Subgroups — #33955626

Order 18688.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623336	C₂	1	2	yes	maximal
2	#22620470	C₂	1	8	yes	maximal, complemented
3	#22620462	C₂	1	8	yes	maximal, complemented
4	#22620455	C₂	1	4	yes	maximal, complemented
5	#22620445	C₂	1	4	yes	maximal, complemented
6	#22618521	C₂	1	2	yes	maximal
7	C₂×#17988625	C₂	1	2	yes	maximal
8	C₂×#405×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989218	—	2	4	no	—
10	#17989215	C₂²	1	16	yes	—
11	#17989213	—	2	4	no	—
12	#17989144	C₂²	1	8	yes	—
13	#17989140	—	2	8	no	—
14	C₂×#14556550	C₂²	1	4	yes	—
15	#17988996	C₂²	1	32	yes	complemented
16	#17988992	C₂²	1	8	yes	complemented
17	#17988989	C₂²	1	8	yes	complemented
18	#17988989	C₂²	1	8	yes	complemented
19	#17988985	C₂²	1	8	yes	complemented
20	#17988629	C₂²	1	16	yes	—
21	#17988625	C₂²	1	4	yes	—
22	#17988625	C₂²	1	8	yes	—
23	#17988625	C₂²	1	4	yes	—
24	#17988625	C₂²	1	4	yes	—
25	#17988625	C₂²	1	4	yes	—
26	C₂×#14556525	C₂²	1	4	yes	—
27	C₂×#14556523	C₂²	1	4	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	4	yes	—
32	#405×C₇₃	C₂²	1	4	yes	—
33	C₂×#97×C₇₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
35	#14556591	C₂³	1	16	yes	—
36	#14556590	—	2	16	no	—
37	#14556583	C₂³	1	16	yes	—
38	#14556582	—	2	16	no	—
39	#14556570	—	2	16	no	—
40	#14556568	—	2	32	no	—
41	C₄×Dic₁₄₆	—	2	16	no	—
42	#14556550	—	2	4	no	—
43	#14556550	—	2	4	no	—
44	#14556550	C₂³	1	8	yes	—
45	#14556550	C₂³	1	8	yes	—
46	C₂×C₄⋊Dic₇₃	C₂³	1	8	yes	—
47	C₂×C₂²⋊Dic₇₃	—	2	4	no	—
48	C₂×C₄×Dic₇₃	—	2	4	no	—
49	#14556525	C₂³	1	8	yes	—
50	#14556525	C₂³	1	8	yes	—
51	#14556525	C₂³	1	16	yes	—
52	#14556523	C₂³	1	8	yes	—
53	#14556523	C₂³	1	8	yes	—
54	#14556523	C₂³	1	16	yes	—
55	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₇₃	Q₈	1	8	yes	complemented
57	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
58	C₂²×C₈×C₇₃	Q₈	1	8	yes	complemented
59	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
64	#117×C₇₃	D₄	1	4	yes	—
65	#117×C₇₃	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
68	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
69	#97×C₇₃	C₂³	1	8	yes	—
70	C₄×C₈×C₇₃	C₂³	1	4	yes	—
71	C₄×C₈×C₇₃	C₂³	1	4	yes	—
72	C₂×Dic₁₄₆	—	2	16	no	—
73	C₂²×Dic₇₃	—	2	8	no	—
74	C₄⋊Dic₇₃	—	2	8	no	—
75	C₄⋊Dic₇₃	—	2	16	no	—
76	C₄⋊Dic₇₃	C₂⁴	1	16	yes	—
77	#11924007	—	2	32	no	—
78	C₂²⋊Dic₇₃	—	2	8	no	—
79	C₂²⋊Dic₇₃	—	2	8	no	—
80	C₄×Dic₇₃	—	2	8	no	—
81	C₄×Dic₇₃	—	2	8	no	—
82	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
90	M₁₆×C₇₃	C₂×Q₈	1	16	yes	—
91	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
92	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
93	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
94	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
95	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
96	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
99	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
100	C₄⋊C₄×C₇₃	C₂×D₄	1	8	yes	—
101	C₂²⋊C₄×C₇₃	C₂×D₄	1	8	yes	—
102	C₄²×C₇₃	C₂×D₄	1	4	yes	—
103	C₄²×C₇₃	C₂⁴	1	4	yes	—
104	C₂³×C₇₃	—	4	2	no	—
105	C₂³×C₇₃	C₄⋊Q₈	1	4	yes	—
106	C₂³×C₇₃	C₂²×D₄	1	2	yes	—
