Subgroups — C₂²×#17989453 #34019808

Order 18688.

3086 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#10596833	C₂	1	2	yes	maximal
2	C₂×#17989453	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10596833	—	2	8	no	—
5	C₂³×#10596833	—	2	8	no	—
6	C₂³×#10596833	C₄	1	16	yes	complemented
7	C₂³×#10596833	C₂²	1	12	yes	—
8	#17989453	C₂²	1	16	yes	direct product factor
9	C₂⁴×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂⁴×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₇₃	—	2	8	no	—
12	C₂²×C₄×Dic₇₃	C₄	1	6	yes	—
13	C₂²×C₄×Dic₇₃	C₂²	1	6	yes	—
14	C₂⁴×C₄×C₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596833	—	2	64	no	—
16	C₂²×#10596833	—	2	24	no	—
17	C₂²×#10596833	—	2	24	no	—
18	C₂²×#10596833	C₂×C₄	1	48	yes	complemented
19	C₂²×#10596833	C₂³	1	8	yes	—
20	C₂³×Dic₇₃	—	2	4	no	—
21	C₂³×Dic₇₃	—	2	4	no	—
22	C₂³×Dic₇₃	C₈	1	8	yes	complemented
23	C₂³×Dic₇₃	C₂×C₄	1	8	yes	—
24	C₂³×Dic₇₃	C₂×C₄	1	6	yes	—
25	C₂³×Dic₇₃	C₂×C₄	1	3	yes	—
26	C₂³×Dic₇₃	D₄	1	8	yes	—
27	C₂³×Dic₇₃	D₄	1	4	yes	—
28	C₂³×Dic₇₃	Q₈	1	4	yes	—
29	C₂³×Dic₇₃	C₂³	1	3	yes	—
30	C₂×C₄×Dic₇₃	—	2	24	no	—
31	C₂×C₄×Dic₇₃	—	2	24	no	—
32	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
34	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₃	—	2	4	no	—
36	C₂³×C₄×C₇₃	—	2	4	no	—
37	C₂³×C₄×C₇₃	C₈	1	8	yes	complemented
38	C₂³×C₄×C₇₃	C₂×C₄	1	6	yes	—
39	C₂×#10596833	—	2	96	no	—
40	C₂×#10596833	—	2	8	no	—
41	C₂×#10596833	—	2	8	no	—
42	C₂×#10596833	C₂²×C₄	1	16	yes	complemented
43	C₂²×Dic₇₃	—	2	32	no	—
44	C₂²×Dic₇₃	—	2	12	no	—
45	C₂²×Dic₇₃	—	2	12	no	—
46	C₂²×Dic₇₃	C₄²	1	12	yes	—
47	C₂²×Dic₇₃	C₂²⋊C₄	1	12	yes	—
48	C₂²×Dic₇₃	C₂²⋊C₄	1	32	yes	—
49	C₂²×Dic₇₃	C₄⋊C₄	1	12	yes	—
50	C₂²×Dic₇₃	C₄⋊C₄	1	32	yes	—
51	C₂²×Dic₇₃	C₂×C₈	1	24	yes	complemented
52	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
53	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
54	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
55	C₂²×Dic₇₃	C₂×D₄	1	6	yes	—
56	C₂²×Dic₇₃	C₂×D₄	1	12	yes	—
57	C₂²×Dic₇₃	C₂×Q₈	1	6	yes	—
58	C₂²×Dic₇₃	C₂⁴	1	1	yes	characteristic
59	C₄×Dic₇₃	—	2	24	no	—
60	C₄×Dic₇₃	—	2	8	no	—
61	C₂⁴×C₇₃	C₄²	1	4	yes	—
62	C₂⁴×C₇₃	C₂²⋊C₄	1	4	yes	—
63	C₂⁴×C₇₃	C₄⋊C₄	1	4	yes	—
64	C₂⁴×C₇₃	C₂×C₈	1	4	yes	—
65	C₂⁴×C₇₃	C₂×C₈	1	4	yes	—
66	C₂⁴×C₇₃	M₁₆	1	4	yes	—
67	C₂⁴×C₇₃	M₁₆	1	4	yes	—
68	C₂⁴×C₇₃	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₇₃	—	2	32	no	—
70	C₂²×C₄×C₇₃	—	2	12	no	—
71	C₂²×C₄×C₇₃	—	2	12	no	—
72	C₂²×C₄×C₇₃	C₂×C₈	1	24	yes	complemented
