Subgroups — #33955529

Order 18688.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989392	C₂	1	4	yes	maximal, complemented
2	#22623709	C₂	1	2	yes	maximal, complemented
3	#22620417	C₂	1	12	yes	maximal, complemented
4	#22620383	C₂	1	6	yes	maximal, complemented
5	#22620359	C₂	1	3	yes	maximal, complemented
6	C₂×#17988964	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556637	C₂²	1	4	yes	—
9	#17989392	—	2	32	no	—
10	C₂×#14556619	C₂²	1	24	yes	complemented
11	C₂×#14556613	C₂²	1	12	yes	complemented
12	C₂²×D₅₈₄	C₂²	1	12	yes	complemented
13	#17989259	—	2	4	no	—
14	D₄×D₂₉₂	—	2	12	no	—
15	#17989156	C₂²	1	12	yes	complemented
16	#17989137	C₂²	1	6	yes	complemented
17	C₂×#14556588	C₂²	1	6	yes	complemented
18	#17988976	C₂²	1	24	yes	complemented
19	#17988971	C₂²	1	12	yes	complemented
20	#17988967	C₂²	1	1	yes	characteristic, complemented
21	#17988964	C₂²	1	12	yes	complemented
22	#17988964	C₂²	1	3	yes	complemented
23	#17988964	C₂²	1	6	yes	complemented
24	#17988960	C₂²	1	6	yes	complemented
25	C₂×#131×C₇₃	C₂²	1	2	yes	—
26	C₄×#41×C₇₃	C₂²	1	1	yes	characteristic
27	#405×C₇₃	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₇₃	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	complemented
30	#14556637	—	2	32	no	—
31	C₂×#11924029	—	2	4	no	—
32	C₂×#11924028	C₂³	1	12	yes	—
33	C₂²×D₂₉₂	—	2	12	no	—
34	C₂²×D₂₉₂	C₂³	1	12	yes	—
35	C₂×D₄×D₇₃	—	2	12	no	—
36	#14556619	—	2	96	no	—
37	#14556613	—	2	48	no	—
38	C₂×Dic₂₉₂	C₂³	1	4	yes	—
39	C₂×#11924017	C₂³	1	24	yes	—
40	C₂×D₅₈₄	—	2	48	no	—
41	C₂×D₅₈₄	C₂³	1	24	yes	—
42	C₂×D₅₈₄	C₂³	1	12	yes	—
43	#14556591	C₂³	1	2	yes	—
44	#14556588	—	2	12	no	—
45	#14556588	C₂³	1	6	yes	—
46	#14556588	C₂³	1	12	yes	—
47	#14556579	C₂³	1	12	yes	—
48	#14556566	—	2	24	no	—
49	#14556565	—	2	12	no	—
50	#14556561	—	2	24	no	—
51	C₄×D₂₉₂	—	2	12	no	—
52	C₂×#41×C₇₃	C₂³	1	1	yes	characteristic
53	#131×C₇₃	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₇₃	C₂³	1	6	yes	—
55	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
56	C₄×Q₈×C₇₃	D₄	1	2	yes	—
57	C₄×D₄×C₇₃	D₄	1	6	yes	—
58	#117×C₇₃	C₂³	1	3	yes	—
59	C₂×C₄²×C₇₃	C₂³	1	3	yes	—
60	#97×C₇₃	C₂³	1	6	yes	—
61	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₇₃	C₂³	1	3	yes	—
63	C₄×C₈×C₇₃	C₂³	1	6	yes	—
64	C₂³×D₇₃	—	2	12	no	—
65	#11924029	—	4	16	no	—
66	#11924028	—	2	48	no	—
67	C₂×Dic₁₄₆	C₂⁴	1	4	yes	—
68	C₂×D₂₉₂	—	2	48	no	—
