Subgroups — C₂³×#14556519 #33949573

Order 18688.

5758 conjugacy classes of subgroups.

The subgroup lattice has 272 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
2	C₂³×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14556519	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#11924028	—	2	7	no	—
6	C₂²×#11924028	—	2	7	no	—
7	C₂²×#11924028	C₄	1	14	yes	complemented
8	C₂²×#11924028	C₂²	1	14	yes	—
9	C₂³×Dic₁₄₆	C₄	1	1	yes	characteristic, complemented
10	C₂³×D₂₉₂	C₄	1	1	yes	characteristic, complemented
11	C₂³×C₂²⋊D₇₃	—	2	1	no	—
12	C₂³×C₄×D₇₃	—	2	1	no	—
13	C₂²×#11924012	—	2	7	no	—
14	C₂²×#11924012	C₂²	1	14	yes	complemented
15	C₂³×#10596831	—	2	1	no	—
16	C₂×#14556519	C₂²	1	112	yes	direct product factor
17	C₂⁴×C₄×C₇₃	—	2	1	no	—
18	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₇₃	—	2	7	no	—
20	C₂²×C₄²×C₇₃	C₂²	1	14	yes	complemented
21	C₂⁴×D₇₃	—	2	1	no	—
22	C₂×#11924028	—	2	84	no	—
23	C₂×#11924028	—	2	42	no	—
24	C₂×#11924028	—	2	42	no	—
25	C₂×#11924028	C₂×C₄	1	84	yes	complemented
26	C₂×#11924028	C₂³	1	28	yes	—
27	C₂²×Dic₁₄₆	—	2	7	no	—
28	C₂²×Dic₁₄₆	C₂×C₄	1	14	yes	complemented
29	C₂²×D₂₉₂	—	2	7	no	—
30	C₂²×D₂₉₂	C₂×C₄	1	14	yes	complemented
31	C₂²×C₂²⋊D₇₃	—	2	14	no	—
32	C₂²×C₂²⋊D₇₃	—	2	14	no	—
33	C₂³×Dic₇₃	—	2	1	no	—
34	C₂²×C₄×D₇₃	—	2	14	no	—
35	C₂²×C₄×D₇₃	—	2	14	no	—
36	C₂×#11924012	—	2	42	no	—
37	C₂×#11924012	C₂³	1	28	yes	complemented
38	C₂²×#10596831	—	2	14	no	—
39	#14556519	C₂³	1	64	yes	direct product factor
40	C₂⁵×C₇₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₇₃	—	2	16	no	—
42	C₂³×C₄×C₇₃	—	2	14	no	—
43	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₇₃	C₂×C₄	1	7	yes	—
45	C₂³×C₄×C₇₃	D₄	1	7	yes	—
46	C₂³×C₄×C₇₃	D₄	1	7	yes	—
47	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₇₃	C₂³	1	7	yes	—
49	C₂×C₄²×C₇₃	—	2	42	no	—
50	C₂×C₄²×C₇₃	C₂³	1	28	yes	complemented
51	C₂³×D₇₃	—	2	14	no	—
52	C₂³×D₇₃	—	4	8	no	—
53	#11924028	—	2	168	no	—
54	#11924028	—	2	28	no	—
55	#11924028	—	2	28	no	—
56	#11924028	C₂²×C₄	1	56	yes	complemented
57	#11924028	C₂⁴	1	8	yes	—
58	C₂×Dic₁₄₆	—	2	42	no	—
59	C₂×Dic₁₄₆	C₂²×C₄	1	28	yes	complemented
60	C₂×D₂₉₂	—	2	42	no	—
61	C₂×D₂₉₂	C₂²×C₄	1	28	yes	complemented
62	C₂×C₂²⋊D₇₃	—	2	84	no	—
63	C₂×C₂²⋊D₇₃	—	2	28	no	—
64	C₂²×Dic₇₃	—	2	14	no	—
65	C₂×C₄×D₇₃	—	2	84	no	—
66	C₂×C₄×D₇₃	—	2	28	no	—
67	#11924012	—	2	28	no	—
68	#11924012	C₂⁴	1	8	yes	complemented
69	C₂×#10596831	—	2	28	no	—
70	C₂⁴×C₇₃	—	2	8	no	—
71	C₂⁴×C₇₃	C₂²⋊C₄	1	7	yes	—
72	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂⁴×C₇₃	C₂×D₄	1	7	yes	—
74	C₂²×C₄×C₇₃	—	2	112	no	—
75	C₂²×C₄×C₇₃	—	2	28	no	—
76	C₂²×C₄×C₇₃	C₂²⋊C₄	1	7	yes	—
