Subgroups — C₂²×#17989588 #34031184

Order 18688.

3870 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	C₂³×#11924036	C₂	1	2	yes	maximal
2	C₂²×#14556658	C₂	1	2	yes	maximal, complemented
3	C₂×#17989588	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#10596833	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924036	—	2	16	no	—
7	C₂²×#11924036	C₂²	1	24	yes	—
8	C₂³×#10596833	C₂²	1	4	yes	—
9	C₂×#14556658	C₂²	1	24	yes	complemented
10	#17989588	C₂²	1	64	yes	direct product factor
11	C₂×C₄⋊#10596833	C₂²	1	24	yes	complemented
12	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
13	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
14	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×D₇₃	C₄	1	12	yes	—
19	C₂×C₄²×D₇₃	C₂²	1	12	yes	—
20	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
21	C₂×#11924036	—	2	96	no	—
22	C₂×#11924036	C₂³	1	32	yes	—
23	C₂²×#10596833	—	2	16	no	—
24	C₂²×#10596833	C₂³	1	24	yes	—
25	#14556658	C₂³	1	32	yes	complemented
26	C₄⋊#10596833	C₂³	1	32	yes	complemented
27	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×D₇₃	—	2	8	no	—
32	C₂²×C₄×D₇₃	C₈	1	16	yes	complemented
33	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
34	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
35	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
36	C₂²×C₄×D₇₃	D₄	1	8	yes	—
37	C₂²×C₄×D₇₃	Q₈	1	8	yes	—
38	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
39	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
40	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
41	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
42	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
43	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
44	C₄²×D₇₃	C₂×C₄	1	48	yes	—
45	C₄²×D₇₃	C₂³	1	16	yes	—
46	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	C₂×C₄	1	12	yes	—
50	#11924036	—	2	64	no	—
51	C₂×#10596833	—	2	48	no	—
52	C₂×#10596833	C₂⁴	1	16	yes	—
53	C₂³×D₇₃	C₄⋊C₄	1	8	yes	—
54	C₂³×D₇₃	C₂×C₈	1	8	yes	—
55	C₂³×D₇₃	M₁₆	1	8	yes	—
56	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
57	C₂²×Dic₇₃	—	2	4	no	—
58	C₂²×Dic₇₃	C₂×C₈	1	8	yes	complemented
59	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
60	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
61	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
62	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
63	C₂²×Dic₇₃	C₂×Q₈	1	4	yes	—
64	C₂²×Dic₇₃	C₂⁴	1	3	yes	—
65	C₂×C₄×D₇₃	—	2	48	no	—
66	C₂×C₄×D₇₃	C₄⋊C₄	1	48	yes	—
67	C₂×C₄×D₇₃	C₂×C₈	1	96	yes	complemented
68	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