107	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	—	4	4	no	—
109	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₇₃	C₄⋊Q₈	1	8	yes	—
111	C₂×C₄×C₇₃	C₂²×D₄	1	8	yes	—
112	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
116	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	16	no	—
120	C₂×Dic₇₃	—	8	8	no	—
121	Dic₁₄₆	—	8	16	no	—
122	C₈×C₇₃	C₂²×Q₈	1	2	yes	—
123	C₈×C₇₃	C₂²×Q₈	1	2	yes	—
124	C₈×C₇₃	C₂²×Q₈	1	4	yes	—
125	C₈×C₇₃	C₂²×Q₈	1	8	yes	—
126	C₂²×C₇₃	—	2	4	no	—
127	C₂²×C₇₃	—	2	2	no	—
128	C₂²×C₇₃	—	4	4	no	—
129	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
130	C₂²×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
131	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₇₃	—	2	8	no	—
133	C₄×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
134	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
135	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
136	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
137	Dic₇₃	—	4	16	no	—
138	#60660	—	73	1	no	maximal, Sylow
139	C₂×C₇₃	—	2	4	no	—
140	C₂×C₇₃	—	2	2	no	—
141	C₂×C₇₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₇₃	#3584	1	2	yes	—
143	#3458	—	73	2	no	—
144	#3164	—	73	8	no	—
145	#3163	—	73	8	no	—
146	#3162	—	73	4	no	—
147	#3160	—	73	4	no	—
148	#2911	—	73	2	no	—
149	C₂×#427	—	73	2	no	—
150	C₂×#405	D₇₃	1	1	yes	characteristic
151	C₇₃	#60660	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
153	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
154	#544	—	73	16	no	—
155	#544	—	146	4	no	—
156	#543	—	146	4	no	—
157	#533	—	73	8	no	—
158	#533	—	146	8	no	—
159	C₂×#117	—	73	4	no	—
160	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
161	#501	—	73	32	no	—
162	#500	—	73	8	no	—
163	#499	—	73	8	no	—
164	#499	—	73	8	no	—
165	#498	—	73	8	no	—
166	#428	—	73	16	no	—
167	#427	—	73	4	no	—
168	#427	—	73	8	no	—
169	#427	—	73	4	no	—
170	#427	—	73	4	no	—
171	#427	—	73	4	no	—
172	C₂×#107	—	73	4	no	—
173	C₂×#106	—	73	4	no	—
174	#405	C₂×D₇₃	1	4	yes	—
175	#405	C₂×D₇₃	1	4	yes	—
176	C₂×#97	C₂×D₇₃	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
178	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
179	C₂×M₁₆	Dic₁₄₆	1	8	yes	complemented
180	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
181	C₂²×C₈	Dic₁₄₆	1	8	yes	complemented
182	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
186	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
187	C₄⋊Q₈	—	73	16	no	—
188	C₄⋊Q₈	—	146	16	no	—
189	#125	—	73	16	no	—
190	#125	—	146	16	no	—
191	C₂²⋊Q₈	—	146	16	no	—
192	C₂²⋊Q₈	—	146	32	no	—
193	C₄×Q₈	—	146	16	no	—
194	#117	—	73	8	no	—
195	#117	—	73	8	no	—
196	#117	—	146	4	no	—
197	#117	—	146	4	no	—
198	#117	D₂₉₂	1	4	yes	—
199	#117	C₂²×D₇₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	73	8	no	—
201	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
202	C₂×C₂²⋊C₄	—	146	4	no	—
203	C₂×C₂²⋊C₄	D₂₉₂	1	2	yes	—
204	C₂×C₄²	—	146	4	no	—
205	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
206	#107	—	73	8	no	—
207	#107	—	73	8	no	—
208	#107	—	73	16	no	—
209	#106	—	73	8	no	—
210	#106	—	73	8	no	—
