Subgroups — #33057807

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620510	C₂	1	1	yes	maximal, characteristic
2	#22618674	C₂	1	2	yes	maximal, complemented
3	#22608844	C₂	1	2	yes	maximal, complemented
4	#22608794	C₂	1	4	yes	maximal, complemented
5	#22608754	C₂	1	2	yes	maximal, complemented
6	#22608329	C₂	1	2	yes	maximal, complemented
7	#22608220	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556597	—	2	2	no	—
10	#17989137	C₂²	1	2	yes	—
11	#17988992	C₄	1	1	yes	characteristic
12	#17988964	C₄	1	1	yes	characteristic
13	#17988934	C₄	1	2	yes	—
14	#17988823	C₄	1	2	yes	—
15	#17988823	C₄	1	2	yes	—
16	C₄×D₅₈₄	C₂²	1	2	yes	—
17	C₄×D₅₈₄	C₂²	1	2	yes	—
18	#17988664	C₂²	1	4	yes	—
19	#17988657	—	2	4	no	—
20	#17988652	C₂²	1	4	yes	—
21	#17988640	C₂²	1	1	yes	characteristic
22	#17988553	—	2	2	no	—
23	C₄×#11924012	C₂²	1	1	yes	characteristic
24	#17988314	C₂²	1	4	yes	—
25	#17988314	C₂²	1	4	yes	—
26	#17988309	C₂²	1	2	yes	—
27	#17988308	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	#334×C₇₃	C₂²	1	1	yes	characteristic
31	#334×C₇₃	C₂²	1	2	yes	—
32	#334×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×#11924028	—	2	2	no	—
34	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
35	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
36	#14556597	—	4	8	no	—
37	C₂×#11924012	D₄	1	2	yes	—
38	C₂×#11924011	—	2	4	no	—
39	C₂×C₈×D₇₃	—	2	4	no	—
40	#14556591	C₂×C₄	1	2	yes	—
41	#14556588	C₂×C₄	1	2	yes	—
42	#14556579	—	2	2	no	—
43	#14556565	—	2	2	no	—
44	#14556565	C₂×C₄	1	4	yes	—
45	C₄×D₂₉₂	—	2	2	no	—
46	C₄×D₂₉₂	C₂³	1	4	yes	—
47	#14556525	C₂×C₄	1	2	yes	—
48	#14556525	C₂×C₄	1	2	yes	—
49	C₄⋊#10596831	—	2	1	no	—
50	#14556513	C₂×C₄	1	4	yes	—
51	#14556513	C₂×C₄	1	4	yes	—
52	C₂²⋊#10596831	—	2	1	no	—
53	#14556503	—	2	4	no	—
54	#14556503	—	2	4	no	—
55	#14556502	D₄	1	2	yes	—
56	C₄×#10596831	C₂³	1	2	yes	—
57	C₄×#10596831	C₂³	1	2	yes	—
58	C₂²×C₈×C₇₃	—	2	2	no	—
59	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₇₃	—	2	2	no	—
63	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
64	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
65	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	#11924028	—	4	4	no	—
70	#11924028	—	4	4	no	—
71	C₂×Dic₁₄₆	—	2	2	no	—
72	C₂×D₂₉₂	—	2	2	no	—
73	C₂×D₂₉₂	—	2	2	no	—
74	C₂×D₂₉₂	C₂²×C₄	1	4	yes	—
75	C₂×C₂²⋊D₇₃	—	2	4	no	—
76	C₂×C₄×D₇₃	—	2	4	no	—
77	D₅₈₄	—	2	4	no	—
78	D₅₈₄	—	4	2	no	—
79	#11924012	—	2	4	no	—
80	#11924011	—	4	8	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	C₂×D₄	1	4	yes	—
83	C₈×D₇₃	—	4	8	no	—
84	C₄⋊Dic₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
86	#11924005	—	2	4	no	—
87	#11924005	—	2	4	no	—
88	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	4	1	no	—
95	C₂×C₈×C₇₃	—	4	4	no	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₃	#41	1	1	yes	characteristic