73	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
74	C₂³×C₇₃	#95	1	16	yes	—
75	C₂³×C₇₃	C₄×C₈	1	16	yes	—
76	C₂³×C₇₃	#97	1	16	yes	—
77	C₂³×C₇₃	C₂²⋊C₈	1	16	yes	—
78	C₂³×C₇₃	C₂²⋊C₈	1	16	yes	—
79	C₂³×C₇₃	C₄⋊C₈	1	16	yes	—
80	C₂³×C₇₃	C₄⋊C₈	1	16	yes	—
81	C₂³×C₇₃	C₂×C₄²	1	6	yes	—
82	C₂³×C₇₃	C₂×C₂²⋊C₄	1	6	yes	—
83	C₂³×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
84	C₂³×C₇₃	C₂²×C₈	1	6	yes	—
85	C₂³×C₇₃	C₂²×C₈	1	6	yes	—
86	C₂³×C₇₃	C₂×M₁₆	1	6	yes	—
87	C₂³×C₇₃	C₂×M₁₆	1	6	yes	—
88	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₃	—	2	4	no	—
90	C₂×C₄×C₇₃	—	2	4	no	—
91	C₂×C₄×C₇₃	—	2	48	no	—
92	C₂×C₄×C₇₃	C₂²×C₈	1	8	yes	complemented
93	C₂×Dic₇₃	—	2	4	no	—
94	C₂×Dic₇₃	—	2	4	no	—
95	C₂×Dic₇₃	—	2	48	no	—
96	C₂×Dic₇₃	#95	1	48	yes	—
97	C₂×Dic₇₃	C₂×C₄²	1	6	yes	—
98	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	24	yes	—
99	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	6	yes	—
100	C₂×Dic₇₃	C₂×C₄⋊C₄	1	24	yes	—
101	C₂×Dic₇₃	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×Dic₇₃	C₂²×C₈	1	8	yes	complemented
103	C₂×Dic₇₃	C₂³×C₄	1	2	yes	—
104	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂²×D₄	1	2	yes	—
106	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
107	#10596833	—	2	16	no	—
108	C₂²×C₇₃	#336	1	64	yes	complemented
109	C₂²×C₇₃	C₂×#95	1	12	yes	—
110	C₂²×C₇₃	C₂×C₄×C₈	1	12	yes	—
111	C₂²×C₇₃	C₂×#97	1	12	yes	—
112	C₂²×C₇₃	C₂×C₂²⋊C₈	1	12	yes	—
113	C₂²×C₇₃	C₂×C₂²⋊C₈	1	12	yes	—
114	C₂²×C₇₃	C₂×C₄⋊C₈	1	12	yes	—
115	C₂²×C₇₃	C₂×C₄⋊C₈	1	12	yes	—
116	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
117	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
120	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
121	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
122	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
123	C₄×C₇₃	—	2	8	no	—
124	Dic₇₃	—	2	8	no	—
125	Dic₇₃	C₂×#95	1	12	yes	—
126	Dic₇₃	C₂²×C₂²⋊C₄	1	2	yes	—
127	Dic₇₃	C₂²×C₄⋊C₄	1	2	yes	—
128	C₂²×#336	—	73	1	no	maximal, Sylow
129	C₂×C₇₃	C₂×#336	1	24	yes	complemented
130	C₂×C₇₃	C₂²×#95	1	1	yes	characteristic
131	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
132	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
133	C₂×C₇₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₇₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
137	C₂⁴×C₈	—	73	2	no	—
138	C₂³×C₄²	—	73	1	no	—
139	C₂×#336	—	73	12	no	—
140	C₇₃	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
