Subgroups — C₂²×#17988825 #33953079

Order 18688.

3006 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×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556567	C₂	1	2	yes	maximal
3	C₂×#17988825	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14556515	C₂	1	2	yes	maximal, complemented
6	C₂³×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₇₃	—	2	8	no	—
10	C₂²×C₈×D₇₃	C₂²	1	12	yes	—
11	C₂×#14556567	—	2	12	no	—
12	C₂×#14556567	C₂²	1	24	yes	—
13	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
14	C₂²×#11924007	—	2	2	no	—
15	C₂²×#11924005	—	2	2	no	—
16	#17988825	C₂²	1	64	yes	direct product factor
17	C₂×#14556525	C₂²	1	12	yes	complemented
18	C₂×#14556515	C₂²	1	24	yes	complemented
19	C₂³×Q₈×C₇₃	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₇₃	—	2	8	no	—
21	C₂²×Q₁₆×C₇₃	C₂²	1	12	yes	complemented
22	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂⁴×D₇₃	D₄	1	1	yes	characteristic
24	C₂³×Dic₇₃	—	2	2	no	—
25	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×D₇₃	—	2	4	no	—
27	C₂²×C₄×D₇₃	—	2	4	no	—
28	C₂²×C₄×D₇₃	D₄	1	6	yes	—
29	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
30	C₂²×#10596831	—	2	4	no	—
31	C₂²×#10596831	C₂³	1	6	yes	—
32	C₂×C₈×D₇₃	—	2	48	no	—
33	C₂×C₈×D₇₃	C₂³	1	16	yes	—
34	#14556567	—	2	48	no	—
35	#14556567	C₂³	1	32	yes	—
36	C₂×C₄⋊Dic₇₃	—	2	6	no	—
37	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
38	C₂×#11924007	—	2	24	no	—
39	C₂×#11924005	—	2	24	no	—
40	#14556525	C₂³	1	16	yes	complemented
41	#14556515	C₂³	1	32	yes	complemented
42	C₂²×Q₈×C₇₃	—	2	6	no	—
43	C₂²×Q₈×C₇₃	—	2	8	no	—
44	C₂²×Q₈×C₇₃	—	2	8	no	—
45	C₂²×Q₈×C₇₃	C₂³	1	12	yes	—
46	C₂³×C₄×C₇₃	—	2	2	no	—
47	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₇₃	—	2	48	no	—
49	C₂×Q₁₆×C₇₃	C₂³	1	16	yes	complemented
50	C₂²×C₈×C₇₃	D₄	1	8	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
52	C₂³×D₇₃	—	2	4	no	—
53	C₂³×D₇₃	—	2	4	no	—
54	C₂³×D₇₃	Q₁₆	1	8	yes	complemented
55	C₂³×D₇₃	C₂×D₄	1	6	yes	—
56	C₂²×Dic₇₃	—	2	4	no	—
57	C₂²×Dic₇₃	—	2	12	no	—
58	C₂²×Dic₇₃	—	4	8	no	—
59	C₂²×Dic₇₃	C₂×D₄	1	6	yes	—
60	C₂×C₄×D₇₃	—	2	24	no	—
61	C₂×C₄×D₇₃	—	2	12	no	—
62	C₂×C₄×D₇₃	—	2	12	no	—
63	C₂×C₄×D₇₃	C₂×D₄	1	12	yes	—
64	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
65	C₂×#10596831	—	2	12	no	—
66	C₂×#10596831	C₂⁴	1	4	yes	—
67	C₈×D₇₃	—	2	32	no	—
68	C₄⋊Dic₇₃	—	2	12	no	—
69	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
70	#11924007	—	2	32	no	—
