	C₇₃	#12917	1	1	yes	characteristic, complemented, Sylow
166	C₂×#131	—	73	1	no	—
167	#504	—	146	2	no	—
168	#504	—	146	2	no	—
169	C₂×M₃₂	—	146	1	no	—
170	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
171	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
172	C₂²×C₁₆	—	73	1	no	—
173	C₂²×C₁₆	—	73	1	no	—
174	C₂²×C₁₆	—	146	1	no	—
175	C₂×#104	—	73	1	no	—
176	#416	—	73	1	no	—
177	#405	—	73	1	no	—
178	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
179	#364	—	2	2	no	—
180	C₄⋊C₁₆	—	146	1	no	—
181	C₄⋊C₁₆	—	146	1	no	—
182	#350	—	146	2	no	—
183	#341	—	73	1	no	—
184	#339	—	73	1	no	—
185	C₂×#41	—	73	1	no	—
186	#131	—	73	2	no	—
187	#131	—	73	2	no	—
188	#131	—	146	1	no	—
189	#131	—	146	1	no	—
190	C₂×M₁₆	—	73	1	no	—
191	C₂×M₁₆	—	73	1	no	—
192	C₂×M₁₆	—	146	1	no	—
193	C₂²×C₈	—	146	1	no	—
194	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
196	#117	—	73	1	no	—
197	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	146	4	no	—
202	M₃₂	C₂²⋊D₇₃	1	2	yes	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	73	1	no	—
206	C₂×C₁₆	—	73	1	no	—
207	C₂×C₁₆	—	146	4	no	—
208	C₂×C₁₆	—	146	1	no	—
209	C₂×C₁₆	—	146	1	no	—
210	C₂×C₁₆	—	146	1	no	—
211	C₂×C₁₆	—	146	1	no	—
212	C₂×C₁₆	—	146	1	no	—
213	#104	—	146	2	no	—
214	#103	—	73	1	no	—
215	#103	—	73	1	no	—
216	#102	—	73	1	no	—
217	#102	—	73	1	no	—
218	#97	—	146	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	146	1	no	—
222	#41	—	73	2	no	—
223	#41	—	73	2	no	—
224	#41	—	146	1	no	—
225	#41	—	146	1	no	—
226	C₂×Q₈	—	73	1	no	—
227	C₂×D₄	—	73	1	no	—
228	C₂×D₄	—	146	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	146	1	no	—
231	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
232	M₁₆	—	73	2	no	—
233	M₁₆	—	146	1	no	—
234	M₁₆	—	146	1	no	—
235	M₁₆	—	146	1	no	—
236	M₁₆	—	146	2	no	—
237	M₁₆	—	146	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	146	1	no	—
241	C₂×C₈	—	146	1	no	—
242	C₂×C₈	—	146	2	no	—
243	C₂×C₈	—	146	2	no	—
244	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
245	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
246	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
247	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
250	C₂×C₈	#11924028	1	1	yes	characteristic
251	C₂×C₈	#11924029	1	1	yes	characteristic
252	C₄⋊C₄	—	73	1	no	—
253	C₄⋊C₄	—	73	1	no	—
254	C₂²⋊C₄	—	146	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	146	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	292	1	no	—
261	C₁₆	—	146	1	no	—
262	C₁₆	—	146	1	no	—
263	C₂³	—	146	1	no	—
264	C₂³	#14556599	1	1	yes	characteristic
265	Q₈	—	73	2	no	—
266	Q₈	—	146	1	no	—
267	D₄	—	73	2	no	—
268	D₄	—	146	2	no	—
269	D₄	—	146	2	no	—
270	D₄	—	146	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	146	1	no	—
275	C₂×C₄	—	146	1	no	—
276	C₂×C₄	—	146	1	no	—
277	C₂×C₄	—	146	2	no	—
278	C₂×C₄	—	146	2	no	—
279	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
281	C₂×C₄	#14556553	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
284	C₂×C₄	#14556597	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	292	1	no	—
288	C₈	—	146	2	no	—
289	C₈	#14556559	1	1	yes	characteristic
290	C₈	#14556567	1	1	yes	characteristic
291	C₈	#14556569	1	1	yes	characteristic
292	C₈	#14556576	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	292	1	no	—
296	C₂²	—	292	1	no	—
297	C₂²	—	146	2	no	—
298	C₂²	#17988612	1	1	yes	characteristic
299	C₂²	#17988652	1	1	yes	characteristic
300	C₂²	#17988656	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	292	1	no	—
303	C₄	—	292	1	no	—
304	C₄	—	146	2	no	—
305	C₄	#17988458	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
307	C₄	C₈×C₂²⋊D₇₃	1	1	yes	characteristic
308	C₄	#17988657	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	292	2	no	—
312	C₂	#22610107	1	1	yes	characteristic
313	C₂	#22611767	1	1	yes	characteristic
314	C₂	#22611772	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	#12917
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #22611691	C₂	semidirect
5. #22608021	C₂	semidirect
6. #22608009	C₂	semidirect
7. C₂×#364×C₇₃	C₂	semidirect
159. C₂×#364	D₇₃	semidirect
165. C₇₃	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989005	2
#22611691	2
C₂×#17988372	2
#22609768	2
#22608021	2
#22608009	2
C₂×#364×C₇₃	2
#12917	73