69	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
70	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
71	C₂×C₄×D₇₃	C₂×D₄	1	24	yes	—
72	C₂×C₄×D₇₃	C₂×Q₈	1	24	yes	—
73	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
74	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
75	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
76	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
77	C₄×Dic₇₃	C₂⁴	1	4	yes	—
78	C₄×Dic₇₃	C₂⁴	1	4	yes	—
79	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	—	2	4	no	—
81	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
82	C₂²×C₄×C₇₃	C₂×C₈	1	8	yes	complemented
83	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
84	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
85	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
86	C₄²×C₇₃	C₂²×C₄	1	16	yes	—
87	C₂³×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
89	C₂³×C₇₃	C₂×M₁₆	1	4	yes	—
90	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
91	C₂²×D₇₃	C₄⋊C₈	1	64	yes	complemented
92	C₂²×D₇₃	C₂×C₄⋊C₄	1	24	yes	—
93	C₂²×D₇₃	C₂²×C₈	1	24	yes	—
94	C₂²×D₇₃	C₂×M₁₆	1	24	yes	—
95	C₂²×D₇₃	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₇₃	—	2	12	no	—
97	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	24	yes	—
98	C₂×C₄×C₇₃	C₂²×C₈	1	24	yes	complemented
99	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
102	C₂×Dic₇₃	—	2	12	no	—
103	C₂×Dic₇₃	C₂×C₄⋊C₄	1	12	yes	—
104	C₂×Dic₇₃	C₂²×C₈	1	24	yes	complemented
105	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
106	C₂×Dic₇₃	C₂³×C₄	1	3	yes	—
107	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
108	C₂×Dic₇₃	C₂²×D₄	1	6	yes	—
109	C₂×Dic₇₃	C₂²×Q₈	1	6	yes	—
110	C₂×Dic₇₃	C₂⁵	1	1	yes	characteristic
111	C₄×D₇₃	—	2	32	no	—
112	C₄×D₇₃	C₂×C₄⋊C₄	1	48	yes	—
113	C₄×D₇₃	C₂²×C₈	1	64	yes	complemented
114	C₄×D₇₃	C₂²×D₄	1	8	yes	—
115	C₄×D₇₃	C₂²×Q₈	1	8	yes	—
116	#10596833	—	2	16	no	—
117	C₂²×C₇₃	C₂×C₄⋊C₈	1	16	yes	complemented
118	C₂²×C₇₃	C₂²×C₄⋊C₄	1	6	yes	—
119	C₂²×C₇₃	C₂³×C₈	1	6	yes	—
120	C₂²×C₇₃	C₂²×M₁₆	1	6	yes	—
121	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
122	C₂×D₇₃	C₂×C₄⋊C₈	1	96	yes	complemented
123	C₂×D₇₃	C₂²×C₄⋊C₄	1	8	yes	—
124	C₂×D₇₃	C₂³×C₈	1	8	yes	—
125	C₂×D₇₃	C₂²×M₁₆	1	8	yes	—
126	C₄×C₇₃	—	2	4	no	—
127	C₄×C₇₃	C₂²×C₄⋊C₄	1	8	yes	—
128	C₄×C₇₃	C₂³×C₈	1	8	yes	complemented
129	Dic₇₃	—	2	4	no	—
130	Dic₇₃	C₂²×C₄⋊C₄	1	6	yes	—
131	Dic₇₃	C₂³×C₈	1	8	yes	complemented
132	Dic₇₃	C₂³×D₄	1	1	yes	characteristic
133	Dic₇₃	C₂³×Q₈	1	1	yes	characteristic
134	C₂³×C₄⋊C₈	—	73	1	no	maximal, Sylow
135	C₂×C₇₃	C₂²×C₄⋊C₈	1	12	yes	complemented
136	C₂×C₇₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
