Subgroups — C₂²×#17989392 #33999796

Order 18688.

11462 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556637	C₂	1	2	yes	maximal, complemented
2	C₂×#17989392	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#14556619	C₂	1	6	yes	maximal, complemented
4	C₂²×#14556613	C₂	1	3	yes	maximal, complemented
5	C₂³×D₅₈₄	C₂	1	3	yes	maximal, complemented
6	C₂²×#131×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556637	—	2	24	no	—
8	C₂×#14556637	C₂²	1	48	yes	complemented
9	C₂²×#11924029	—	2	2	no	—
10	C₂²×#11924028	C₂²	1	6	yes	complemented
11	C₂³×D₂₉₂	C₂²	1	6	yes	complemented
12	C₂²×D₄×D₇₃	—	2	6	no	—
13	#17989392	C₂²	1	256	yes	direct product factor
14	C₂×#14556619	C₂²	1	144	yes	complemented
15	C₂×#14556613	C₂²	1	72	yes	complemented
16	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic, complemented
17	C₂²×#11924017	C₂²	1	6	yes	complemented
18	C₂²×D₅₈₄	C₂²	1	3	yes	complemented
19	C₂²×D₅₈₄	C₂²	1	6	yes	complemented
20	C₂²×D₅₈₄	C₂²	1	72	yes	complemented
21	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×#131×C₇₃	C₂²	1	24	yes	complemented
23	C₂²×M₁₆×C₇₃	C₂²	1	3	yes	complemented
24	C₂³×C₈×C₇₃	C₂²	1	3	yes	complemented
25	C₂⁴×D₇₃	—	2	6	no	—
26	#14556637	—	2	192	no	—
27	#14556637	C₂³	1	128	yes	complemented
28	C₂×#11924029	—	2	48	no	—
29	C₂×#11924028	—	2	36	no	—
30	C₂×#11924028	C₂³	1	72	yes	complemented
31	C₂²×Dic₁₄₆	C₂³	1	2	yes	—
32	C₂²×D₂₉₂	—	2	6	no	—
33	C₂²×D₂₉₂	—	2	36	no	—
34	C₂²×D₂₉₂	C₂³	1	6	yes	—
35	C₂²×D₂₉₂	C₂³	1	72	yes	complemented
36	C₂²×C₂²⋊D₇₃	—	2	6	no	—
37	C₂×D₄×D₇₃	—	2	144	no	—
38	C₂²×C₄×D₇₃	—	2	6	no	—
39	#14556619	C₂³	1	384	yes	complemented
40	#14556613	C₂³	1	192	yes	complemented
41	C₂×Dic₂₉₂	C₂³	1	12	yes	complemented
42	C₂×#11924017	C₂³	1	72	yes	complemented
43	C₂×D₅₈₄	C₂³	1	72	yes	complemented
44	C₂×D₅₈₄	C₂³	1	36	yes	complemented
45	C₂×D₅₈₄	C₂³	1	192	yes	complemented
46	C₂×#41×C₇₃	D₄	1	12	yes	—
47	C₂×#41×C₇₃	C₂³	1	12	yes	—
48	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₇₃	D₄	1	3	yes	—
50	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
51	#131×C₇₃	C₂³	1	64	yes	complemented
52	C₂×M₁₆×C₇₃	C₂³	1	36	yes	complemented
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
55	C₂²×C₈×C₇₃	C₂³	1	36	yes	complemented
56	C₂³×D₇₃	—	2	6	no	—
57	C₂³×D₇₃	—	2	72	no	—
58	C₂³×D₇₃	—	8	24	no	—
59	#11924029	—	2	128	no	—
60	#11924028	—	2	144	no	—
61	#11924028	C₂⁴	1	96	yes	complemented
62	C₂×Dic₁₄₆	—	2	6	no	—
63	C₂×Dic₁₄₆	C₂⁴	1	12	yes	—
64	C₂×D₂₉₂	—	2	72	no	—
65	C₂×D₂₉₂	—	2	18	no	—
66	C₂×D₂₉₂	—	2	144	no	—
67	C₂×D₂₉₂	C₂⁴	1	36	yes	—
68	C₂×D₂₉₂	C₂⁴	1	96	yes	complemented