211	#106	—	73	16	no	—
212	#97	C₂²×D₇₃	1	8	yes	—
213	C₄×C₈	C₂²×D₇₃	1	4	yes	—
214	C₄×C₈	C₂²×D₇₃	1	4	yes	—
215	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
216	C₂×Q₈	—	146	16	no	—
217	C₂²×C₄	—	146	8	no	—
218	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
219	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
222	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
223	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
224	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
225	M₁₆	C₂×Dic₁₄₆	1	16	yes	—
226	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
227	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
228	C₂×C₈	C₂×Dic₁₄₆	1	8	yes	—
229	C₂×C₈	C₂×Dic₁₄₆	1	8	yes	—
230	C₂×C₈	C₂×Dic₁₄₆	1	8	yes	—
231	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₇₃	1	4	yes	—
234	C₂×C₈	C₂³×D₇₃	1	2	yes	—
235	C₄⋊C₄	—	73	16	no	—
236	C₄⋊C₄	—	146	32	no	—
237	C₄⋊C₄	—	146	8	no	—
238	C₄⋊C₄	—	146	16	no	—
239	C₄⋊C₄	C₂×D₂₉₂	1	8	yes	—
240	C₂²⋊C₄	—	146	8	no	—
241	C₂²⋊C₄	—	146	8	no	—
242	C₂²⋊C₄	C₂×D₂₉₂	1	8	yes	—
243	C₄²	—	146	8	no	—
244	C₄²	—	146	8	no	—
245	C₄²	C₂×D₂₉₂	1	4	yes	—
246	C₄²	C₂³×D₇₃	1	4	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#14556591	1	4	yes	—
249	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
250	C₂³	C₂²×D₂₉₂	1	2	yes	—
251	Q₈	—	584	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	584	8	no	—
254	C₂×C₄	—	146	16	no	—
255	C₂×C₄	#14556591	1	4	yes	—
256	C₂×C₄	#14556591	1	8	yes	—
257	C₂×C₄	C₂²×D₂₉₂	1	8	yes	—
258	C₂×C₄	C₂²×D₂₉₂	1	2	yes	—
259	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₁₄₆	1	4	yes	—
261	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
262	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
263	C₂×C₄	C₂²×Dic₁₄₆	1	4	yes	—
264	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
265	C₈	C₂²×Dic₁₄₆	1	2	yes	—
266	C₈	C₂²×Dic₁₄₆	1	2	yes	—
267	C₈	C₂²×Dic₁₄₆	1	4	yes	—
268	C₈	C₂²×Dic₁₄₆	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#14556591	1	2	yes	—
273	C₂²	C₂×#14556591	1	4	yes	—
274	C₂²	C₂³×D₂₉₂	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	292	16	no	—
277	C₄	C₂×#14556591	1	4	yes	—
278	C₄	C₂×#14556591	1	2	yes	—
279	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
280	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#14556591	1	1	yes	characteristic
284	C₂	#22624705	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂×#405×C₇₃	C₂
socle	C₂²×C₇₃	C₂³×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂×#405	D₇₃
hypocenter	C₇₃	#60660
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22620470	C₂	semidirect
3. #22620462	C₂	semidirect
4. #22620455	C₂	semidirect
5. #22620445	C₂	semidirect
15. #17988996	C₂²	semidirect
16. #17988992	C₂²	semidirect
17. #17988989	C₂²	semidirect
18. #17988989	C₂²	semidirect
19. #17988985	C₂²	semidirect
56. C₂×M₁₆×C₇₃	Q₈	semidirect
58. C₂²×C₈×C₇₃	Q₈	semidirect
151. C₇₃	#60660	semidirect
179. C₂×M₁₆	Dic₁₄₆	semidirect
181. C₂²×C₈	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
#22620470	2
#22620462	2
#22620455	2
#22620445	2
#22618521	2
C₂×#17988625	2
C₂×#405×C₇₃	2
#60660	73