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	7	yes	—
78	C₂²×C₄×C₇₃	C₂²⋊C₄	1	42	yes	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	7	yes	—
80	C₂²×C₄×C₇₃	C₂²×C₄	1	21	yes	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	21	yes	—
82	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
83	C₂²×C₄×C₇₃	C₂×D₄	1	21	yes	—
84	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
85	C₄²×C₇₃	—	2	28	no	—
86	C₄²×C₇₃	C₂⁴	1	8	yes	complemented
87	C₂³×C₇₃	—	2	28	no	—
88	C₂³×C₇₃	—	2	28	no	—
89	C₂³×C₇₃	#104	1	8	yes	complemented
90	C₂³×C₇₃	C₂×C₂²⋊C₄	1	21	yes	—
91	C₂³×C₇₃	C₂×C₂²⋊C₄	1	7	yes	—
92	C₂³×C₇₃	C₂²×D₄	1	7	yes	—
93	C₂²×D₇₃	—	2	28	no	—
94	C₂²×D₇₃	—	4	56	no	—
95	C₂×C₄×C₇₃	—	2	112	no	—
96	C₂×C₄×C₇₃	—	2	8	no	—
97	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	21	yes	—
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	21	yes	—
99	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	42	yes	—
100	C₂×C₄×C₇₃	C₂³×C₄	1	7	yes	—
101	C₂×C₄×C₇₃	C₂³×C₄	1	7	yes	—
102	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
103	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
104	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
105	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
106	C₂²⋊D₇₃	—	2	56	no	—
107	C₂²⋊D₇₃	—	2	8	no	—
108	C₂×Dic₇₃	—	2	28	no	—
109	D₂₉₂	—	2	28	no	—
110	D₂₉₂	C₂³×C₄	1	8	yes	complemented
111	C₄×D₇₃	—	2	56	no	—
112	C₄×D₇₃	—	2	8	no	—
113	Dic₁₄₆	—	2	28	no	—
114	Dic₁₄₆	C₂³×C₄	1	8	yes	complemented
115	#10596831	—	2	8	no	—
116	C₂²×C₇₃	—	2	14	no	—
117	C₂²×C₇₃	—	2	42	no	—
118	C₂²×C₇₃	C₂×#104	1	28	yes	complemented
119	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
120	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
121	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
122	C₂×D₇₃	—	2	8	no	—
123	C₂×D₇₃	—	4	56	no	—
124	C₄×C₇₃	—	2	16	no	—
125	C₄×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
126	C₄×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
127	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
128	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
129	Dic₇₃	—	2	8	no	—
130	C₂³×#104	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	—	2	1	no	—
132	C₂×C₇₃	—	2	7	no	—
133	C₂×C₇₃	C₂²×#104	1	14	yes	complemented
134	C₂×C₇₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
135	D₇₃	—	4	8	no	—
136	C₂³×#41	—	73	1	no	—
137	C₂³×M₁₆	—	73	1	no	—
138	C₂³×C₄²	D₇₃	1	1	yes	characteristic, complemented
139	C₂²×#104	—	73	28	no	—
140	C₇₃	C₂³×#104	1	1	yes	characteristic, complemented, Sylow
141	C₂²×#41	—	73	14	no	—
142	C₂²×#41	—	73	14	no	—
143	C₂²×#41	—	146	7	no	—
