Subgroups — C₂³×#14556617 #33999776

Order 18688.

13310 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924028	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556617	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11924016	C₂	1	2	yes	maximal, complemented
4	C₂³×#11924013	C₂	1	2	yes	maximal, complemented
5	C₂³×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11924028	—	2	14	no	—
8	C₂²×#11924028	C₂²	1	28	yes	complemented
9	C₂³×Dic₁₄₆	C₂²	1	1	yes	characteristic, complemented
10	C₂³×D₂₉₂	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₂²⋊D₇₃	—	2	1	no	—
12	C₂³×C₄×D₇₃	—	2	1	no	—
13	C₂×#14556617	C₂²	1	448	yes	direct product factor
14	C₂²×#11924016	C₂²	1	56	yes	complemented
15	C₂²×#11924013	C₂²	1	56	yes	complemented
16	C₂²×#11924012	C₂²	1	28	yes	complemented
17	C₂³×#10596831	C₂²	1	2	yes	complemented
18	C₂⁶×C₇₃	—	2	1	no	—
19	C₂³×D₄×C₇₃	—	2	1	no	—
20	C₂³×D₄×C₇₃	—	2	14	no	—
21	C₂³×D₄×C₇₃	C₂²	1	2	yes	—
22	C₂³×D₄×C₇₃	C₂²	1	28	yes	complemented
23	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
24	C₂⁴×D₇₃	—	2	1	no	—
25	C₂×#11924028	—	2	168	no	—
26	C₂×#11924028	C₂³	1	112	yes	complemented
27	C₂²×Dic₁₄₆	—	2	7	no	—
28	C₂²×Dic₁₄₆	C₂³	1	14	yes	complemented
29	C₂²×D₂₉₂	—	2	7	no	—
30	C₂²×D₂₉₂	C₂³	1	14	yes	complemented
31	C₂²×C₂²⋊D₇₃	—	2	28	no	—
32	C₂³×Dic₇₃	—	2	1	no	—
33	C₂²×C₄×D₇₃	—	2	28	no	—
34	#14556617	C₂³	1	512	yes	direct product factor
35	C₂×#11924016	C₂³	1	224	yes	complemented
36	C₂×#11924013	C₂³	1	224	yes	complemented
37	C₂×#11924012	C₂³	1	112	yes	complemented
38	C₂²×#10596831	C₂³	1	28	yes	complemented
39	C₂⁵×C₇₃	—	2	2	no	—
40	C₂⁵×C₇₃	—	2	28	no	—
41	C₂⁵×C₇₃	—	4	16	no	—
42	C₂⁵×C₇₃	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₇₃	—	2	14	no	—
44	C₂²×D₄×C₇₃	—	2	28	no	—
45	C₂²×D₄×C₇₃	—	2	168	no	—
46	C₂²×D₄×C₇₃	C₂³	1	28	yes	—
47	C₂²×D₄×C₇₃	C₂³	1	112	yes	complemented
48	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₇₃	D₄	1	14	yes	—
50	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
52	C₂³×D₇₃	—	2	14	no	—
53	C₂³×D₇₃	—	4	8	no	—
54	#11924028	—	2	224	no	—
55	#11924028	C₂⁴	1	64	yes	complemented
56	C₂×Dic₁₄₆	—	2	42	no	—
57	C₂×Dic₁₄₆	C₂⁴	1	28	yes	complemented
58	C₂×D₂₉₂	—	2	42	no	—
59	C₂×D₂₉₂	C₂⁴	1	28	yes	complemented
60	C₂×C₂²⋊D₇₃	—	2	112	no	—
61	C₂²×Dic₇₃	—	2	14	no	—
62	C₂×C₄×D₇₃	—	2	112	no	—
63	#11924016	C₂⁴	1	128	yes	complemented
64	#11924013	C₂⁴	1	128	yes	complemented
65	#11924012	C₂⁴	1	64	yes	complemented
66	C₂×#10596831	C₂⁴	1	56	yes	complemented
67	C₂⁴×C₇₃	—	2	8	no	—
68	C₂⁴×C₇₃	—	2	28	no	—
69	C₂⁴×C₇₃	—	2	112	no	—