69	C₂×D₂₉₂	—	2	12	no	—
70	C₂×D₂₉₂	—	2	12	no	—
71	C₂×D₂₉₂	—	2	24	no	—
72	C₂×D₂₉₂	—	8	24	no	—
73	C₂×D₂₉₂	C₂⁴	1	12	yes	—
74	C₂×C₂²⋊D₇₃	—	2	12	no	—
75	D₄×D₇₃	—	4	48	no	—
76	C₂×C₄×D₇₃	—	2	12	no	—
77	Dic₂₉₂	—	2	8	no	—
78	#11924017	—	2	48	no	—
79	D₅₈₄	—	2	24	no	—
80	D₅₈₄	—	2	48	no	—
81	C₄⋊Dic₇₃	—	2	4	no	—
82	#11924005	—	2	24	no	—
83	#41×C₇₃	C₂×D₄	1	8	yes	—
84	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂×D₄×C₇₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
88	M₁₆×C₇₃	C₂×D₄	1	24	yes	complemented
89	C₂×C₈×C₇₃	C₂×D₄	1	24	yes	complemented
90	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
91	C₂×C₈×C₇₃	C₂⁴	1	6	yes	—
92	C₄⋊C₄×C₇₃	C₂×D₄	1	6	yes	—
93	C₂²⋊C₄×C₇₃	C₂×D₄	1	6	yes	—
94	C₄²×C₇₃	C₂×D₄	1	6	yes	—
95	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
96	C₄²×C₇₃	C₂⁴	1	3	yes	—
97	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
98	C₂²×D₇₃	—	2	12	no	—
99	C₂²×D₇₃	—	4	24	no	—
100	C₂²×D₇₃	—	8	24	no	—
101	Q₈×C₇₃	#127	1	4	yes	—
102	D₄×C₇₃	#127	1	12	yes	—
103	C₂×C₄×C₇₃	—	2	12	no	—
104	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₇₃	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₇₃	—	4	24	no	—
110	C₂×Dic₇₃	—	2	4	no	—
111	D₂₉₂	—	2	24	no	—
112	D₂₉₂	—	4	24	no	—
113	D₂₉₂	—	4	24	no	—
114	D₂₉₂	—	8	24	no	—
115	C₄×D₇₃	—	4	24	no	—
116	Dic₁₄₆	—	2	8	no	—
117	C₈×C₇₃	C₂²×D₄	1	4	yes	—
118	C₈×C₇₃	C₂²×D₄	1	12	yes	—
119	C₂²×C₇₃	—	2	6	no	—
120	C₂²×C₇₃	C₂×#127	1	6	yes	—
121	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
122	C₂×D₇₃	—	4	12	no	—
123	C₂×D₇₃	—	4	24	no	—
124	C₂×D₇₃	—	16	24	no	—
125	C₄×C₇₃	—	2	6	no	—
126	C₄×C₇₃	C₂×#127	1	6	yes	—
127	C₄×C₇₃	#576	1	4	yes	complemented
128	C₄×C₇₃	C₂³×D₄	1	2	yes	—
129	Dic₇₃	—	4	4	no	—
130	#60647	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	—	2	6	no	—
132	C₂×C₇₃	C₂²×#127	1	1	yes	characteristic
133	C₂×C₇₃	C₂×#576	1	2	yes	—
134	D₇₃	—	8	12	no	—
135	C₂×#576	—	73	4	no	—
136	#3498	—	73	2	no	—
137	#3155	—	73	12	no	—
138	#3151	—	73	6	no	—
139	#3147	—	73	3	no	—
140	C₂×#493	—	73	3	no	—
141	C₄×#131	D₇₃	1	1	yes	characteristic, complemented
142	C₇₃	#60647	1	1	yes	characteristic, complemented, Sylow
143	C₂×H(5,𝔽₂)	—	73	4	no	—
144	#576	—	146	32	no	—
145	C₂×AGL(1,ℤ/8)	—	73	24	no	—
146	C₂×#135	—	73	12	no	—
147	C₂²×D₈	—	73	12	no	—
148	C₂×#131	C₂×D₇₃	1	2	yes	—
149	#550	—	146	4	no	—
150	D₄²	—	146	12	no	—
151	#535	—	73	12	no	—
152	#532	—	73	6	no	—
