Subgroups — C₂²×#17988638 #33949689

Order 18688.

3086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988638	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556520	C₂	1	2	yes	maximal, complemented
4	C₂²×#14556503	C₂	1	2	yes	maximal
5	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
10	C₂³×#10596831	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₇₃	C₄	1	12	yes	—
15	C₂×C₄²×D₇₃	C₂²	1	12	yes	—
16	#17988638	C₂²	1	64	yes	direct product factor
17	C₂×#14556520	C₂²	1	24	yes	complemented
18	C₂×#14556503	C₂²	1	24	yes	—
19	C₂×C₄×#10596831	C₂²	1	12	yes	complemented
20	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	12	yes	complemented
23	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₇₃	—	2	8	no	—
28	C₂²×C₄×D₇₃	—	2	8	no	—
29	C₂²×C₄×D₇₃	C₈	1	16	yes	complemented
30	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
31	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
32	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
33	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
34	C₂²×#10596831	—	2	8	no	—
35	C₂²×#10596831	C₂³	1	12	yes	—
36	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
37	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
39	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
40	C₄²×D₇₃	C₂×C₄	1	48	yes	—
41	C₄²×D₇₃	C₂³	1	16	yes	—
42	#14556520	C₂³	1	32	yes	complemented
43	#14556503	C₂³	1	32	yes	—
44	C₄×#10596831	C₂³	1	16	yes	complemented
45	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	—	2	8	no	—
49	C₂²×C₈×C₇₃	C₂³	1	12	yes	—
50	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
51	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
52	C₄×C₈×C₇₃	C₂³	1	16	yes	complemented
53	C₂³×D₇₃	—	2	4	no	—
54	C₂³×D₇₃	—	2	4	no	—
55	C₂³×D₇₃	C₂×C₈	1	8	yes	—
56	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
57	C₂²×Dic₇₃	—	2	4	no	—
58	C₂²×Dic₇₃	—	2	4	no	—
59	C₂²×Dic₇₃	C₂×C₈	1	8	yes	—
60	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
61	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
62	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
63	C₂×C₄×D₇₃	—	2	24	no	—
64	C₂×C₄×D₇₃	—	2	48	no	—
65	C₂×C₄×D₇₃	—	2	24	no	—
66	C₂×C₄×D₇₃	C₂×C₈	1	96	yes	complemented
67	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
68	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
69	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
70	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
71	C₂×#10596831	—	2	24	no	—
72	C₂×#10596831	C₂⁴	1	8	yes	—
73	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
74	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
75	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
76	C₄×Dic₇₃	C₂⁴	1	4	yes	—
77	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×C₈	1	8	yes	—
79	C₂²×C₄×C₇₃	M₁₆	1	8	yes	—
80	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
83	C₂²×C₄×C₇₃	#41	1	8	yes	—
84	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
85	C₂×C₈×C₇₃	—	2	24	no	—
86	C₂×C₈×C₇₃	C₂⁴	1	8	yes	—
87	C₄²×C₇₃	C₂²×C₄	1	12	yes	—
88	C₄²×C₇₃	C₂⁴	1	4	yes	—
89	C₂³×C₇₃	#117	1	4	yes	—
90	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
91	C₂³×C₇₃	C₂×M₁₆	1	4	yes	—
92	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
93	C₂²×D₇₃	—	2	12	no	—
94	C₂²×D₇₃	—	2	12	no	—
95	C₂²×D₇₃	—	2	32	no	—