141	C₂⁴×C₄	—	73	1	no	—
142	C₂⁴×C₄	—	73	1	no	—
143	C₂⁴×C₄	#64408	1	1	yes	characteristic
144	C₂³×C₈	—	73	16	no	—
145	C₂³×C₈	—	73	12	no	—
146	C₂³×C₈	—	146	8	no	—
147	C₂³×C₈	—	146	8	no	—
148	C₂²×C₄²	—	73	6	no	—
149	C₂²×C₄²	—	73	6	no	—
150	C₂²×C₄²	—	146	8	no	—
151	#336	—	73	16	no	—
152	C₂⁵	C₂×#64408	1	1	yes	characteristic
153	C₂³×C₄	—	2	4	no	—
154	C₂³×C₄	—	2	4	no	—
155	C₂³×C₄	—	73	8	no	—
156	C₂³×C₄	—	73	8	no	—
157	C₂³×C₄	—	73	4	no	—
158	C₂³×C₄	—	73	4	no	—
159	C₂³×C₄	—	73	3	no	—
160	C₂³×C₄	—	73	3	no	—
161	C₂³×C₄	—	73	8	no	—
162	C₂³×C₄	—	73	6	no	—
163	C₂³×C₄	—	146	4	no	—
164	C₂³×C₄	—	146	4	no	—
165	C₂³×C₄	#10596833	1	8	yes	complemented
166	C₂³×C₄	C₂×#64408	1	6	yes	—
167	C₂²×C₈	—	73	48	no	—
168	C₂²×C₈	—	73	8	no	—
169	C₂²×C₈	—	146	64	no	—
170	C₂²×C₈	—	146	24	no	—
171	C₂²×C₈	—	146	24	no	—
172	C₂×C₄²	—	73	4	no	—
173	C₂×C₄²	—	73	12	no	—
174	C₂×C₄²	—	146	24	no	—
175	C₂×C₄²	—	146	24	no	—
176	C₂⁴	C₄×#64408	1	4	yes	—
177	C₂⁴	C₂²⋊#64408	1	4	yes	—
178	C₂⁴	C₄⋊#64408	1	4	yes	—
179	C₂⁴	C₂×#10596833	1	4	yes	—
180	C₂⁴	#11924036	1	4	yes	—
181	C₂⁴	#11924037	1	4	yes	—
182	C₂⁴	#11924038	1	4	yes	—
183	C₂⁴	C₂²×#64408	1	3	yes	—
184	C₂²×C₄	—	2	12	no	—
185	C₂²×C₄	—	2	32	no	—
186	C₂²×C₄	—	2	12	no	—
187	C₂²×C₄	—	73	3	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂²×C₄	—	73	32	no	—
190	C₂²×C₄	—	73	32	no	—
191	C₂²×C₄	—	73	12	no	—
192	C₂²×C₄	—	73	12	no	—
193	C₂²×C₄	—	73	12	no	—
194	C₂²×C₄	—	73	12	no	—
195	C₂²×C₄	—	73	12	no	—
196	C₂²×C₄	—	73	6	no	—
197	C₂²×C₄	—	73	6	no	—
198	C₂²×C₄	—	73	24	no	—
199	C₂²×C₄	—	73	4	no	—
200	C₂²×C₄	—	146	12	no	—
201	C₂²×C₄	—	146	32	no	—
202	C₂²×C₄	—	146	12	no	—
203	C₂²×C₄	C₂×#10596833	1	24	yes	complemented
204	C₂²×C₄	C₂²×#64408	1	4	yes	—
205	C₂×C₈	—	73	16	no	—
206	C₂×C₈	—	146	8	no	—
207	C₂×C₈	—	146	8	no	—
208	C₂×C₈	—	146	96	no	—
209	C₄²	—	146	8	no	—
210	C₄²	—	146	24	no	—
211	C₂³	(C₂×#64408)⋊C₄	1	16	yes	—
212	C₂³	C₄×#10596833	1	16	yes	—
213	C₂³	#14556646	1	16	yes	—
214	C₂³	C₂²⋊#10596833	1	16	yes	—
215	C₂³	#14556648	1	16	yes	—
216	C₂³	C₄⋊#10596833	1	16	yes	—
217	C₂³	#14556658	1	16	yes	—
218	C₂³	C₂×C₄×#64408	1	6	yes	—
219	C₂³	C₂×C₂²⋊#64408	1	6	yes	—
220	C₂³	C₂×C₄⋊#64408	1	6	yes	—
221	C₂³	C₂²×#10596833	1	6	yes	—
222	C₂³	C₂×#11924036	1	6	yes	—
223	C₂³	C₂×#11924037	1	6	yes	—
224	C₂³	C₂×#11924038	1	6	yes	—
225	C₂³	C₂³×#64408	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	73	2	no	—
230	C₂×C₄	—	73	6	no	—
231	C₂×C₄	—	73	2	no	—
232	C₂×C₄	—	73	6	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	6	no	—