71	#11924005	—	2	32	no	—
72	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₇₃	—	2	12	no	—
74	C₂×Q₈×C₇₃	—	2	24	no	—
75	C₂×Q₈×C₇₃	—	2	24	no	—
76	C₂×Q₈×C₇₃	—	4	24	no	—
77	C₂×Q₈×C₇₃	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₇₃	—	2	12	no	—
79	C₂²×C₄×C₇₃	—	4	8	no	—
80	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₇₃	#41	1	4	yes	—
83	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
84	Q₁₆×C₇₃	—	2	32	no	—
85	C₂×C₈×C₇₃	C₂×D₄	1	24	yes	—
86	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
87	C₂³×C₇₃	#121	1	4	yes	—
88	C₂³×C₇₃	C₂×Q₁₆	1	4	yes	—
89	C₂³×C₇₃	#135	1	4	yes	—
90	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
91	C₂²×D₇₃	—	2	12	no	—
92	C₂²×D₇₃	—	2	12	no	—
93	C₂²×D₇₃	—	2	32	no	—
94	C₂²×D₇₃	C₂×Q₁₆	1	24	yes	complemented
95	C₂²×D₇₃	C₂²×D₄	1	4	yes	—
96	Q₈×C₇₃	—	2	8	no	—
97	Q₈×C₇₃	—	2	8	no	—
98	Q₈×C₇₃	—	4	24	no	—
99	C₂×C₄×C₇₃	—	2	8	no	—
100	C₂×C₄×C₇₃	—	4	24	no	—
101	C₂×C₄×C₇₃	#121	1	12	yes	—
102	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
103	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
104	C₂×C₄×C₇₃	C₂×#41	1	6	yes	—
105	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
106	C₂×Dic₇₃	—	2	12	no	—
107	C₂×Dic₇₃	—	2	8	no	—
108	C₂×Dic₇₃	—	4	24	no	—
109	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
110	C₄×D₇₃	—	2	24	no	—
111	C₄×D₇₃	—	2	4	no	—
112	C₄×D₇₃	—	2	4	no	—
113	C₈×C₇₃	C₂²×D₄	1	8	yes	—
114	#10596831	—	2	4	no	—
115	C₂²×C₇₃	#467	1	16	yes	complemented
116	C₂²×C₇₃	C₂×#121	1	6	yes	—
117	C₂²×C₇₃	C₂²×Q₁₆	1	6	yes	—
118	C₂²×C₇₃	C₂×#135	1	6	yes	—
119	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
120	C₂×D₇₃	—	2	4	no	—
121	C₂×D₇₃	—	2	48	no	—
122	C₂×D₇₃	—	2	4	no	—
123	C₂×D₇₃	C₂²×Q₁₆	1	8	yes	complemented
124	C₄×C₇₃	—	4	8	no	—
125	C₄×C₇₃	C₂×#121	1	6	yes	—
126	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
127	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
128	Dic₇₃	—	2	4	no	—
129	Dic₇₃	—	4	8	no	—
130	C₂²×#467	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	C₂×#467	1	12	yes	complemented
132	C₂×C₇₃	C₂²×#121	1	1	yes	characteristic
133	C₂×C₇₃	C₂³×Q₁₆	1	1	yes	characteristic
134	C₂×C₇₃	C₂²×#135	1	1	yes	characteristic
135	D₇₃	—	2	8	no	—
136	C₂³×Q₁₆	D₇₃	1	1	yes	characteristic, complemented
137	C₂⁴×C₈	—	73	1	no	—
138	C₂²×C₂²⋊Q₈	—	73	2	no	—
139	C₂×#467	—	73	24	no	—
140	C₂²×#107	—	73	1	no	—
141	C₂²×#103	—	73	2	no	—
142	C₇₃	C₂²×#467	1	1	yes	characteristic, complemented, Sylow