96	C₂²×D₇₃	C₂²×C₈	1	24	yes	—
97	C₂²×D₇₃	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₇₃	#117	1	12	yes	—
99	C₂×C₄×C₇₃	C₂²×C₈	1	24	yes	—
100	C₂×C₄×C₇₃	C₂×M₁₆	1	24	yes	—
101	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
102	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₇₃	C₂×#41	1	12	yes	—
105	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
106	C₂×Dic₇₃	—	2	12	no	—
107	C₂×Dic₇₃	—	2	12	no	—
108	C₂×Dic₇₃	C₂²×C₈	1	24	yes	—
109	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
110	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
111	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
112	C₄×D₇₃	—	2	24	no	—
113	C₄×D₇₃	—	2	32	no	—
114	C₄×D₇₃	—	2	8	no	—
115	C₄×D₇₃	C₂²×C₈	1	64	yes	complemented
116	C₈×C₇₃	—	2	8	no	—
117	#10596831	—	2	8	no	—
118	C₂²×C₇₃	#431	1	16	yes	complemented
119	C₂²×C₇₃	C₂×#117	1	6	yes	—
120	C₂²×C₇₃	C₂³×C₈	1	6	yes	—
121	C₂²×C₇₃	C₂²×M₁₆	1	6	yes	—
122	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
123	C₂×D₇₃	—	2	4	no	—
124	C₂×D₇₃	—	2	48	no	—
125	C₂×D₇₃	—	2	4	no	—
126	C₂×D₇₃	C₂³×C₈	1	8	yes	—
127	C₄×C₇₃	C₂×#117	1	6	yes	—
128	C₄×C₇₃	C₂³×C₈	1	8	yes	—
129	C₄×C₇₃	C₂²×M₁₆	1	8	yes	—
130	C₄×C₇₃	C₂²×#41	1	2	yes	—
131	Dic₇₃	—	2	4	no	—
132	Dic₇₃	—	2	4	no	—
133	Dic₇₃	C₂³×C₈	1	8	yes	—
134	C₂²×#431	—	73	1	no	maximal, Sylow
135	C₂×C₇₃	C₂×#431	1	12	yes	complemented
136	C₂×C₇₃	C₂²×#117	1	1	yes	characteristic
137	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
138	C₂×C₇₃	C₂³×M₁₆	1	1	yes	characteristic
139	D₇₃	—	2	8	no	—
140	C₂³×C₄²	—	73	1	no	—
141	C₂×#431	—	73	24	no	—
142	C₂²×C₄⋊C₈	—	73	2	no	—
143	C₂²×C₂²⋊C₈	—	73	2	no	—
144	C₂²×C₄×C₈	—	73	1	no	—
145	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic, complemented
146	C₇₃	C₂²×#431	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴×C₄	—	73	1	no	—
148	C₂⁴×C₄	—	73	1	no	—
149	C₂⁴×C₄	—	73	1	no	—
150	C₂³×C₈	—	73	2	no	—
151	C₂³×C₈	C₂×D₇₃	1	2	yes	—
152	C₂²×C₄²	—	73	1	no	—
153	C₂²×C₄²	—	73	1	no	—
154	C₂²×C₄²	—	73	1	no	—
155	C₂²×C₄²	—	73	12	no	—
156	C₂²×C₄²	—	73	12	no	—
157	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
158	#431	—	73	64	no	—
159	C₂×C₄⋊C₈	—	73	24	no	—
160	C₂×C₂²⋊C₈	—	73	24	no	—
161	C₂×C₄×C₈	—	73	12	no	—
162	C₂×C₄×C₈	C₂×D₇₃	1	12	yes	complemented
163	C₂⁵	—	73	1	no	—
164	C₂³×C₄	—	73	1	no	—
165	C₂³×C₄	—	73	16	no	—
166	C₂³×C₄	—	73	12	no	—
167	C₂³×C₄	—	73	1	no	—
168	C₂³×C₄	—	73	1	no	—
169	C₂³×C₄	—	73	6	no	—
170	C₂³×C₄	—	73	12	no	—
171	C₂³×C₄	—	73	6	no	—
172	C₂³×C₄	—	146	8	no	—
173	C₂³×C₄	—	146	8	no	—
174	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
175	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	—	73	12	no	—
179	C₂²×C₈	—	146	8	no	—
180	C₂²×C₈	C₂²×D₇₃	1	12	yes	—
181	C₂×C₄²	—	73	6	no	—
182	C₂×C₄²	—	73	12	no	—
183	C₂×C₄²	—	73	12	no	—
184	C₂×C₄²	—	73	6	no	—
185	C₂×C₄²	—	73	48	no	—
186	C₂×C₄²	—	73	16	no	—
187	C₂×C₄²	C₄×D₇₃	1	6	yes	—
188	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
189	C₄⋊C₈	—	73	32	no	—
190	C₂²⋊C₈	—	73	32	no	—
191	C₄×C₈	—	73	16	no	—
192	C₄×C₈	C₂²×D₇₃	1	16	yes	complemented
193	C₂⁴	—	73	8	no	—
194	C₂⁴	—	73	6	no	—
195	C₂⁴	—	146	4	no	—
196	C₂⁴	—	146	4	no	—
197	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
198	C₂²×C₄	—	73	8	no	—
199	C₂²×C₄	—	73	6	no	—
200	C₂²×C₄	—	73	96	no	—
201	C₂²×C₄	—	73	16	no	—
202	C₂²×C₄	—	73	6	no	—
203	C₂²×C₄	—	73	6	no	—