235	C₂×C₄	—	73	1	no	—
236	C₂×C₄	—	73	24	no	—
237	C₂×C₄	—	73	48	no	—
238	C₂×C₄	—	73	24	no	—
239	C₂×C₄	—	73	8	no	—
240	C₂×C₄	—	146	4	no	—
241	C₂×C₄	—	146	4	no	—
242	C₂×C₄	—	146	48	no	—
243	C₂×C₄	C₂²×#10596833	1	8	yes	complemented
244	C₈	—	146	16	no	—
245	C₂²	#17989453	1	64	yes	direct product factor
246	C₂²	C₂×(C₂×#64408)⋊C₄	1	12	yes	—
247	C₂²	C₂×C₄×#10596833	1	12	yes	—
248	C₂²	C₂×#14556646	1	12	yes	—
249	C₂²	C₂×C₂²⋊#10596833	1	12	yes	—
250	C₂²	C₂×#14556648	1	12	yes	—
251	C₂²	C₂×C₄⋊#10596833	1	12	yes	—
252	C₂²	C₂×#14556658	1	12	yes	—
253	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
254	C₂²	C₂²×C₂²⋊#64408	1	1	yes	characteristic
255	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
256	C₂²	C₂³×#10596833	1	1	yes	characteristic
257	C₂²	C₂²×#11924036	1	1	yes	characteristic
258	C₂²	C₂²×#11924037	1	1	yes	characteristic
259	C₂²	C₂²×#11924038	1	1	yes	characteristic
260	C₄	—	2	8	no	—
261	C₄	—	73	2	no	—
262	C₄	—	73	2	no	—
263	C₄	—	73	12	no	—
264	C₄	—	146	8	no	—
265	C₂	C₂×#17989453	1	24	yes	direct product factor
266	C₂	C₂²×(C₂×#64408)⋊C₄	1	1	yes	characteristic
267	C₂	C₂²×C₄×#10596833	1	1	yes	characteristic
268	C₂	C₂²×#14556646	1	1	yes	characteristic
269	C₂	C₂²×C₂²⋊#10596833	1	1	yes	characteristic
270	C₂	C₂²×#14556648	1	1	yes	characteristic
271	C₂	C₂²×C₄⋊#10596833	1	1	yes	characteristic
272	C₂	C₂²×#14556658	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₂³	C₂³×#64408
Fitting	C₂⁴×C₄×C₇₃	C₄
socle, prime radical	C₂⁵×C₇₃	C₂×C₄
cosocle	C₂²×Dic₇₃	C₂⁴
hypercenter	C₂⁴×C₄	#64408
hypocenter	C₇₃	C₂²×#336
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#17989453	C₂	direct (N × H)
6. C₂³×#10596833	C₄	semidirect
8. #17989453	C₂²	direct (N × H)
18. C₂²×#10596833	C₂×C₄	semidirect
22. C₂³×Dic₇₃	C₈	semidirect
37. C₂³×C₄×C₇₃	C₈	semidirect
42. C₂×#10596833	C₂²×C₄	semidirect
51. C₂²×Dic₇₃	C₂×C₈	semidirect
72. C₂²×C₄×C₇₃	C₂×C₈	semidirect
92. C₂×C₄×C₇₃	C₂²×C₈	semidirect
102. C₂×Dic₇₃	C₂²×C₈	semidirect
108. C₂²×C₇₃	#336	semidirect
129. C₂×C₇₃	C₂×#336	semidirect
140. C₇₃	C₂²×#336	semidirect
165. C₂³×C₄	#10596833	semidirect
203. C₂²×C₄	C₂×#10596833	semidirect
243. C₂×C₄	C₂²×#10596833	semidirect
245. C₂²	#17989453	direct (N × H)
265. C₂	C₂×#17989453	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#10596833	2
C₂×#17989453	2
C₂³×C₄×Dic₇₃	2
C₂²×#336	73

Subgroups — C22×#17989453 #34019808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989453 #34019808