69	C₂×C₂²⋊D₇₃	—	2	72	no	—
70	D₄×D₇₃	—	2	384	no	—
71	C₂²×Dic₇₃	—	2	2	no	—
72	C₂×C₄×D₇₃	—	2	72	no	—
73	Dic₂₉₂	C₂⁴	1	16	yes	complemented
74	#11924017	C₂⁴	1	96	yes	complemented
75	D₅₈₄	C₂⁴	1	48	yes	complemented
76	D₅₈₄	C₂⁴	1	96	yes	complemented
77	C₂⁴×C₇₃	C₂×D₄	1	3	yes	—
78	#41×C₇₃	C₂×D₄	1	48	yes	—
79	#41×C₇₃	C₂⁴	1	16	yes	—
80	C₂×Q₈×C₇₃	C₂×D₄	1	12	yes	—
81	C₂×D₄×C₇₃	C₂×D₄	1	36	yes	—
82	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₇₃	C₂×D₄	1	18	yes	—
84	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂⁴	1	18	yes	—
86	M₁₆×C₇₃	C₂⁴	1	48	yes	complemented
87	C₂×C₈×C₇₃	C₂⁴	1	6	yes	—
88	C₂×C₈×C₇₃	C₂⁴	1	18	yes	—
89	C₂×C₈×C₇₃	C₂⁴	1	48	yes	complemented
90	C₂³×C₇₃	—	2	12	no	—
91	C₂³×C₇₃	C₂²×D₄	1	18	yes	—
92	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×D₇₃	—	2	36	no	—
94	C₂²×D₇₃	—	2	96	no	—
95	C₂²×D₇₃	—	4	24	no	—
96	C₂²×D₇₃	—	8	144	no	—
97	Q₈×C₇₃	C₂²×D₄	1	16	yes	—
98	D₄×C₇₃	C₂²×D₄	1	48	yes	—
99	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₇₃	C₂²×D₄	1	18	yes	—
101	C₂×C₄×C₇₃	C₂²×D₄	1	36	yes	—
102	C₂×C₄×C₇₃	C₂⁵	1	3	yes	—
103	C₂×C₄×C₇₃	C₂⁵	1	12	yes	—
104	C₂²⋊D₇₃	—	2	96	no	—
105	C₂×Dic₇₃	—	2	12	no	—
106	D₂₉₂	—	2	96	no	—
107	D₂₉₂	—	2	36	no	—
108	D₂₉₂	C₂⁵	1	24	yes	—
109	C₄×D₇₃	—	2	96	no	—
110	Dic₁₄₆	—	2	12	no	—
111	Dic₁₄₆	C₂⁵	1	8	yes	—
112	C₈×C₇₃	C₂⁵	1	4	yes	—
113	C₈×C₇₃	C₂⁵	1	12	yes	—
114	C₂²×C₇₃	—	2	36	no	—
115	C₂²×C₇₃	#576	1	4	yes	complemented
116	C₂²×C₇₃	C₂³×D₄	1	12	yes	—
117	C₂²×C₇₃	C₂³×D₄	1	3	yes	—
118	C₂×D₇₃	—	2	24	no	—
119	C₂×D₇₃	—	4	72	no	—
120	C₂×D₇₃	—	8	96	no	—
121	C₄×C₇₃	C₂³×D₄	1	3	yes	—
122	C₄×C₇₃	C₂³×D₄	1	12	yes	—
123	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
124	Dic₇₃	—	2	8	no	—
125	C₂²×#576	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	—	2	12	no	—
127	C₂×C₇₃	C₂×#576	1	6	yes	complemented
128	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
129	D₇₃	—	4	24	no	—
130	C₂²×H(5,𝔽₂)	—	73	2	no	—
131	C₂×#576	—	73	48	no	—
132	C₂²×AGL(1,ℤ/8)	—	73	6	no	—
133	C₂²×#135	—	73	3	no	—
134	C₂³×D₈	—	73	3	no	—
135	C₂²×#131	D₇₃	1	1	yes	characteristic, complemented
136	C₇₃	C₂²×#576	1	1	yes	characteristic, complemented, Sylow
137	C₂×H(5,𝔽₂)	—	73	48	no	—
138	C₂×H(5,𝔽₂)	—	146	24	no	—
139	C₂²×#41	—	73	6	no	—
140	C₂²×#41	—	146	2	no	—
141	C₂²×#41	C₂×D₇₃	1	1	yes	characteristic
142	C₂³×D₄	—	73	6	no	—
143	C₂³×D₄	—	146	6	no	—
144	#576	—	73	256	no	—
145	C₂×AGL(1,ℤ/8)	—	73	144	no	—
146	C₂×#135	—	73	72	no	—
147	C₂²×Q₁₆	—	73	1	no	—
148	C₂²×SD₁₆	—	73	6	no	—
149	C₂²×D₈	—	73	3	no	—
150	C₂²×D₈	—	73	6	no	—
151	C₂²×D₈	—	73	72	no	—
152	C₂×#131	C₂×D₇₃	1	24	yes	complemented