143	C₂³×Q₈	C₂×D₇₃	1	2	yes	—
144	C₂⁴×C₄	—	73	1	no	—
145	C₂²×Q₁₆	—	2	8	no	—
146	C₂²×Q₁₆	C₂×D₇₃	1	12	yes	complemented
147	C₂³×C₈	—	73	1	no	—
148	C₂³×C₈	—	73	12	no	—
149	C₂³×C₈	—	146	8	no	—
150	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
151	C₂×C₂²⋊Q₈	—	73	24	no	—
152	C₂×C₂²⋊Q₈	—	146	12	no	—
153	C₂²×C₄⋊C₄	—	73	2	no	—
154	C₂²×C₄⋊C₄	—	146	2	no	—
155	C₂²×C₂²⋊C₄	—	146	2	no	—
156	#467	—	73	64	no	—
157	C₂×#107	—	73	12	no	—
158	C₂×#103	—	73	24	no	—
159	C₂⁵	—	73	1	no	—
160	C₂²×Q₈	—	2	6	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	—	2	8	no	—
163	C₂²×Q₈	C₂²×D₇₃	1	12	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	—	73	1	no	—
166	C₂³×C₄	—	73	6	no	—
167	C₂³×C₄	—	73	6	no	—
168	C₂³×C₄	—	146	2	no	—
169	C₂³×C₄	—	146	4	no	—
170	C₂³×C₄	—	146	4	no	—
171	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
172	C₂×Q₁₆	—	2	48	no	—
173	C₂×Q₁₆	C₂²×D₇₃	1	16	yes	complemented
174	C₂²×C₈	—	73	6	no	—
175	C₂²×C₈	—	73	16	no	—
176	C₂²×C₈	—	146	4	no	—
177	C₂²×C₈	—	146	48	no	—
178	C₂²×C₈	C₂²⋊D₇₃	1	8	yes	—
179	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
180	C₂²⋊Q₈	—	73	32	no	—
181	C₂²⋊Q₈	—	146	48	no	—
182	C₂×C₄⋊C₄	—	73	12	no	—
183	C₂×C₄⋊C₄	—	146	24	no	—
184	C₂×C₄⋊C₄	—	146	6	no	—
185	C₂×C₂²⋊C₄	—	146	24	no	—
186	#107	—	73	16	no	—
187	#103	—	73	32	no	—
188	C₂⁴	—	73	8	no	—
189	C₂⁴	—	73	6	no	—
190	C₂⁴	—	146	4	no	—
191	C₂⁴	—	146	4	no	—
192	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
193	C₂×Q₈	—	2	12	no	—
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	4	24	no	—
197	C₂×Q₈	C₂³×D₇₃	1	8	yes	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	292	8	no	—
201	C₂²×C₄	—	73	6	no	—
202	C₂²×C₄	—	73	12	no	—
203	C₂²×C₄	—	73	4	no	—
204	C₂²×C₄	—	146	12	no	—
205	C₂²×C₄	—	146	4	no	—
206	C₂²×C₄	—	146	24	no	—
207	C₂²×C₄	—	146	12	no	—
208	C₂²×C₄	—	146	12	no	—
209	C₂²×C₄	D₄×D₇₃	1	3	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
211	C₂²×C₄	#11924027	1	4	yes	—
212	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
213	Q₁₆	—	2	32	no	—
214	C₂×C₈	—	73	4	no	—
215	C₂×C₈	—	146	12	no	—
216	C₂×C₈	—	146	32	no	—
217	C₂×C₈	C₂×C₂²⋊D₇₃	1	24	yes	—
218	C₂×C₈	C₂³×D₇₃	1	4	yes	—
219	C₄⋊C₄	—	73	8	no	—
220	C₄⋊C₄	—	146	32	no	—
221	C₄⋊C₄	—	146	12	no	—
222	C₂²⋊C₄	—	146	32	no	—
223	C₂³	—	73	4	no	—
224	C₂³	—	73	24	no	—
225	C₂³	—	146	12	no	—
226	C₂³	—	146	32	no	—
227	C₂³	—	146	12	no	—
228	C₂³	#14556562	1	4	yes	—
229	C₂³	Q₁₆×D₇₃	1	4	yes	—
230	C₂³	#14556614	1	4	yes	—