138	C₂×C₇₃	C₂³×M₁₆	1	1	yes	characteristic
139	D₇₃	C₂²×C₄⋊C₈	1	16	yes	complemented
140	C₂⁴×C₈	—	73	2	no	—
141	C₂³×C₄²	—	73	1	no	—
142	C₂²×C₄⋊C₈	—	73	2	no	—
143	C₂²×C₄⋊C₈	—	73	24	no	—
144	C₂²×C₄⋊C₈	—	73	2	no	—
145	C₇₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴×C₄	—	73	1	no	—
147	C₂⁴×C₄	—	73	1	no	—
148	C₂⁴×C₄	—	73	1	no	—
149	C₂³×C₈	—	73	4	no	—
150	C₂³×C₈	—	73	24	no	—
151	C₂³×C₈	—	146	16	no	—
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	#64408	1	1	yes	characteristic
158	C₂×C₄⋊C₈	—	73	24	no	—
159	C₂×C₄⋊C₈	—	73	64	no	—
160	C₂×C₄⋊C₈	—	73	24	no	—
161	C₂⁵	—	73	1	no	—
162	C₂³×C₄	—	73	1	no	—
163	C₂³×C₄	—	73	1	no	—
164	C₂³×C₄	—	73	8	no	—
165	C₂³×C₄	—	73	8	no	—
166	C₂³×C₄	—	73	16	no	—
167	C₂³×C₄	—	73	6	no	—
168	C₂³×C₄	—	73	12	no	—
169	C₂³×C₄	—	73	6	no	—
170	C₂³×C₄	—	73	1	no	—
171	C₂³×C₄	—	73	12	no	—
172	C₂³×C₄	—	146	8	no	—
173	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
174	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
175	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
176	C₂²×C₈	—	73	24	no	—
177	C₂²×C₈	—	73	32	no	—
178	C₂²×C₈	—	146	16	no	—
179	C₂²×C₈	—	146	96	no	—
180	C₂×C₄²	—	73	6	no	—
181	C₂×C₄²	—	73	6	no	—
182	C₂×C₄²	—	73	12	no	—
183	C₂×C₄²	—	73	6	no	—
184	C₂×C₄²	—	73	6	no	—
185	C₂×C₄²	—	73	48	no	—
186	C₂×C₄²	—	73	16	no	—
187	C₂×C₄²	C₂×#64408	1	12	yes	—
188	C₄⋊C₈	—	73	32	no	—
189	C₄⋊C₈	—	73	32	no	—
190	C₂⁴	—	73	8	no	—
191	C₂⁴	—	73	8	no	—
192	C₂⁴	—	73	6	no	—
193	C₂⁴	—	73	8	no	—
194	C₂⁴	C₂²×#64408	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	73	8	no	—
197	C₂²×C₄	—	73	3	no	—
198	C₂²×C₄	—	73	6	no	—
199	C₂²×C₄	—	73	3	no	—
200	C₂²×C₄	—	73	4	no	—
201	C₂²×C₄	—	73	4	no	—
202	C₂²×C₄	—	73	96	no	—
203	C₂²×C₄	—	73	12	no	—
204	C₂²×C₄	—	73	16	no	—
205	C₂²×C₄	—	73	4	no	—
206	C₂²×C₄	—	73	48	no	—
207	C₂²×C₄	—	73	24	no	—
208	C₂²×C₄	—	73	24	no	—
209	C₂²×C₄	—	73	6	no	—
210	C₂²×C₄	—	73	16	no	—
211	C₂²×C₄	—	146	48	no	—
212	C₂²×C₄	—	146	4	no	—
213	C₂²×C₄	C₄⋊#64408	1	8	yes	—
214	C₂²×C₄	#11924036	1	8	yes	complemented
215	C₂²×C₄	C₂²×#64408	1	6	yes	—
216	C₂²×C₄	C₂²×#64408	1	6	yes	—
217	C₂²×C₄	C₂²×#64408	1	6	yes	—
218	C₂×C₈	—	73	16	no	—
219	C₂×C₈	—	146	48	no	—
220	C₂×C₈	—	146	64	no	—
221	C₄²	—	73	12	no	—
222	C₄²	—	73	16	no	—
223	C₄²	—	73	12	no	—
224	C₄²	—	73	4	no	—
225	C₄²	—	73	4	no	—
226	C₄²	C₂²×#64408	1	16	yes	—
227	C₂³	—	73	24	no	—
228	C₂³	—	73	24	no	—
229	C₂³	—	73	24	no	—
230	C₂³	—	73	4	no	—
231	C₂³	—	73	64	no	—
232	C₂³	C₂×C₄⋊#64408	1	4	yes	—
233	C₂³	C₂×#11924036	1	4	yes	—
234	C₂³	D₇₃⋊M₁₆	1	4	yes	—
235	C₂³	C₂³×#64408	1	3	yes	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	73	6	no	—