153	C₂²×M₁₆	C₂×D₇₃	1	3	yes	complemented
154	C₂³×C₈	C₂×D₇₃	1	3	yes	complemented
155	C₂⁵	—	146	6	no	—
156	H(5,𝔽₂)	—	73	128	no	—
157	H(5,𝔽₂)	—	146	192	no	—
158	C₂×#41	—	73	72	no	—
159	C₂×#41	—	146	48	no	—
160	C₂×#41	—	146	36	no	—
161	C₂×#41	D₂₉₂	1	12	yes	—
162	C₂×#41	C₂²×D₇₃	1	12	yes	—
163	C₂²×Q₈	—	73	2	no	—
164	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
165	C₂²×D₄	—	73	6	no	—
166	C₂²×D₄	—	73	72	no	—
167	C₂²×D₄	—	146	6	no	—
168	C₂²×D₄	—	146	144	no	—
169	C₂²×D₄	—	146	6	no	—
170	C₂²×D₄	—	146	36	no	—
171	C₂²×D₄	D₂₉₂	1	3	yes	—
172	C₂³×C₄	—	146	6	no	—
173	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
174	AGL(1,ℤ/8)	—	73	384	no	—
175	#135	—	73	192	no	—
176	C₂×Q₁₆	—	73	12	no	—
177	C₂×SD₁₆	—	73	72	no	—
178	C₂×D₈	—	73	36	no	—
179	C₂×D₈	—	73	72	no	—
180	C₂×D₈	—	73	192	no	—
181	#131	C₂²×D₇₃	1	64	yes	complemented
182	C₂×M₁₆	C₂²×D₇₃	1	36	yes	complemented
183	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
185	C₂²×C₈	C₂²×D₇₃	1	36	yes	complemented
186	C₂⁴	—	584	24	no	—
187	C₂⁴	—	146	6	no	—
188	C₂⁴	—	146	72	no	—
189	C₂⁴	C₂×D₂₉₂	1	3	yes	—
190	#41	—	73	96	no	—
191	#41	—	146	128	no	—
192	#41	—	146	144	no	—
193	#41	C₂×D₂₉₂	1	48	yes	—
194	#41	C₂³×D₇₃	1	16	yes	—
195	C₂×Q₈	—	73	12	no	—
196	C₂×Q₈	—	146	6	no	—
197	C₂×Q₈	C₂×D₂₉₂	1	12	yes	—
198	C₂×D₄	—	73	36	no	—
199	C₂×D₄	—	73	96	no	—
200	C₂×D₄	—	146	72	no	—
201	C₂×D₄	—	146	384	no	—
202	C₂×D₄	—	146	72	no	—
203	C₂×D₄	—	146	18	no	—
204	C₂×D₄	—	146	144	no	—
205	C₂×D₄	C₂×D₂₉₂	1	36	yes	—
206	C₂²×C₄	—	146	2	no	—
207	C₂²×C₄	—	146	72	no	—
208	C₂²×C₄	C₂×D₂₉₂	1	18	yes	—
209	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
210	C₂²×C₄	C₂³×D₇₃	1	18	yes	—
211	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
212	Q₁₆	—	73	16	no	—
213	SD₁₆	—	73	96	no	—
214	D₈	—	73	48	no	—
215	D₈	—	73	96	no	—
216	M₁₆	C₂³×D₇₃	1	48	yes	complemented
217	C₂×C₈	C₂³×D₇₃	1	6	yes	—
218	C₂×C₈	C₂³×D₇₃	1	18	yes	—
219	C₂×C₈	C₂³×D₇₃	1	48	yes	complemented
220	C₂³	—	2	12	no	—
221	C₂³	—	292	24	no	—
222	C₂³	—	584	144	no	—
223	C₂³	—	146	36	no	—
224	C₂³	—	146	96	no	—
225	C₂³	C₂²×D₂₉₂	1	18	yes	—
226	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
227	Q₈	—	73	8	no	—
228	Q₈	—	146	12	no	—
229	Q₈	C₂²×D₂₉₂	1	16	yes	—
230	D₄	—	73	24	no	—
231	D₄	—	146	96	no	—
232	D₄	—	146	96	no	—
233	D₄	—	146	36	no	—
234	D₄	C₂²×D₂₉₂	1	48	yes	—
235	C₂×C₄	—	146	12	no	—
236	C₂×C₄	—	146	96	no	—
237	C₂×C₄	C₂²×D₂₉₂	1	18	yes	—
238	C₂×C₄	C₂²×D₂₉₂	1	3	yes	—
239	C₂×C₄	C₂²×D₂₉₂	1	36	yes	—
240	C₂×C₄	C₂⁴×D₇₃	1	3	yes	—
241	C₂×C₄	C₂⁴×D₇₃	1	12	yes	—
242	C₈	C₂⁴×D₇₃	1	4	yes	—
243	C₈	C₂⁴×D₇₃	1	12	yes	—