70	C₂⁴×C₇₃	—	4	224	no	—
71	C₂⁴×C₇₃	—	4	16	no	—
72	C₂⁴×C₇₃	C₂×D₄	1	14	yes	—
73	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₇₃	—	2	84	no	—
75	C₂×D₄×C₇₃	—	2	112	no	—
76	C₂×D₄×C₇₃	—	2	224	no	—
77	C₂×D₄×C₇₃	C₂⁴	1	56	yes	—
78	C₂×D₄×C₇₃	C₂⁴	1	64	yes	complemented
79	C₂²×C₄×C₇₃	C₂×D₄	1	14	yes	—
80	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	84	yes	—
82	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
83	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
84	C₂³×C₇₃	—	2	56	no	—
85	C₂³×C₇₃	—	2	56	no	—
86	C₂³×C₇₃	—	2	64	no	—
87	C₂³×C₇₃	—	4	448	no	—
88	C₂³×C₇₃	—	4	112	no	—
89	C₂³×C₇₃	AGL(1,ℤ/8)	1	8	yes	complemented
90	C₂³×C₇₃	C₂²×D₄	1	28	yes	—
91	C₂³×C₇₃	C₂²×D₄	1	7	yes	—
92	C₂²×D₇₃	—	2	28	no	—
93	C₂²×D₇₃	—	4	56	no	—
94	D₄×C₇₃	—	2	56	no	—
95	D₄×C₇₃	—	2	64	no	—
96	D₄×C₇₃	C₂⁵	1	16	yes	—
97	C₂×C₄×C₇₃	C₂²×D₄	1	28	yes	—
98	C₂×C₄×C₇₃	C₂²×D₄	1	21	yes	—
99	C₂×C₄×C₇₃	C₂²×D₄	1	56	yes	—
100	C₂×C₄×C₇₃	C₂⁵	1	7	yes	—
101	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
102	C₂²⋊D₇₃	—	2	64	no	—
103	C₂×Dic₇₃	—	2	28	no	—
104	D₂₉₂	—	2	28	no	—
105	D₂₉₂	C₂⁵	1	8	yes	complemented
106	C₄×D₇₃	—	2	64	no	—
107	Dic₁₄₆	—	2	28	no	—
108	Dic₁₄₆	C₂⁵	1	8	yes	complemented
109	#10596831	C₂⁵	1	16	yes	complemented
110	C₂²×C₇₃	—	2	56	no	—
111	C₂²×C₇₃	—	2	16	no	—
112	C₂²×C₇₃	—	4	128	no	—
113	C₂²×C₇₃	—	4	112	no	—
114	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	28	yes	complemented
115	C₂²×C₇₃	C₂³×D₄	1	8	yes	—
116	C₂²×C₇₃	C₂³×D₄	1	7	yes	—
117	C₂×D₇₃	—	2	8	no	—
118	C₂×D₇₃	—	4	56	no	—
119	C₄×C₇₃	C₂³×D₄	1	7	yes	—
120	C₄×C₇₃	C₂³×D₄	1	8	yes	—
121	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
122	Dic₇₃	—	2	8	no	—
123	C₂³×AGL(1,ℤ/8)	—	73	1	no	maximal, Sylow
124	C₂×C₇₃	—	2	8	no	—
125	C₂×C₇₃	—	4	16	no	—
126	C₂×C₇₃	C₂²×AGL(1,ℤ/8)	1	14	yes	complemented
127	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
128	D₇₃	—	4	8	no	—
129	C₂³×#41	—	73	1	no	—
130	C₂⁴×D₄	D₇₃	1	1	yes	characteristic, complemented
131	C₂²×AGL(1,ℤ/8)	—	73	56	no	—
132	C₂³×SD₁₆	—	73	2	no	—
133	C₂³×D₈	—	73	2	no	—
134	C₂³×M₁₆	—	73	1	no	—
135	C₇₃	C₂³×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
136	C₂⁶	—	2	1	no	—
137	C₂²×#41	—	73	28	no	—
138	C₂²×#41	—	146	14	no	—
139	C₂³×Q₈	—	73	1	no	—
140	C₂³×D₄	—	2	14	no	—
141	C₂³×D₄	—	2	1	no	—
142	C₂³×D₄	—	73	1	no	—
143	C₂³×D₄	—	146	1	no	—
144	C₂³×D₄	C₂×D₇₃	1	2	yes	—
145	C₂³×D₄	C₂×D₇₃	1	28	yes	complemented
146	C₂⁴×C₄	—	146	1	no	—
147	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
148	C₂×AGL(1,ℤ/8)	—	73	448	no	—
149	C₂²×SD₁₆	—	73	56	no	—
