Subgroups — #33950393

Order 18688.

1810 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623336	C₂	1	1	yes	maximal, characteristic
2	#22618875	C₂	1	16	yes	maximal, complemented
3	#22618871	C₂	1	4	yes	maximal, complemented
4	#22618865	C₂	1	4	yes	maximal, complemented
5	#22618495	C₂	1	1	yes	maximal, characteristic
6	C₂×#17988613	C₂	1	2	yes	maximal
7	#22618349	C₂	1	2	yes	maximal
8	C₂×#405×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11924012	C₂²	1	2	yes	—
10	#17989218	C₄	1	4	yes	—
11	#17989215	C₄	1	8	yes	—
12	#17989213	C₄	1	4	yes	—
13	#17989144	C₂²	1	4	yes	—
14	#17989140	C₂²	1	8	yes	—
15	C₂×#14556550	C₂²	1	2	yes	—
16	#17988717	C₂²	1	16	yes	complemented
17	#17988715	C₂²	1	32	yes	complemented
18	C₈×Dic₁₄₆	C₂²	1	16	yes	complemented
19	#17988614	C₂²	1	8	yes	—
20	#17988613	C₂²	1	4	yes	—
21	#17988613	C₂²	1	4	yes	—
22	#17988613	C₂²	1	8	yes	—
23	#17988611	C₂²	1	4	yes	—
24	C₂×#14556520	C₂²	1	8	yes	—
25	#17988545	C₂²	1	8	yes	—
26	#17988542	C₂²	1	8	yes	—
27	M₁₆×Dic₇₃	C₂²	1	8	yes	—
28	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
31	#405×C₇₃	C₂²	1	8	yes	—
32	C₂×#97×C₇₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
34	C₂×#11924012	—	2	8	no	—
35	C₂×#11924012	C₂³	1	4	yes	—
36	C₂×#11924012	C₂³	1	4	yes	—
37	C₂²×#10596831	C₂³	1	4	yes	—
38	#14556591	C₂×C₄	1	8	yes	—
39	#14556590	C₂×C₄	1	16	yes	—
40	#14556583	C₂×C₄	1	8	yes	—
41	#14556582	C₂×C₄	1	16	yes	—
42	#14556570	C₂×C₄	1	16	yes	—
43	#14556568	C₂×C₄	1	32	yes	—
44	C₄×Dic₁₄₆	C₂³	1	16	yes	—
45	#14556550	C₂×C₄	1	4	yes	—
46	#14556550	C₂×C₄	1	4	yes	—
47	#14556550	C₂³	1	4	yes	—
48	#14556550	C₂³	1	4	yes	—
49	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
52	C₄⋊#10596831	C₂³	1	16	yes	—
53	#14556520	C₂³	1	16	yes	—
54	#14556520	C₂³	1	16	yes	—
55	#14556501	C₂³	1	16	yes	—
56	C₈×Dic₇₃	C₂³	1	16	yes	—
57	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₇₃	Q₈	1	8	yes	complemented
59	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₇₃	—	2	4	no	—
61	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
65	#117×C₇₃	C₂×C₄	1	4	yes	—
66	#117×C₇₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
70	#97×C₇₃	C₂³	1	8	yes	—
71	C₄×C₈×C₇₃	C₂³	1	8	yes	—
72	C₂×Dic₁₄₆	C₂²×C₄	1	16	yes	—
73	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
74	#11924012	—	2	8	no	—
75	#11924012	—	2	8	no	—
76	C₂×#10596831	—	2	8	no	—
77	C₂×#10596831	C₂⁴	1	8	yes	—
78	C₄⋊Dic₇₃	C₂²×C₄	1	8	yes	—
79	C₄⋊Dic₇₃	C₂²×C₄	1	8	yes	—
80	C₄⋊Dic₇₃	C₂²×C₄	1	16	yes	—
81	#11924007	C₂²×C₄	1	32	yes	—
82	C₂²⋊Dic₇₃	C₂²×C₄	1	8	yes	—
83	C₂²⋊Dic₇₃	C₂²×C₄	1	8	yes	—
84	C₄×Dic₇₃	C₂²×C₄	1	8	yes	—
85	C₄×Dic₇₃	C₂⁴	1	8	yes	—
86	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₇₃	#41	1	4	yes	—
92	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
94	M₁₆×C₇₃	C₂×Q₈	1	16	yes	—
95	C₂×C₈×C₇₃	—	2	8	no	—
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	—	2	2	no	—
98	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
99	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