244	C₂²	—	2	36	no	—
245	C₂²	—	292	72	no	—
246	C₂²	—	584	96	no	—
247	C₂²	—	146	24	no	—
248	C₂²	#17989392	1	4	yes	direct product factor
249	C₂²	C₂³×D₂₉₂	1	12	yes	—
250	C₂²	C₂³×D₂₉₂	1	3	yes	—
251	C₄	—	146	8	no	—
252	C₄	C₂³×D₂₉₂	1	12	yes	—
253	C₄	C₂³×D₂₉₂	1	3	yes	—
254	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
255	C₂	—	2	12	no	—
256	C₂	—	292	24	no	—
257	C₂	C₂×#17989392	1	6	yes	direct product factor
258	C₂	C₂⁴×D₂₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂²×#131×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
hypercenter	C₂²×#131	D₇₃
hypocenter	C₇₃	C₂²×#576
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556637	C₂	semidirect
2. C₂×#17989392	C₂	direct (N × H)
3. C₂²×#14556619	C₂	semidirect
4. C₂²×#14556613	C₂	semidirect
5. C₂³×D₅₈₄	C₂	semidirect
6. C₂²×#131×C₇₃	C₂	semidirect
8. C₂×#14556637	C₂²	semidirect
10. C₂²×#11924028	C₂²	semidirect
11. C₂³×D₂₉₂	C₂²	semidirect
13. #17989392	C₂²	direct (N × H)
14. C₂×#14556619	C₂²	semidirect
15. C₂×#14556613	C₂²	semidirect
16. C₂²×Dic₂₉₂	C₂²	semidirect
17. C₂²×#11924017	C₂²	semidirect
18. C₂²×D₅₈₄	C₂²	semidirect
19. C₂²×D₅₈₄	C₂²	semidirect
20. C₂²×D₅₈₄	C₂²	semidirect
22. C₂×#131×C₇₃	C₂²	semidirect
23. C₂²×M₁₆×C₇₃	C₂²	semidirect
24. C₂³×C₈×C₇₃	C₂²	semidirect
27. #14556637	C₂³	semidirect
30. C₂×#11924028	C₂³	semidirect
35. C₂²×D₂₉₂	C₂³	semidirect
39. #14556619	C₂³	semidirect
40. #14556613	C₂³	semidirect
41. C₂×Dic₂₉₂	C₂³	semidirect
42. C₂×#11924017	C₂³	semidirect
43. C₂×D₅₈₄	C₂³	semidirect
44. C₂×D₅₈₄	C₂³	semidirect
45. C₂×D₅₈₄	C₂³	semidirect
51. #131×C₇₃	C₂³	semidirect
52. C₂×M₁₆×C₇₃	C₂³	semidirect
55. C₂²×C₈×C₇₃	C₂³	semidirect
61. #11924028	C₂⁴	semidirect
68. C₂×D₂₉₂	C₂⁴	semidirect
73. Dic₂₉₂	C₂⁴	semidirect
74. #11924017	C₂⁴	semidirect
75. D₅₈₄	C₂⁴	semidirect
76. D₅₈₄	C₂⁴	semidirect
86. M₁₆×C₇₃	C₂⁴	semidirect
89. C₂×C₈×C₇₃	C₂⁴	semidirect
115. C₂²×C₇₃	#576	semidirect
127. C₂×C₇₃	C₂×#576	semidirect
135. C₂²×#131	D₇₃	semidirect
136. C₇₃	C₂²×#576	semidirect
152. C₂×#131	C₂×D₇₃	semidirect
153. C₂²×M₁₆	C₂×D₇₃	semidirect
154. C₂³×C₈	C₂×D₇₃	semidirect
181. #131	C₂²×D₇₃	semidirect
182. C₂×M₁₆	C₂²×D₇₃	semidirect
185. C₂²×C₈	C₂²×D₇₃	semidirect
216. M₁₆	C₂³×D₇₃	semidirect
219. C₂×C₈	C₂³×D₇₃	semidirect
248. C₂²	#17989392	direct (N × H)
257. C₂	C₂×#17989392	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556637	2
C₂×#17989392	2
C₂²×#14556619	2
C₂²×#14556613	2
C₂³×D₅₈₄	2
C₂²×#131×C₇₃	2
C₂²×#576	73

Subgroups — C22×#17989392 #33999796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989392 #33999796