231	C₂³	C₂×D₄×D₇₃	1	3	yes	—
232	Q₈	—	2	8	no	—
233	Q₈	—	2	8	no	—
234	Q₈	—	4	24	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	24	no	—
237	C₂×C₄	—	292	24	no	—
238	C₂×C₄	—	73	4	no	—
239	C₂×C₄	—	146	8	no	—
240	C₂×C₄	—	146	12	no	—
241	C₂×C₄	—	146	4	no	—
242	C₂×C₄	—	146	4	no	—
243	C₂×C₄	—	146	24	no	—
244	C₂×C₄	#14556562	1	12	yes	—
245	C₂×C₄	C₂×D₄×D₇₃	1	3	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₇₃	1	6	yes	—
247	C₂×C₄	C₂×#11924027	1	6	yes	—
248	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
249	C₈	—	146	4	no	—
250	C₈	C₂²×C₂²⋊D₇₃	1	8	yes	—
251	C₂²	—	73	8	no	—
252	C₂²	—	146	48	no	—
253	C₂²	—	146	4	no	—
254	C₂²	—	146	4	no	—
255	C₂²	#17988825	1	16	yes	direct product factor
256	C₂²	C₂×#14556562	1	6	yes	—
257	C₂²	C₂×Q₁₆×D₇₃	1	6	yes	—
258	C₂²	C₂×#14556614	1	6	yes	—
259	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
260	C₄	—	4	8	no	—
261	C₄	—	292	8	no	—
262	C₄	—	146	4	no	—
263	C₄	C₂×#14556562	1	6	yes	—
264	C₄	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
265	C₄	C₂²×#11924027	1	1	yes	characteristic
266	C₂	—	146	8	no	—
267	C₂	C₂×#17988825	1	12	yes	direct product factor
268	C₂	C₂²×#14556562	1	1	yes	characteristic
269	C₂	C₂²×Q₁₆×D₇₃	1	1	yes	characteristic
270	C₂	C₂²×#14556614	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂³×Q₁₆×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
prime radical	C₂⁴×D₇₃	D₄
hypercenter	C₂³×Q₁₆	D₇₃
hypocenter	C₇₃	C₂²×#467
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#17988825	C₂	direct (N × H)
4. C₂²×#14556525	C₂	semidirect
5. C₂²×#14556515	C₂	semidirect
6. C₂³×Q₁₆×C₇₃	C₂	semidirect
16. #17988825	C₂²	direct (N × H)
17. C₂×#14556525	C₂²	semidirect
18. C₂×#14556515	C₂²	semidirect
21. C₂²×Q₁₆×C₇₃	C₂²	semidirect
40. #14556525	C₂³	semidirect
41. #14556515	C₂³	semidirect
49. C₂×Q₁₆×C₇₃	C₂³	semidirect
54. C₂³×D₇₃	Q₁₆	semidirect
94. C₂²×D₇₃	C₂×Q₁₆	semidirect
115. C₂²×C₇₃	#467	semidirect
123. C₂×D₇₃	C₂²×Q₁₆	semidirect
131. C₂×C₇₃	C₂×#467	semidirect
136. C₂³×Q₁₆	D₇₃	semidirect
142. C₇₃	C₂²×#467	semidirect
146. C₂²×Q₁₆	C₂×D₇₃	semidirect
173. C₂×Q₁₆	C₂²×D₇₃	semidirect
255. C₂²	#17988825	direct (N × H)
267. C₂	C₂×#17988825	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×C₈×D₇₃	2
C₂²×#14556567	2
C₂×#17988825	2
C₂²×#14556525	2
C₂²×#14556515	2
C₂³×Q₁₆×C₇₃	2
C₂²×#467	73

Subgroups — C22×#17988825 #33953079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988825 #33953079