100	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
101	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
102	C₄⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
103	C₂²⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
105	C₄²×C₇₃	C₂⁴	1	4	yes	—
106	C₂³×C₇₃	—	4	2	no	—
107	C₂³×C₇₃	C₄×Q₈	1	4	yes	—
108	C₂³×C₇₃	C₂³×C₄	1	2	yes	—
109	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	—	4	4	no	—
111	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₇₃	C₄×Q₈	1	8	yes	—
113	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
119	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
121	C₂×Dic₇₃	—	4	8	no	—
122	C₂×Dic₇₃	C₂³×C₄	1	16	yes	—
123	Dic₁₄₆	—	4	16	no	—
124	C₈×C₇₃	—	2	2	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	C₂²×Q₈	1	8	yes	—
127	#10596831	—	2	8	no	—
128	C₂²×C₇₃	—	2	4	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	4	4	no	—
131	C₂²×C₇₃	C₂×C₄×Q₈	1	2	yes	—
132	C₂²×C₇₃	C₂×C₄×Q₈	1	4	yes	—
133	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₇₃	—	2	8	no	—
135	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₇₃	C₂×C₄×Q₈	1	4	yes	—
137	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
138	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
139	Dic₇₃	—	2	16	no	—
140	#60193	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	4	no	—
142	C₂×C₇₃	—	2	2	no	—
143	C₂×C₇₃	C₂²×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₇₃	#3573	1	2	yes	—
145	#3458	—	73	1	no	—
146	#2962	—	73	16	no	—
147	#2961	—	73	4	no	—
148	#2960	—	73	4	no	—
149	#2905	—	73	1	no	—
150	C₂×#424	—	73	2	no	—
151	#2873	—	73	2	no	—
152	C₂×#405	D₇₃	1	1	yes	characteristic
153	C₇₃	#60193	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	73	2	no	—
155	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
157	#544	—	73	8	no	—
158	#544	—	73	4	no	—
159	#543	—	73	4	no	—
160	#533	—	73	8	no	—
161	#533	—	73	4	no	—
162	C₂×#117	—	73	2	no	—
163	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
164	#446	—	73	32	no	—
165	#446	—	73	16	no	—
166	C₈×Q₈	—	73	16	no	—
167	#424	—	73	4	no	—
168	#424	—	73	4	no	—
169	#424	—	73	8	no	—
170	#424	—	73	8	no	—
171	#423	—	73	4	no	—
172	C₂×C₄⋊C₈	—	73	8	no	—
173	#405	—	73	8	no	—
174	#405	—	73	8	no	—
175	#405	C₂×D₇₃	1	8	yes	—
176	C₄×M₁₆	—	73	8	no	—
177	C₂×#97	C₂×D₇₃	1	2	yes	—
178	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
179	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
180	C₂×M₁₆	—	73	4	no	—
181	C₂×M₁₆	—	73	4	no	—
182	C₂×M₁₆	—	146	8	no	—
183	C₂×M₁₆	Dic₁₄₆	1	8	yes	complemented
184	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	73	4	no	—
187	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
190	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
191	C₄⋊Q₈	—	73	16	no	—
192	C₄⋊Q₈	—	73	8	no	—
193	#125	—	73	16	no	—
194	#125	—	73	8	no	—
195	C₂²⋊Q₈	—	73	16	no	—
196	C₂²⋊Q₈	—	73	32	no	—
197	C₄×Q₈	—	73	16	no	—
198	#117	—	73	4	no	—
199	#117	—	73	4	no	—
200	#117	—	73	4	no	—
201	#117	—	73	4	no	—
202	#117	C₄×D₇₃	1	4	yes	—
203	#117	C₂²×D₇₃	1	4	yes	—
204	C₂×C₄⋊C₄	—	73	4	no	—