238	C₂×C₄	—	73	12	no	—
239	C₂×C₄	—	73	6	no	—
240	C₂×C₄	—	73	24	no	—
241	C₂×C₄	—	73	1	no	—
242	C₂×C₄	—	73	3	no	—
243	C₂×C₄	—	73	4	no	—
244	C₂×C₄	—	73	8	no	—
245	C₂×C₄	—	73	48	no	—
246	C₂×C₄	—	73	8	no	—
247	C₂×C₄	—	73	64	no	—
248	C₂×C₄	—	73	4	no	—
249	C₂×C₄	—	146	32	no	—
250	C₂×C₄	—	146	12	no	—
251	C₂×C₄	C₂×C₄⋊#64408	1	24	yes	—
252	C₂×C₄	C₂×#11924036	1	24	yes	complemented
253	C₂×C₄	C₂³×#64408	1	4	yes	—
254	C₂×C₄	C₂³×#64408	1	4	yes	—
255	C₂×C₄	C₂³×#64408	1	4	yes	—
256	C₈	—	146	16	no	—
257	C₂²	—	73	96	no	—
258	C₂²	—	73	8	no	—
259	C₂²	—	73	8	no	—
260	C₂²	—	73	8	no	—
261	C₂²	#17989588	1	16	yes	direct product factor
262	C₂²	C₂²×C₄⋊#64408	1	6	yes	—
263	C₂²	C₂²×#11924036	1	6	yes	—
264	C₂²	C₂×D₇₃⋊M₁₆	1	6	yes	—
265	C₂²	C₂⁴×#64408	1	1	yes	characteristic
266	C₄	—	2	4	no	—
267	C₄	—	73	1	no	—
268	C₄	—	73	6	no	—
269	C₄	—	73	1	no	—
270	C₄	—	73	8	no	—
271	C₄	—	146	4	no	—
272	C₄	C₂²×C₄⋊#64408	1	8	yes	—
273	C₄	C₂²×#11924036	1	8	yes	complemented
274	C₂	—	73	16	no	—
275	C₂	C₂×#17989588	1	12	yes	direct product factor
276	C₂	C₂³×C₄⋊#64408	1	1	yes	characteristic
277	C₂	C₂³×#11924036	1	1	yes	characteristic
278	C₂	C₂²×D₇₃⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#64408
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#64408
Fitting	C₂²×C₄²×C₇₃	C₄
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₂×Dic₇₃	C₂⁵
hypercenter	C₂²×C₄²	#64408
hypocenter	C₇₃	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×#14556658	C₂	semidirect
3. C₂×#17989588	C₂	direct (N × H)
4. C₂²×C₄⋊#10596833	C₂	semidirect
9. C₂×#14556658	C₂²	semidirect
10. #17989588	C₂²	direct (N × H)
11. C₂×C₄⋊#10596833	C₂²	semidirect
25. #14556658	C₂³	semidirect
26. C₄⋊#10596833	C₂³	semidirect
32. C₂²×C₄×D₇₃	C₈	semidirect
58. C₂²×Dic₇₃	C₂×C₈	semidirect
67. C₂×C₄×D₇₃	C₂×C₈	semidirect
82. C₂²×C₄×C₇₃	C₂×C₈	semidirect
91. C₂²×D₇₃	C₄⋊C₈	semidirect
98. C₂×C₄×C₇₃	C₂²×C₈	semidirect
104. C₂×Dic₇₃	C₂²×C₈	semidirect
113. C₄×D₇₃	C₂²×C₈	semidirect
117. C₂²×C₇₃	C₂×C₄⋊C₈	semidirect
122. C₂×D₇₃	C₂×C₄⋊C₈	semidirect
128. C₄×C₇₃	C₂³×C₈	semidirect
131. Dic₇₃	C₂³×C₈	semidirect
135. C₂×C₇₃	C₂²×C₄⋊C₈	semidirect
139. D₇₃	C₂²×C₄⋊C₈	semidirect
145. C₇₃	C₂³×C₄⋊C₈	semidirect
214. C₂²×C₄	#11924036	semidirect
252. C₂×C₄	C₂×#11924036	semidirect
261. C₂²	#17989588	direct (N × H)
273. C₄	C₂²×#11924036	semidirect
275. C₂	C₂×#17989588	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×#11924036	2
C₂²×#14556658	2
C₂×#17989588	2
C₂²×C₄⋊#10596833	2
C₂²×C₄²×D₇₃	2
C₂³×C₄⋊C₈	73

Subgroups — C22×#17989588 #34031184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989588 #34031184