144	C₂²×#41	—	146	7	no	—
145	C₂³×Q₈	—	73	1	no	—
146	C₂³×D₄	—	73	1	no	—
147	C₂³×D₄	—	146	1	no	—
148	C₂⁴×C₄	—	2	1	no	—
149	C₂⁴×C₄	—	146	1	no	—
150	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
151	C₂²×M₁₆	—	73	14	no	—
152	C₂²×M₁₆	—	146	7	no	—
153	C₂³×C₈	—	146	1	no	—
154	C₂²×C₄²	—	2	7	no	—
155	C₂²×C₄²	C₂×D₇₃	1	14	yes	complemented
156	C₂×#104	—	73	112	no	—
157	C₂⁵	—	146	1	no	—
158	C₂⁵	C₂²⋊D₇₃	1	1	yes	characteristic
159	C₂×#41	—	73	84	no	—
160	C₂×#41	—	73	28	no	—
161	C₂×#41	—	146	84	no	—
162	C₂×#41	—	146	42	no	—
163	C₂×#41	—	146	42	no	—
164	C₂²×Q₈	—	73	14	no	—
165	C₂²×Q₈	—	146	7	no	—
166	C₂²×D₄	—	73	14	no	—
167	C₂²×D₄	—	146	14	no	—
168	C₂²×D₄	—	146	14	no	—
169	C₂²×D₄	—	146	7	no	—
170	C₂³×C₄	—	2	16	no	—
171	C₂³×C₄	—	2	14	no	—
172	C₂³×C₄	—	146	1	no	—
173	C₂³×C₄	—	146	14	no	—
174	C₂³×C₄	—	146	14	no	—
175	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
176	C₂³×C₄	C₄×D₇₃	1	7	yes	—
177	C₂³×C₄	D₂₉₂	1	7	yes	—
178	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
179	C₂³×C₄	C₂²⋊D₇₃	1	7	yes	—
180	C₂³×C₄	C₂²×D₇₃	1	7	yes	—
181	C₂×M₁₆	—	73	28	no	—
182	C₂×M₁₆	—	146	42	no	—
183	C₂²×C₈	—	146	14	no	—
184	C₂×C₄²	—	2	42	no	—
185	C₂×C₄²	C₂²×D₇₃	1	28	yes	complemented
186	#104	—	73	64	no	—
187	C₂⁴	—	2	8	no	—
188	C₂⁴	—	292	8	no	—
189	C₂⁴	—	146	14	no	—
190	C₂⁴	#11924005	1	7	yes	—
191	C₂⁴	#11924005	1	1	yes	characteristic
192	C₂⁴	C₂×C₂²⋊D₇₃	1	7	yes	—
193	#41	—	73	56	no	—
194	#41	—	73	8	no	—
195	#41	—	146	168	no	—
196	#41	—	146	28	no	—
197	#41	—	146	28	no	—
198	C₂×Q₈	—	73	28	no	—
199	C₂×Q₈	—	146	42	no	—
200	C₂×D₄	—	73	28	no	—
201	C₂×D₄	—	146	84	no	—
202	C₂×D₄	—	146	28	no	—
203	C₂×D₄	—	146	42	no	—
204	C₂²×C₄	—	2	112	no	—
205	C₂²×C₄	—	2	28	no	—
206	C₂²×C₄	—	146	14	no	—
207	C₂²×C₄	—	146	84	no	—
208	C₂²×C₄	—	146	28	no	—
209	C₂²×C₄	#11924005	1	7	yes	—
210	C₂²×C₄	#11924005	1	7	yes	—
211	C₂²×C₄	#11924005	1	42	yes	—
212	C₂²×C₄	C₂×C₄×D₇₃	1	7	yes	—
213	C₂²×C₄	C₂×C₄×D₇₃	1	21	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₇₃	1	21	yes	—
215	C₂²×C₄	C₂×D₂₉₂	1	7	yes	—
216	C₂²×C₄	C₂×D₂₉₂	1	21	yes	—
217	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
218	M₁₆	—	73	8	no	—
219	M₁₆	—	146	28	no	—
220	C₂×C₈	—	146	28	no	—
221	C₄²	—	2	28	no	—
222	C₄²	C₂³×D₇₃	1	8	yes	complemented
223	C₂³	—	2	28	no	—
224	C₂³	—	2	28	no	—
225	C₂³	—	292	56	no	—
226	C₂³	—	146	28	no	—
227	C₂³	#14556519	1	8	yes	direct product factor
228	C₂³	C₂×#11924005	1	21	yes	—
229	C₂³	C₂×#11924005	1	7	yes	—
230	C₂³	C₂²×C₂²⋊D₇₃	1	7	yes	—
231	Q₈	—	73	8	no	—
232	Q₈	—	146	28	no	—
233	D₄	—	73	8	no	—
234	D₄	—	146	8	no	—
235	D₄	—	146	56	no	—
236	D₄	—	146	28	no	—