150	C₂²×D₈	—	73	56	no	—
151	C₂²×M₁₆	—	73	28	no	—
152	C₂³×C₈	—	73	2	no	—
153	C₂⁵	—	2	28	no	—
154	C₂⁵	—	2	2	no	—
155	C₂⁵	—	4	16	no	—
156	C₂⁵	—	146	1	no	—
157	C₂⁵	C₂²⋊D₇₃	1	1	yes	characteristic
158	C₂×#41	—	73	112	no	—
159	C₂×#41	—	146	168	no	—
160	C₂²×Q₈	—	73	14	no	—
161	C₂²×Q₈	—	146	7	no	—
162	C₂²×D₄	—	2	14	no	—
163	C₂²×D₄	—	2	28	no	—
164	C₂²×D₄	—	2	168	no	—
165	C₂²×D₄	—	73	14	no	—
166	C₂²×D₄	—	146	28	no	—
167	C₂²×D₄	—	146	7	no	—
168	C₂²×D₄	C₂²×D₇₃	1	28	yes	—
169	C₂²×D₄	C₂²×D₇₃	1	112	yes	complemented
170	C₂³×C₄	—	146	1	no	—
171	C₂³×C₄	—	146	28	no	—
172	C₂³×C₄	C₂²⋊D₇₃	1	14	yes	—
173	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
175	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
176	AGL(1,ℤ/8)	—	73	512	no	—
177	C₂×SD₁₆	—	73	224	no	—
178	C₂×D₈	—	73	224	no	—
179	C₂×M₁₆	—	73	112	no	—
180	C₂²×C₈	—	73	28	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	28	no	—
183	C₂⁴	—	2	112	no	—
184	C₂⁴	—	4	224	no	—
185	C₂⁴	—	4	16	no	—
186	C₂⁴	—	292	8	no	—
187	C₂⁴	—	146	14	no	—
188	C₂⁴	C₂×C₂²⋊D₇₃	1	14	yes	—
189	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
190	#41	—	73	64	no	—
191	#41	—	146	224	no	—
192	C₂×Q₈	—	73	28	no	—
193	C₂×Q₈	—	146	42	no	—
194	C₂×D₄	—	2	84	no	—
195	C₂×D₄	—	2	112	no	—
196	C₂×D₄	—	2	224	no	—
197	C₂×D₄	—	73	28	no	—
198	C₂×D₄	—	146	112	no	—
199	C₂×D₄	—	146	42	no	—
200	C₂×D₄	C₂³×D₇₃	1	56	yes	—
201	C₂×D₄	C₂³×D₇₃	1	64	yes	complemented
202	C₂²×C₄	—	146	14	no	—
203	C₂²×C₄	—	146	112	no	—
204	C₂²×C₄	C₂×C₂²⋊D₇₃	1	14	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₇₃	1	7	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₇₃	1	84	yes	—
207	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
208	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
209	SD₁₆	—	73	128	no	—
210	D₈	—	73	128	no	—
211	M₁₆	—	73	64	no	—
212	C₂×C₈	—	73	56	no	—
213	C₂³	—	2	56	no	—
214	C₂³	—	2	56	no	—
215	C₂³	—	2	64	no	—
216	C₂³	—	4	448	no	—
217	C₂³	—	4	112	no	—
218	C₂³	—	292	56	no	—
219	C₂³	—	146	28	no	—
220	C₂³	#14556617	1	8	yes	direct product factor
221	C₂³	C₂²×C₂²⋊D₇₃	1	28	yes	—
222	C₂³	C₂²×C₂²⋊D₇₃	1	7	yes	—
223	Q₈	—	73	8	no	—
224	Q₈	—	146	28	no	—
225	D₄	—	2	56	no	—
226	D₄	—	2	64	no	—
227	D₄	—	73	8	no	—
228	D₄	—	146	64	no	—
229	D₄	—	146	28	no	—
230	D₄	C₂⁴×D₇₃	1	16	yes	—
231	C₂×C₄	—	146	28	no	—
232	C₂×C₄	—	146	64	no	—
233	C₂×C₄	C₂²×C₂²⋊D₇₃	1	28	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₇₃	1	21	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₇₃	1	56	yes	—
236	C₂×C₄	C₂⁴×D₇₃	1	7	yes	—
237	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