103	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₇₃	—	2	4	no	—
106	C₂×C₄×C₇₃	—	2	1	no	—
107	C₂×C₄×C₇₃	—	2	1	no	—
108	C₂×C₄×C₇₃	—	4	1	no	—
109	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
116	C₂²⋊D₇₃	—	4	8	no	—
117	C₂×Dic₇₃	—	2	4	no	—
118	D₂₉₂	—	2	4	no	—
119	D₂₉₂	—	2	4	no	—
120	D₂₉₂	—	4	4	no	—
121	D₂₉₂	—	8	2	no	—
122	C₄×D₇₃	—	4	4	no	—
123	C₄×D₇₃	—	4	4	no	—
124	Dic₁₄₆	—	4	4	no	—
125	C₈×C₇₃	—	2	1	no	—
126	C₈×C₇₃	—	2	1	no	—
127	C₈×C₇₃	—	2	1	no	—
128	C₈×C₇₃	—	4	2	no	—
129	C₈×C₇₃	—	4	2	no	—
130	C₈×C₇₃	#131	1	2	yes	complemented
131	#10596831	—	2	4	no	—
132	#10596831	—	4	2	no	—
133	C₂²×C₇₃	—	2	1	no	—
134	C₂²×C₇₃	—	2	1	no	—
135	C₂²×C₇₃	#442	1	2	yes	—
136	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×D₇₃	—	4	4	no	—
138	C₂×D₇₃	—	4	4	no	—
139	C₂×D₇₃	—	4	4	no	—
140	C₄×C₇₃	—	2	1	no	—
141	C₄×C₇₃	—	2	1	no	—
142	C₄×C₇₃	—	2	1	no	—
143	C₄×C₇₃	#442	1	2	yes	—
144	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
145	C₄×C₇₃	#568	1	1	yes	characteristic
146	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	Dic₇₃	—	4	4	no	—
148	#18073	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	#2935	1	1	yes	characteristic
151	C₂×C₇₃	C₂×#442	1	1	yes	characteristic
152	C₂×C₇₃	#2957	1	1	yes	characteristic
153	D₇₃	—	8	4	no	—
154	#3169	—	73	1	no	—
155	#2935	—	73	2	no	—
156	#1594	—	73	2	no	—
157	#1586	—	73	4	no	—
158	#1580	—	73	2	no	—
159	C₂×#334	D₇₃	1	1	yes	characteristic, complemented
160	#1480	—	73	2	no	—
161	#1451	—	73	1	no	—
162	C₇₃	#18073	1	1	yes	characteristic, complemented, Sylow
163	C₂×#131	—	146	2	no	—
164	#532	—	73	2	no	—
165	#500	—	73	1	no	—
166	#493	—	73	1	no	—
167	#486	—	73	2	no	—
168	#466	—	73	2	no	—
169	#466	—	73	2	no	—
170	C₄×D₈	—	73	2	no	—
171	C₄×D₈	—	73	2	no	—
172	#436	—	73	4	no	—
173	#435	—	146	4	no	—
174	C₈×D₄	—	73	4	no	—
175	#431	—	73	1	no	—
176	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
177	#408	—	146	2	no	—
178	C₄×M₁₆	—	73	1	no	—
179	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
180	#334	C₂×D₇₃	1	1	yes	characteristic
181	#334	C₂×D₇₃	1	2	yes	—
182	#334	C₂×D₇₃	1	1	yes	characteristic
183	D₄⋊C₈	—	73	4	no	—
184	D₄⋊C₈	—	73	4	no	—
185	#322	—	73	2	no	—
186	#322	—	73	2	no	—
187	C₂×#41	—	146	2	no	—
188	C₂×D₈	—	73	2	no	—
189	C₂×D₈	—	73	2	no	—
190	#131	—	292	8	no	—
191	C₂×M₁₆	—	73	2	no	—
192	C₂×M₁₆	—	146	4	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	146	4	no	—
195	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	73	2	no	—
198	#127	—	73	2	no	—
199	#124	—	146	2	no	—
200	#121	—	73	4	no	—
201	#121	—	146	2	no	—
202	C₄×D₄	—	73	4	no	—
203	C₄×D₄	—	146	2	no	—
204	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
205	#107	—	73	2	no	—
206	#107	—	73	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	146	1	no	—
209	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
210	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