205	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
206	C₂×C₂²⋊C₄	—	73	4	no	—
207	C₂×C₂²⋊C₄	C₄×D₇₃	1	2	yes	—
208	C₂×C₄²	—	73	4	no	—
209	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
210	C₄⋊C₈	—	73	16	no	—
211	C₄⋊C₈	—	73	16	no	—
212	C₄⋊C₈	—	73	16	no	—
213	#97	—	73	16	no	—
214	#97	C₂²×D₇₃	1	8	yes	—
215	C₄×C₈	—	73	16	no	—
216	C₄×C₈	C₂²×D₇₃	1	8	yes	—
217	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
218	C₂×Q₈	—	73	16	no	—
219	C₂²×C₄	—	73	8	no	—
220	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
221	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
224	C₂²×C₄	#11924028	1	4	yes	—
225	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
226	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
227	M₁₆	—	146	8	no	—
228	M₁₆	—	146	8	no	—
229	M₁₆	C₂×Dic₁₄₆	1	16	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	73	8	no	—
234	C₂×C₈	—	146	8	no	—
235	C₂×C₈	C₂×Dic₁₄₆	1	8	yes	—
236	C₂×C₈	C₂³×D₇₃	1	2	yes	—
237	C₂×C₈	C₂³×D₇₃	1	2	yes	—
238	C₂×C₈	C₂³×D₇₃	1	4	yes	—
239	C₄⋊C₄	—	73	32	no	—
240	C₄⋊C₄	—	73	8	no	—
241	C₄⋊C₄	—	73	8	no	—
242	C₄⋊C₄	—	73	16	no	—
243	C₄⋊C₄	C₂×C₄×D₇₃	1	8	yes	—
244	C₂²⋊C₄	—	73	8	no	—
245	C₂²⋊C₄	—	73	8	no	—
246	C₂²⋊C₄	C₂×C₄×D₇₃	1	8	yes	—
247	C₄²	—	73	8	no	—
248	C₄²	—	73	8	no	—
249	C₄²	C₂×C₄×D₇₃	1	4	yes	—
250	C₄²	C₂³×D₇₃	1	4	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄×Dic₁₄₆	1	4	yes	—
253	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
254	C₂³	C₂²×C₄×D₇₃	1	2	yes	—
255	Q₈	—	292	16	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	292	8	no	—
258	C₂×C₄	—	73	16	no	—
259	C₂×C₄	C₄×Dic₁₄₆	1	8	yes	—
260	C₂×C₄	C₄×Dic₁₄₆	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₇₃	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
265	C₂×C₄	C₂²×Dic₁₄₆	1	4	yes	—
266	C₂×C₄	C₂×#11924028	1	4	yes	—
267	C₂×C₄	C₂×#11924028	1	2	yes	—
268	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	—	146	8	no	—
272	C₈	C₂²×Dic₁₄₆	1	8	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×C₄×Dic₁₄₆	1	2	yes	—
277	C₂²	C₂×C₄×Dic₁₄₆	1	4	yes	—
278	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	146	16	no	—
281	C₄	C₂×C₄×Dic₁₄₆	1	4	yes	—
282	C₄	C₂×C₄×Dic₁₄₆	1	2	yes	—
283	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
284	C₄	C₂²×#11924028	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×C₄×Dic₁₄₆	1	1	yes	characteristic
288	C₂	#22624642	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇₃
commutator, socle	C₂²×C₇₃	C₂⁴×C₄
Fitting	C₂×#405×C₇₃	C₂
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂×#405	D₇₃
hypocenter	C₇₃	#60193
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22618875	C₂	semidirect
3. #22618871	C₂	semidirect
4. #22618865	C₂	semidirect
16. #17988717	C₂²	semidirect
17. #17988715	C₂²	semidirect
18. C₈×Dic₁₄₆	C₂²	semidirect
58. C₂×M₁₆×C₇₃	Q₈	semidirect
153. C₇₃	#60193	semidirect
183. C₂×M₁₆	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
#22623336	2
#22618875	2
#22618871	2
#22618865	2
#22618495	2
C₂×#17988613	2
#22618349	2
C₂×#405×C₇₃	2
#60193	73