Subgroups — C₂²×#17988438 #33244161

Order 18688.

5758 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924007	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₇₃	C₂	1	2	yes	maximal
3	C₂³×C₄×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988438	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#14556498	C₂	1	1	yes	maximal, characteristic
6	C₂²×#95×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₇₃	C₂²	1	2	yes	—
8	C₂⁴×Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂²×#11924007	C₄	1	32	yes	complemented
11	C₂²×#11924007	C₂²	1	24	yes	—
12	C₂²×C₄×Dic₇₃	—	2	16	no	—
13	C₂²×C₄×Dic₇₃	C₄	1	16	yes	complemented
14	C₂²×C₄×Dic₇₃	C₂²	1	24	yes	—
15	C₂²×C₄×Dic₇₃	C₂²	1	12	yes	—
16	#17988438	C₂²	1	64	yes	direct product factor
17	C₂×#14556498	C₂²	1	12	yes	—
18	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₇₃	C₂²	1	2	yes	—
20	C₂×#95×C₇₃	C₂²	1	12	yes	—
21	C₂³×Dic₇₃	—	2	8	no	—
22	C₂³×Dic₇₃	—	2	8	no	—
23	C₂³×Dic₇₃	C₂×C₄	1	16	yes	—
24	C₂³×Dic₇₃	C₂×C₄	1	8	yes	—
25	C₂³×Dic₇₃	C₂×C₄	1	16	yes	—
26	C₂³×Dic₇₃	C₂×C₄	1	8	yes	—
27	C₂³×Dic₇₃	D₄	1	8	yes	—
28	C₂³×Dic₇₃	Q₈	1	8	yes	—
29	C₂³×Dic₇₃	C₂³	1	12	yes	—
30	C₂³×Dic₇₃	C₂³	1	6	yes	—
31	C₂³×Dic₇₃	C₂³	1	6	yes	—
32	C₂×#11924007	C₂×C₄	1	192	yes	complemented
33	C₂×#11924007	C₂³	1	32	yes	—
34	C₂×C₄×Dic₇₃	—	2	96	no	—
35	C₂×C₄×Dic₇₃	C₂×C₄	1	96	yes	complemented
36	C₂×C₄×Dic₇₃	C₂³	1	32	yes	—
37	C₂×C₄×Dic₇₃	C₂³	1	16	yes	—
38	#14556498	C₂³	1	16	yes	—
39	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₇₃	—	2	8	no	—
41	C₂³×C₄×C₇₃	—	2	8	no	—
42	C₂³×C₄×C₇₃	—	2	8	no	—
43	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₇₃	C₂×C₄	1	16	yes	—
45	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
46	C₂³×C₄×C₇₃	C₂³	1	12	yes	—
47	#95×C₇₃	C₂³	1	16	yes	—
48	C₂²×Dic₇₃	—	2	64	no	—
49	C₂²×Dic₇₃	—	2	24	no	—
50	C₂²×Dic₇₃	—	2	24	no	—
51	C₂²×Dic₇₃	C₄²	1	64	yes	complemented
52	C₂²×Dic₇₃	C₄⋊C₄	1	64	yes	complemented
53	C₂²×Dic₇₃	C₂²×C₄	1	48	yes	—
54	C₂²×Dic₇₃	C₂²×C₄	1	24	yes	—
55	C₂²×Dic₇₃	C₂²×C₄	1	48	yes	—
56	C₂²×Dic₇₃	C₂²×C₄	1	24	yes	—
57	C₂²×Dic₇₃	C₂×D₄	1	24	yes	—
58	C₂²×Dic₇₃	C₂×Q₈	1	24	yes	—
59	C₂²×Dic₇₃	C₂⁴	1	8	yes	—
60	C₂²×Dic₇₃	C₂⁴	1	4	yes	—
61	C₂²×Dic₇₃	C₂⁴	1	4	yes	—
62	#11924007	C₂²×C₄	1	128	yes	complemented
63	C₄×Dic₇₃	—	2	64	no	—
64	C₄×Dic₇₃	C₂²×C₄	1	64	yes	complemented
65	C₂⁴×C₇₃	C₂²×C₄	1	4	yes	—
66	C₂⁴×C₇₃	C₂²×C₄	1	8	yes	—
67	C₂⁴×C₇₃	C₂×D₄	1	4	yes	—
68	C₂⁴×C₇₃	C₂×Q₈	1	4	yes	—
69	C₂⁴×C₇₃	#41	1	8	yes	—
70	C₂⁴×C₇₃	C₂⁴	1	3	yes	—
71	C₂²×C₄×C₇₃	—	2	32	no	—
72	C₂²×C₄×C₇₃	—	2	64	no	—
73	C₂²×C₄×C₇₃	—	2	24	no	—
74	C₂²×C₄×C₇₃	—	2	24	no	—