237	C₂×C₄	—	2	112	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	146	28	no	—
240	C₂×C₄	—	146	8	no	—
241	C₂×C₄	—	146	56	no	—
242	C₂×C₄	C₂×#11924005	1	21	yes	—
243	C₂×C₄	C₂×#11924005	1	21	yes	—
244	C₂×C₄	C₂×#11924005	1	42	yes	—
245	C₂×C₄	C₂²×C₄×D₇₃	1	7	yes	—
246	C₂×C₄	C₂²×C₄×D₇₃	1	7	yes	—
247	C₂×C₄	C₂²×C₂²⋊D₇₃	1	7	yes	—
248	C₂×C₄	C₂²×D₂₉₂	1	7	yes	—
249	C₂×C₄	C₂²×D₂₉₂	1	7	yes	—
250	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
251	C₈	—	146	8	no	—
252	C₂²	—	2	14	no	—
253	C₂²	—	2	42	no	—
254	C₂²	—	292	56	no	—
255	C₂²	—	146	8	no	—
256	C₂²	C₂×#14556519	1	28	yes	direct product factor
257	C₂²	C₂²×#11924005	1	7	yes	—
258	C₂²	C₂²×#11924005	1	7	yes	—
259	C₂²	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	146	8	no	—
262	C₄	C₂²×#11924005	1	7	yes	—
263	C₄	C₂²×#11924005	1	7	yes	—
264	C₄	C₂³×C₄×D₇₃	1	1	yes	characteristic
265	C₄	C₂³×D₂₉₂	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	7	no	—
268	C₂	—	292	8	no	—
269	C₂	C₂²×#14556519	1	14	yes	direct product factor
270	C₂	C₂³×#11924005	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₉₂
commutator	C₄×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂³×C₄²×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²⋊C₄
prime radical	C₂³×#11924028	C₂
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂³×C₄²	D₇₃
hypocenter	C₇₃	C₂³×#104
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂³×#11924012	C₂	semidirect
3. C₂²×#14556519	C₂	direct (N × H)
4. C₂³×C₄²×C₇₃	C₂	semidirect
7. C₂²×#11924028	C₄	semidirect
9. C₂³×Dic₁₄₆	C₄	semidirect
10. C₂³×D₂₉₂	C₄	semidirect
14. C₂²×#11924012	C₂²	semidirect
16. C₂×#14556519	C₂²	direct (N × H)
20. C₂²×C₄²×C₇₃	C₂²	semidirect
25. C₂×#11924028	C₂×C₄	semidirect
28. C₂²×Dic₁₄₆	C₂×C₄	semidirect
30. C₂²×D₂₉₂	C₂×C₄	semidirect
37. C₂×#11924012	C₂³	semidirect
39. #14556519	C₂³	direct (N × H)
50. C₂×C₄²×C₇₃	C₂³	semidirect
56. #11924028	C₂²×C₄	semidirect
59. C₂×Dic₁₄₆	C₂²×C₄	semidirect
61. C₂×D₂₉₂	C₂²×C₄	semidirect
68. #11924012	C₂⁴	semidirect
86. C₄²×C₇₃	C₂⁴	semidirect
89. C₂³×C₇₃	#104	semidirect
110. D₂₉₂	C₂³×C₄	semidirect
114. Dic₁₄₆	C₂³×C₄	semidirect
118. C₂²×C₇₃	C₂×#104	semidirect
133. C₂×C₇₃	C₂²×#104	semidirect
138. C₂³×C₄²	D₇₃	semidirect
140. C₇₃	C₂³×#104	semidirect
155. C₂²×C₄²	C₂×D₇₃	semidirect
185. C₂×C₄²	C₂²×D₇₃	semidirect
222. C₄²	C₂³×D₇₃	semidirect
227. C₂³	#14556519	direct (N × H)
256. C₂²	C₂×#14556519	direct (N × H)
269. C₂	C₂²×#14556519	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
C₂³×#11924012	2
C₂²×#14556519	2
C₂³×C₄²×C₇₃	2
C₂³×#104	73

Subgroups — C23×#14556519 #33949573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556519 #33949573