153	C₂×#127	—	73	6	no	—
154	C₄×#41	C₂×D₇₃	1	1	yes	characteristic
155	#496	—	73	24	no	—
156	#495	—	73	12	no	—
157	#494	—	73	1	no	—
158	#493	—	73	3	no	—
159	#493	—	73	6	no	—
160	#493	—	73	12	no	—
161	#492	—	73	6	no	—
162	#405	C₂×D₇₃	1	6	yes	complemented
163	C₄×M₁₆	C₂×D₇₃	1	3	yes	complemented
164	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	complemented
165	H(5,𝔽₂)	—	146	32	no	—
166	C₂×#41	—	73	12	no	—
167	C₂×#41	—	146	4	no	—
168	C₂×#41	C₂²×D₇₃	1	1	yes	characteristic
169	C₂²×D₄	—	73	12	no	—
170	C₂²×D₄	—	146	12	no	—
171	C₂²×D₄	—	146	12	no	—
172	AGL(1,ℤ/8)	—	146	96	no	—
173	#135	—	146	48	no	—
174	C₂×Q₁₆	—	73	4	no	—
175	C₂×SD₁₆	—	73	24	no	—
176	C₂×D₈	—	73	12	no	—
177	C₂×D₈	—	73	24	no	—
178	C₂×D₈	—	146	48	no	—
179	#131	D₂₉₂	1	16	yes	complemented
180	C₂×M₁₆	C₂²×D₇₃	1	6	yes	—
181	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
182	C₄⋊Q₈	—	73	2	no	—
183	#127	—	73	6	no	—
184	#127	—	73	12	no	—
185	#127	—	146	12	no	—
186	#124	—	73	12	no	—
187	#121	—	146	12	no	—
188	#121	—	146	24	no	—
189	#120	—	146	24	no	—
190	C₄×Q₈	D₂₉₂	1	2	yes	—
191	C₄×D₄	—	146	12	no	—
192	C₄×D₄	D₂₉₂	1	6	yes	—
193	#117	C₂²×D₇₃	1	3	yes	—
194	C₂×C₄²	C₂²×D₇₃	1	3	yes	—
195	#97	C₂²×D₇₃	1	6	yes	—
196	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₇₃	1	3	yes	—
198	C₄×C₈	C₂²×D₇₃	1	6	yes	—
199	C₂⁴	—	146	12	no	—
200	#41	—	292	16	no	—
201	#41	—	146	48	no	—
202	#41	C₂×D₂₉₂	1	8	yes	—
203	C₂×Q₈	—	73	4	no	—
204	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
205	C₂×D₄	—	292	48	no	—
206	C₂×D₄	—	584	24	no	—
207	C₂×D₄	—	73	12	no	—
208	C₂×D₄	—	146	12	no	—
209	C₂×D₄	—	146	12	no	—
210	C₂×D₄	—	146	12	no	—
211	C₂×D₄	—	146	48	no	—
212	C₂×D₄	—	146	24	no	—
213	C₂×D₄	C₂×D₂₉₂	1	3	yes	—
214	C₂²×C₄	—	146	12	no	—
215	C₂²×C₄	C₂×D₂₉₂	1	6	yes	—
216	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
217	Q₁₆	—	146	8	no	—
218	SD₁₆	—	146	48	no	—
219	D₈	—	146	24	no	—
220	D₈	—	146	48	no	—
221	M₁₆	C₂×D₂₉₂	1	24	yes	complemented
222	C₂×C₈	C₂×D₂₉₂	1	24	yes	complemented
223	C₂×C₈	C₂³×D₇₃	1	2	yes	—
224	C₂×C₈	C₂³×D₇₃	1	6	yes	—
225	C₄⋊C₄	—	146	4	no	—
226	C₄⋊C₄	C₂×D₂₉₂	1	6	yes	—
227	C₂²⋊C₄	—	146	24	no	—
228	C₂²⋊C₄	C₂×D₂₉₂	1	6	yes	—
229	C₄²	C₂×D₂₉₂	1	6	yes	—
230	C₄²	C₂³×D₇₃	1	1	yes	characteristic
231	C₄²	C₂³×D₇₃	1	3	yes	—
232	C₂³	—	292	24	no	—
233	C₂³	—	584	24	no	—
234	C₂³	—	146	12	no	—
235	C₂³	C₂²×D₂₉₂	1	3	yes	—
236	Q₈	—	146	8	no	—
