Subgroups — #33238631

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989003	C₂	1	2	yes	maximal
2	C₄×#14556597	C₂	1	1	yes	maximal, characteristic
3	#22611529	C₂	1	4	yes	maximal, complemented
4	#22611527	C₂	1	2	yes	maximal, complemented
5	C₄×#14556531	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611524	C₂	1	2	yes	maximal, complemented
7	#346×D₇₃	C₂	1	2	yes	maximal, complemented
8	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556597	C₄	1	1	yes	characteristic
10	C₂×#14556597	C₂²	1	1	yes	characteristic
11	C₄×#11924028	C₄	1	1	yes	characteristic
12	#17989003	C₄	1	16	yes	complemented
13	C₂×#14556531	C₂²	1	2	yes	—
14	C₂×#14556530	C₂²	1	4	yes	—
15	C₂×C₁₆×D₇₃	C₂²	1	4	yes	—
16	#17988542	C₄	1	1	yes	characteristic
17	#17988542	C₂²	1	1	yes	characteristic
18	#17988541	C₄	1	2	yes	—
19	#17988541	C₂²	1	2	yes	—
20	C₄×#11924012	C₄	1	1	yes	characteristic
21	C₄×#11924011	C₄	1	2	yes	—
22	C₄×C₈×D₇₃	C₂²	1	2	yes	—
23	#17988363	C₂²	1	2	yes	—
24	#17988362	C₂²	1	4	yes	—
25	C₄×#11924003	C₂²	1	2	yes	—
26	C₁₆×Dic₇₃	C₂²	1	4	yes	—
27	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	#346×C₇₃	C₂²	1	1	yes	characteristic
30	#346×C₇₃	C₂²	1	1	yes	characteristic
31	#346×C₇₃	C₂²	1	2	yes	—
32	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
33	#14556597	C₈	1	8	yes	—
34	#14556597	C₂×C₄	1	8	yes	—
35	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924011	C₂×C₄	1	2	yes	—
38	C₂×#11924011	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₃	C₂³	1	2	yes	—
41	C₄×Dic₁₄₆	C₈	1	2	yes	—
42	C₄×D₂₉₂	C₈	1	2	yes	—
43	C₄×C₂²⋊D₇₃	C₈	1	4	yes	—
44	#14556550	C₂×C₄	1	1	yes	characteristic
45	#14556548	C₂×C₄	1	2	yes	—
46	C₄²×D₇₃	C₂×C₄	1	2	yes	—
47	#14556531	C₂×C₄	1	8	yes	complemented
48	#14556530	C₂×C₄	1	16	yes	complemented
49	C₂×#11924003	C₂³	1	4	yes	—
50	C₁₆×D₇₃	C₂×C₄	1	16	yes	complemented
51	#14556502	C₂×C₄	1	2	yes	—
52	#14556501	C₂×C₄	1	2	yes	—
53	#14556501	C₂×C₄	1	2	yes	—
54	C₄×#10596831	C₂×C₄	1	2	yes	—
55	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₇₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₇₃	C₂×C₄	1	8	yes	complemented
61	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
62	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
63	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	#11924028	C₄²	1	4	yes	—
68	#11924028	C₂×C₈	1	4	yes	—
69	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
70	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
71	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
73	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
74	#11924012	C₄²	1	4	yes	—
75	#11924012	C₂×C₈	1	4	yes	—
76	#11924011	C₄²	1	8	yes	—
77	#11924011	C₂×C₈	1	8	yes	—
78	C₂×#10596831	C₂²×C₄	1	2	yes	—
79	C₂×#10596831	C₂²×C₄	1	2	yes	—
80	C₈×D₇₃	C₂×C₈	1	8	yes	—
81	C₈×D₇₃	C₂²×C₄	1	8	yes	—
82	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
83	#11924007	C₂×C₈	1	4	yes	—
84	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
85	#11924005	C₂×C₈	1	4	yes	—
86	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
87	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
88	#11924003	C₂²×C₄	1	8	yes	—
89	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
92	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×C₈	1	2	yes	—
98	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
101	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
102	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇₃	—	4	2	no	—
105	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₇₃	C₄×C₈	1	8	yes	—
112	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
113	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
114	D₂₉₂	C₄×C₈	1	4	yes	—
115	C₄×D₇₃	—	4	4	no	—
116	C₄×D₇₃	C₂×C₄²	1	4	yes	—
117	C₄×D₇₃	C₂²×C₈	1	4	yes	—
118	Dic₁₄₆	C₄×C₈	1	4	yes	—
119	C₈×C₇₃	C₂×C₄²	1	2	yes	—
120	C₈×C₇₃	C₂²×C₈	1	2	yes	—
121	C₈×C₇₃	C₂²×C₈	1	2	yes	—
122	C₈×C₇₃	C₂³×C₄	1	2	yes	—
123	#10596831	C₂×C₄²	1	4	yes	—
124	#10596831	C₂²×C₈	1	4	yes	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
127	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
128	C₂×D₇₃	—	2	4	no	—
129	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₇₃	—	2	2	no	—
131	C₄×C₇₃	—	2	2	no	—
132	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
135	Dic₇₃	—	2	4	no	—
136	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
137	#33326	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	—	2	2	no	—