211	#102	—	73	4	no	—
212	#102	—	73	4	no	—
213	C₂²⋊C₈	—	146	4	no	—
214	C₂²⋊C₈	—	146	4	no	—
215	C₂²⋊C₈	—	146	1	no	—
216	#97	—	73	2	no	—
217	C₄×C₈	—	73	2	no	—
218	C₄×C₈	—	73	2	no	—
219	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
220	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
223	#41	—	292	4	no	—
224	#41	—	292	4	no	—
225	C₂×Q₈	—	146	2	no	—
226	C₂×D₄	—	73	4	no	—
227	C₂×D₄	—	146	4	no	—
228	C₂×D₄	—	146	2	no	—
229	C₂×D₄	—	146	2	no	—
230	C₂²×C₄	—	146	4	no	—
231	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
233	C₂²×C₄	#11924029	1	1	yes	characteristic
234	D₈	—	292	2	no	—
235	D₈	—	146	4	no	—
236	M₁₆	—	292	8	no	—
237	M₁₆	—	146	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	292	8	no	—
244	C₂×C₈	—	73	4	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
250	C₄⋊C₄	—	73	4	no	—
251	C₄⋊C₄	—	146	2	no	—
252	C₂²⋊C₄	—	146	4	no	—
253	C₂²⋊C₄	—	146	4	no	—
254	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
255	C₄²	D₄×D₇₃	1	1	yes	characteristic
256	C₄²	#11924029	1	1	yes	characteristic
257	C₄²	C₂³×D₇₃	1	1	yes	characteristic
258	C₂³	—	146	4	no	—
259	C₂³	#14556553	1	1	yes	characteristic
260	Q₈	—	292	4	no	—
261	D₄	—	292	8	no	—
262	D₄	—	292	4	no	—
263	D₄	—	584	2	no	—
264	D₄	—	146	4	no	—
265	D₄	—	146	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	292	4	no	—
270	C₂×C₄	—	292	4	no	—
271	C₂×C₄	—	146	4	no	—
272	C₂×C₄	#14556553	1	1	yes	characteristic
273	C₂×C₄	#14556553	1	1	yes	characteristic
274	C₂×C₄	#14556553	1	1	yes	characteristic
275	C₂×C₄	#14556618	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	292	2	no	—
285	C₈	—	146	4	no	—
286	C₈	#14556597	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	292	4	no	—
290	C₂²	—	292	4	no	—
291	C₂²	—	292	4	no	—
292	C₂²	#17988698	1	2	yes	—
293	C₂²	C₂×#14556553	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	292	4	no	—
298	C₄	#17988698	1	2	yes	—
299	C₄	C₂×#14556597	1	1	yes	characteristic
300	C₄	#17989353	1	1	yes	characteristic
301	C₄	C₂×#14556618	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	584	4	no	—
304	C₂	#22618674	1	1	yes	characteristic
305	C₂	C₂×#17988698	1	1	yes	characteristic
306	C₂	#22618847	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#22620510	C₂
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#18073
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. #22618674	C₂	semidirect
3. #22608844	C₂	semidirect
4. #22608794	C₂	semidirect
5. #22608754	C₂	semidirect
6. #22608329	C₂	semidirect
7. #22608220	C₂	semidirect
8. C₂×#334×C₇₃	C₂	semidirect
130. C₈×C₇₃	#131	semidirect
159. C₂×#334	D₇₃	semidirect
162. C₇₃	#18073	semidirect
286. C₈	#14556597	semidirect

Maximal subgroups

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

subgroup	index
#22620510	2
#22618674	2
#22608844	2
#22608794	2
#22608754	2
#22608329	2
#22608220	2
C₂×#334×C₇₃	2
#18073	73