237	Q₈	#14556588	1	4	yes	—
238	D₄	—	292	24	no	—
239	D₄	—	292	24	no	—
240	D₄	—	292	24	no	—
241	D₄	—	584	24	no	—
242	D₄	—	146	24	no	—
243	D₄	#14556588	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	292	24	no	—
246	C₂×C₄	—	146	4	no	—
247	C₂×C₄	C₂²×D₂₉₂	1	6	yes	—
248	C₂×C₄	C₂²×D₂₉₂	1	3	yes	—
249	C₂×C₄	C₂²×D₂₉₂	1	12	yes	—
250	C₂×C₄	C₂²×D₂₉₂	1	2	yes	—
251	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
252	C₈	C₂²×D₂₉₂	1	4	yes	—
253	C₈	C₂²×D₂₉₂	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	292	24	no	—
256	C₂²	—	292	12	no	—
257	C₂²	—	1168	24	no	—
258	C₂²	C₂×#14556588	1	6	yes	—
259	C₂²	C₂³×D₂₉₂	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	292	4	no	—
262	C₄	C₂×#14556588	1	6	yes	—
263	C₄	#17989392	1	4	yes	complemented
264	C₄	C₂³×D₂₉₂	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	584	12	no	—
267	C₂	C₂²×#14556588	1	1	yes	characteristic
268	C₂	C₂×#17989392	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₉₂
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₄×#131×C₇₃	C₂
socle	C₂²×C₇₃	C₂³×D₄
hypercenter	C₄×#131	D₇₃
hypocenter	C₇₃	#60647
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#17989392	C₂	semidirect
2. #22623709	C₂	semidirect
3. #22620417	C₂	semidirect
4. #22620383	C₂	semidirect
5. #22620359	C₂	semidirect
6. C₂×#17988964	C₂	semidirect
7. C₄×#131×C₇₃	C₂	semidirect
10. C₂×#14556619	C₂²	semidirect
11. C₂×#14556613	C₂²	semidirect
12. C₂²×D₅₈₄	C₂²	semidirect
15. #17989156	C₂²	semidirect
16. #17989137	C₂²	semidirect
17. C₂×#14556588	C₂²	semidirect
18. #17988976	C₂²	semidirect
19. #17988971	C₂²	semidirect
20. #17988967	C₂²	semidirect
21. #17988964	C₂²	semidirect
22. #17988964	C₂²	semidirect
23. #17988964	C₂²	semidirect
24. #17988960	C₂²	semidirect
27. #405×C₇₃	C₂²	semidirect
28. C₄×M₁₆×C₇₃	C₂²	semidirect
29. C₂×C₄×C₈×C₇₃	C₂²	semidirect
53. #131×C₇₃	D₄	semidirect
88. M₁₆×C₇₃	C₂×D₄	semidirect
89. C₂×C₈×C₇₃	C₂×D₄	semidirect
127. C₄×C₇₃	#576	semidirect
141. C₄×#131	D₇₃	semidirect
142. C₇₃	#60647	semidirect
162. #405	C₂×D₇₃	semidirect
163. C₄×M₁₆	C₂×D₇₃	semidirect
164. C₂×C₄×C₈	C₂×D₇₃	semidirect
179. #131	D₂₉₂	semidirect
221. M₁₆	C₂×D₂₉₂	semidirect
222. C₂×C₈	C₂×D₂₉₂	semidirect
263. C₄	#17989392	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989392	2
#22623709	2
#22620417	2
#22620383	2
#22620359	2
C₂×#17988964	2
C₄×#131×C₇₃	2
#60647	73