238	C₈	—	73	16	no	—
239	C₂²	—	2	56	no	—
240	C₂²	—	2	16	no	—
241	C₂²	—	4	128	no	—
242	C₂²	—	4	112	no	—
243	C₂²	—	292	56	no	—
244	C₂²	—	146	8	no	—
245	C₂²	C₂×#14556617	1	28	yes	direct product factor
246	C₂²	C₂³×C₂²⋊D₇₃	1	8	yes	—
247	C₂²	C₂³×C₂²⋊D₇₃	1	7	yes	—
248	C₄	—	146	8	no	—
249	C₄	C₂³×C₂²⋊D₇₃	1	8	yes	—
250	C₄	C₂³×C₂²⋊D₇₃	1	7	yes	—
251	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
252	C₂	—	2	8	no	—
253	C₂	—	4	16	no	—
254	C₂	—	292	8	no	—
255	C₂	C₂²×#14556617	1	14	yes	direct product factor
256	C₂	C₂⁴×C₂²⋊D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₃
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂⁴×D₄×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂⁴×D₄	D₇₃
hypocenter	C₇₃	C₂³×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11924028	C₂	semidirect
2. C₂²×#14556617	C₂	direct (N × H)
3. C₂³×#11924016	C₂	semidirect
4. C₂³×#11924013	C₂	semidirect
5. C₂³×#11924012	C₂	semidirect
6. C₂⁴×D₄×C₇₃	C₂	semidirect
8. C₂²×#11924028	C₂²	semidirect
9. C₂³×Dic₁₄₆	C₂²	semidirect
10. C₂³×D₂₉₂	C₂²	semidirect
13. C₂×#14556617	C₂²	direct (N × H)
14. C₂²×#11924016	C₂²	semidirect
15. C₂²×#11924013	C₂²	semidirect
16. C₂²×#11924012	C₂²	semidirect
17. C₂³×#10596831	C₂²	semidirect
22. C₂³×D₄×C₇₃	C₂²	semidirect
26. C₂×#11924028	C₂³	semidirect
28. C₂²×Dic₁₄₆	C₂³	semidirect
30. C₂²×D₂₉₂	C₂³	semidirect
34. #14556617	C₂³	direct (N × H)
35. C₂×#11924016	C₂³	semidirect
36. C₂×#11924013	C₂³	semidirect
37. C₂×#11924012	C₂³	semidirect
38. C₂²×#10596831	C₂³	semidirect
47. C₂²×D₄×C₇₃	C₂³	semidirect
55. #11924028	C₂⁴	semidirect
57. C₂×Dic₁₄₆	C₂⁴	semidirect
59. C₂×D₂₉₂	C₂⁴	semidirect
63. #11924016	C₂⁴	semidirect
64. #11924013	C₂⁴	semidirect
65. #11924012	C₂⁴	semidirect
66. C₂×#10596831	C₂⁴	semidirect
78. C₂×D₄×C₇₃	C₂⁴	semidirect
89. C₂³×C₇₃	AGL(1,ℤ/8)	semidirect
105. D₂₉₂	C₂⁵	semidirect
108. Dic₁₄₆	C₂⁵	semidirect
109. #10596831	C₂⁵	semidirect
114. C₂²×C₇₃	C₂×AGL(1,ℤ/8)	semidirect
126. C₂×C₇₃	C₂²×AGL(1,ℤ/8)	semidirect
130. C₂⁴×D₄	D₇₃	semidirect
135. C₇₃	C₂³×AGL(1,ℤ/8)	semidirect
145. C₂³×D₄	C₂×D₇₃	semidirect
169. C₂²×D₄	C₂²×D₇₃	semidirect
201. C₂×D₄	C₂³×D₇₃	semidirect
220. C₂³	#14556617	direct (N × H)
245. C₂²	C₂×#14556617	direct (N × H)
255. C₂	C₂²×#14556617	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11924028	2
C₂²×#14556617	2
C₂³×#11924016	2
C₂³×#11924013	2
C₂³×#11924012	2
C₂⁴×D₄×C₇₃	2
C₂³×AGL(1,ℤ/8)	73

Subgroups — C23×#14556617 #33999776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556617 #33999776