139	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
140	C₂×C₇₃	#3407	1	2	yes	—
141	D₇₃	—	2	4	no	—
142	C₂×#504	—	73	2	no	—
143	C₄×#131	—	73	1	no	—
144	#2108	—	73	2	no	—
145	#2108	—	73	4	no	—
146	C₄×M₃₂	—	73	1	no	—
147	C₄×M₃₂	—	73	2	no	—
148	C₂×#346	—	73	2	no	—
149	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
150	C₇₃	#33326	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	73	1	no	—
152	C₂×#131	—	73	1	no	—
153	C₄×#41	—	73	1	no	—
154	#504	—	73	16	no	—
155	C₂×M₃₂	—	73	2	no	—
156	C₂×M₃₂	—	73	4	no	—
157	C₂²×C₁₆	—	73	4	no	—
158	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
159	#405	—	73	1	no	—
160	#405	—	73	1	no	—
161	#405	—	73	2	no	—
162	#405	—	73	2	no	—
163	C₄×M₁₆	—	73	1	no	—
164	C₄×M₁₆	—	73	2	no	—
165	C₂×C₄×C₈	—	73	2	no	—
166	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
167	#346	—	73	4	no	—
168	#346	—	73	2	no	—
169	#346	C₂×D₇₃	1	1	yes	characteristic
170	#346	C₂×D₇₃	1	1	yes	characteristic
171	#346	C₂×D₇₃	1	2	yes	—
172	C₄×C₁₆	—	73	4	no	—
173	C₄×C₁₆	—	73	2	no	—
174	C₂×#41	—	73	1	no	—
175	#131	—	73	8	no	—
176	#131	—	73	8	no	—
177	C₂×M₁₆	—	73	1	no	—
178	C₂×M₁₆	—	73	1	no	—
179	C₂×M₁₆	—	73	2	no	—
180	C₂×M₁₆	—	73	2	no	—
181	C₂²×C₈	—	73	2	no	—
182	C₂²×C₈	—	73	2	no	—
183	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
185	C₄×Q₈	—	73	2	no	—
186	C₄×D₄	—	73	4	no	—
187	C₄×D₄	—	73	2	no	—
188	#117	—	73	1	no	—
189	#117	—	73	2	no	—
190	C₂×C₄²	—	73	2	no	—
191	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
192	M₃₂	—	73	8	no	—
193	M₃₂	—	73	16	no	—
194	C₂×C₁₆	—	73	4	no	—
195	C₂×C₁₆	—	73	16	no	—
196	C₂×C₁₆	C₄×D₇₃	1	8	yes	complemented
197	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
198	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
199	#97	—	73	2	no	—
200	#97	—	73	2	no	—
201	#97	—	73	2	no	—
202	C₄×C₈	—	73	2	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
209	#41	—	73	4	no	—
210	#41	—	73	4	no	—
211	C₂×Q₈	—	73	1	no	—
212	C₂×D₄	—	73	2	no	—
213	C₂×D₄	—	73	1	no	—
214	C₂²×C₄	—	73	4	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	C₈×D₇₃	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
218	M₁₆	—	73	4	no	—
219	M₁₆	—	73	4	no	—
220	M₁₆	—	73	8	no	—
221	M₁₆	—	73	8	no	—
222	C₂×C₈	—	73	2	no	—
223	C₂×C₈	—	73	2	no	—
224	C₂×C₈	—	73	8	no	—
225	C₂×C₈	—	73	8	no	—
226	C₂×C₈	C₈×D₇₃	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
229	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
232	C₄⋊C₄	—	73	4	no	—
233	C₄⋊C₄	—	73	2	no	—
234	C₂²⋊C₄	—	73	2	no	—
235	C₂²⋊C₄	—	73	4	no	—
236	C₄²	—	73	4	no	—
237	C₄²	—	73	2	no	—
238	C₄²	C₈×D₇₃	1	2	yes	—
239	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
241	C₁₆	—	73	8	no	—
242	C₁₆	C₂×C₄×D₇₃	1	4	yes	—
243	C₁₆	C₂×C₄×D₇₃	1	4	yes	—
244	C₂³	—	73	2	no	—
245	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
246	Q₈	—	73	4	no	—
247	D₄	—	73	8	no	—
248	D₄	—	73	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	292	4	no	—
251	C₂×C₄	—	73	2	no	—
252	C₂×C₄	—	73	4	no	—
253	C₂×C₄	—	73	4	no	—
254	C₂×C₄	—	73	4	no	—
255	C₂×C₄	C₄²×D₇₃	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
261	C₈	—	73	4	no	—
262	C₈	—	73	4	no	—
263	C₈	C₄²×D₇₃	1	2	yes	—
264	C₈	C₂×C₈×D₇₃	1	2	yes	—
265	C₈	C₂×C₈×D₇₃	1	2	yes	—
266	C₈	C₂²×C₄×D₇₃	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	73	4	no	—
269	C₂²	—	146	4	no	—
270	C₂²	C₄×C₈×D₇₃	1	2	yes	—
271	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	73	4	no	—
275	C₄	—	146	4	no	—
276	C₄	C₄×C₈×D₇₃	1	2	yes	—
277	C₄	C₂×C₄²×D₇₃	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	146	4	no	—
281	C₂	C₂×C₄×C₈×D₇₃	1	1	yes	characteristic
282	C₂	#22623033	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator	C₂×C₇₃	C₂²×C₄×C₈
Fitting	C₂×#346×C₇₃	C₂
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#33326
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #22611529	C₂	semidirect
4. #22611527	C₂	semidirect
5. C₄×#14556531	C₂	semidirect
6. #22611524	C₂	semidirect
7. #346×D₇₃	C₂	semidirect
8. C₂×#346×C₇₃	C₂	semidirect
12. #17989003	C₄	semidirect
47. #14556531	C₂×C₄	semidirect
48. #14556530	C₂×C₄	semidirect
50. C₁₆×D₇₃	C₂×C₄	semidirect
60. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
149. C₂×#346	D₇₃	semidirect
150. C₇₃	#33326	semidirect
196. C₂×C₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989003	2
C₄×#14556597	2
#22611529	2
#22611527	2
C₄×#14556531	2
#22611524	2
#346×D₇₃	2
C₂×#346×C₇₃	2
#33326	73