204	C₂²×C₄	—	73	12	no	—
205	C₂²×C₄	—	73	16	no	—
206	C₂²×C₄	—	73	4	no	—
207	C₂²×C₄	—	146	4	no	—
208	C₂²×C₄	—	146	4	no	—
209	C₂²×C₄	—	146	24	no	—
210	C₂²×C₄	—	146	48	no	—
211	C₂²×C₄	—	146	24	no	—
212	C₂²×C₄	C₈×D₇₃	1	8	yes	—
213	C₂²×C₄	#11924011	1	8	yes	—
214	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
215	C₂²×C₄	C₂×C₄×D₇₃	1	6	yes	—
216	C₂²×C₄	C₂×C₄×D₇₃	1	6	yes	—
217	C₂²×C₄	#11924028	1	8	yes	—
218	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
219	C₂×C₈	—	2	24	no	—
220	C₂×C₈	—	73	8	no	—
221	C₂×C₈	—	146	24	no	—
222	C₂×C₈	C₂³×D₇₃	1	8	yes	—
223	C₄²	—	73	12	no	—
224	C₄²	—	73	16	no	—
225	C₄²	—	73	16	no	—
226	C₄²	—	73	4	no	—
227	C₄²	C₂×C₄×D₇₃	1	12	yes	—
228	C₄²	C₂³×D₇₃	1	4	yes	—
229	C₂³	—	73	4	no	—
230	C₂³	—	73	24	no	—
231	C₂³	—	146	32	no	—
232	C₂³	—	146	12	no	—
233	C₂³	—	146	12	no	—
234	C₂³	#14556548	1	4	yes	—
235	C₂³	C₂×C₈×D₇₃	1	4	yes	—
236	C₂³	C₂×#11924011	1	4	yes	—
237	C₂³	C₂²×C₄×D₇₃	1	3	yes	—
238	C₂×C₄	—	73	4	no	—
239	C₂×C₄	—	73	24	no	—
240	C₂×C₄	—	73	64	no	—
241	C₂×C₄	—	73	4	no	—
242	C₂×C₄	—	73	4	no	—
243	C₂×C₄	—	146	12	no	—
244	C₂×C₄	—	146	12	no	—
245	C₂×C₄	—	146	8	no	—
246	C₂×C₄	—	146	24	no	—
247	C₂×C₄	—	146	32	no	—
248	C₂×C₄	#14556548	1	12	yes	—
249	C₂×C₄	C₂×C₈×D₇₃	1	24	yes	—
250	C₂×C₄	C₂×#11924011	1	24	yes	—
251	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
252	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
254	C₂×C₄	C₂×#11924028	1	12	yes	—
255	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
256	C₈	—	2	8	no	—
257	C₈	—	146	8	no	—
258	C₂²	—	73	8	no	—
259	C₂²	—	146	4	no	—
260	C₂²	—	146	4	no	—
261	C₂²	—	146	48	no	—
262	C₂²	#17988638	1	16	yes	direct product factor
263	C₂²	C₂×#14556548	1	6	yes	—
264	C₂²	C₂²×C₈×D₇₃	1	6	yes	—
265	C₂²	C₂²×#11924011	1	6	yes	—
266	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
267	C₄	—	73	8	no	—
268	C₄	—	146	4	no	—
269	C₄	—	146	4	no	—
270	C₄	C₂×#14556548	1	6	yes	—
271	C₄	C₂²×C₈×D₇₃	1	8	yes	—
272	C₄	C₂²×#11924011	1	8	yes	—
273	C₄	C₂²×#11924028	1	2	yes	—
274	C₂	—	146	8	no	—
275	C₂	C₂×#17988638	1	12	yes	direct product factor
276	C₂	C₂²×#14556548	1	1	yes	characteristic
277	C₂	C₂³×C₈×D₇₃	1	1	yes	characteristic
278	C₂	C₂³×#11924011	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₇₃
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	C₂²×#431
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17988638	C₂	direct (N × H)
3. C₂²×#14556520	C₂	semidirect
5. C₂²×C₄×#10596831	C₂	semidirect
6. C₂²×C₄×C₈×C₇₃	C₂	semidirect
16. #17988638	C₂²	direct (N × H)
17. C₂×#14556520	C₂²	semidirect
19. C₂×C₄×#10596831	C₂²	semidirect
22. C₂×C₄×C₈×C₇₃	C₂²	semidirect
29. C₂²×C₄×D₇₃	C₈	semidirect
42. #14556520	C₂³	semidirect
44. C₄×#10596831	C₂³	semidirect
52. C₄×C₈×C₇₃	C₂³	semidirect
66. C₂×C₄×D₇₃	C₂×C₈	semidirect
115. C₄×D₇₃	C₂²×C₈	semidirect
118. C₂²×C₇₃	#431	semidirect
135. C₂×C₇₃	C₂×#431	semidirect
145. C₂²×C₄×C₈	D₇₃	semidirect
146. C₇₃	C₂²×#431	semidirect
162. C₂×C₄×C₈	C₂×D₇₃	semidirect
192. C₄×C₈	C₂²×D₇₃	semidirect
262. C₂²	#17988638	direct (N × H)
275. C₂	C₂×#17988638	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×C₄²×D₇₃	2
C₂×#17988638	2
C₂²×#14556520	2
C₂²×#14556503	2
C₂²×C₄×#10596831	2
C₂²×C₄×C₈×C₇₃	2
C₂²×#431	73

Subgroups — C22×#17988638 #33949689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988638 #33949689