75	C₂²×C₄×C₇₃	—	2	24	no	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	48	yes	—
78	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₇₃	C₂⁴	1	8	yes	—
80	C₂³×C₇₃	C₂×C₄²	1	16	yes	—
81	C₂³×C₇₃	C₂×C₄⋊C₄	1	16	yes	—
82	C₂³×C₇₃	#117	1	16	yes	—
83	C₂³×C₇₃	C₄×D₄	1	32	yes	—
84	C₂³×C₇₃	C₄×Q₈	1	32	yes	—
85	C₂³×C₇₃	C₂³×C₄	1	6	yes	—
86	C₂³×C₇₃	C₂³×C₄	1	12	yes	—
87	C₂³×C₇₃	C₂²×D₄	1	6	yes	—
88	C₂³×C₇₃	C₂²×Q₈	1	6	yes	—
89	C₂³×C₇₃	C₂×#41	1	12	yes	—
90	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
91	C₂×C₄×C₇₃	—	2	48	no	—
92	C₂×C₄×C₇₃	—	2	96	no	—
93	C₂×C₄×C₇₃	—	2	8	no	—
94	C₂×C₄×C₇₃	—	2	8	no	—
95	C₂×C₄×C₇₃	—	2	8	no	—
96	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₇₃	C₂³×C₄	1	16	yes	—
98	C₂×Dic₇₃	—	2	96	no	—
99	C₂×Dic₇₃	—	2	8	no	—
100	C₂×Dic₇₃	—	2	8	no	—
101	C₂×Dic₇₃	C₂×C₄²	1	96	yes	complemented
102	C₂×Dic₇₃	C₂×C₄⋊C₄	1	96	yes	complemented
103	C₂×Dic₇₃	C₂³×C₄	1	16	yes	—
104	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
105	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
106	C₂×Dic₇₃	C₂³×C₄	1	16	yes	—
107	C₂×Dic₇₃	C₂²×D₄	1	8	yes	—
108	C₂×Dic₇₃	C₂²×Q₈	1	8	yes	—
109	C₂²×C₇₃	C₄×C₄⋊C₄	1	64	yes	complemented
110	C₂²×C₇₃	C₂²×C₄²	1	12	yes	—
111	C₂²×C₇₃	C₂²×C₄⋊C₄	1	12	yes	—
112	C₂²×C₇₃	C₂×#117	1	12	yes	—
113	C₂²×C₇₃	C₂×C₄×D₄	1	24	yes	—
114	C₂²×C₇₃	C₂×C₄×Q₈	1	24	yes	—
115	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
116	C₂²×C₇₃	C₂⁴×C₄	1	2	yes	—
117	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
118	C₂²×C₇₃	C₂³×Q₈	1	1	yes	characteristic
119	C₂²×C₇₃	C₂²×#41	1	2	yes	—
120	C₄×C₇₃	—	2	8	no	—
121	C₄×C₇₃	—	2	16	no	—
122	Dic₇₃	—	2	16	no	—
123	Dic₇₃	C₂²×C₄²	1	16	yes	complemented
124	Dic₇₃	C₂²×C₄⋊C₄	1	16	yes	complemented
125	C₂²×C₄×C₄⋊C₄	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	24	yes	complemented
127	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
128	C₂×C₇₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₇₃	C₂²×#117	1	1	yes	characteristic
130	C₂×C₇₃	C₂²×C₄×D₄	1	2	yes	—
131	C₂×C₇₃	C₂²×C₄×Q₈	1	2	yes	—
132	C₂³×C₄⋊C₄	—	73	2	no	—
133	C₂³×C₄²	—	73	1	no	—
134	C₂³×C₄²	—	73	2	no	—
135	C₂×C₄×C₄⋊C₄	—	73	24	no	—
136	C₂²×#95	—	73	1	no	—
137	C₂²×#95	D₇₃	1	1	yes	characteristic
138	C₇₃	C₂²×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴×C₄	—	73	1	no	—
140	C₂⁴×C₄	—	73	1	no	—
141	C₂⁴×C₄	—	73	2	no	—
142	C₂⁴×C₄	C₂×D₇₃	1	2	yes	—
143	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
144	C₂²×C₄⋊C₄	—	73	32	no	—
145	C₂²×C₄⋊C₄	—	73	24	no	—
146	C₂²×C₄²	—	73	16	no	—
147	C₂²×C₄²	—	73	12	no	—
148	C₂²×C₄²	—	73	24	no	—
149	C₂²×C₄²	—	146	16	no	—
150	C₄×C₄⋊C₄	—	73	64	no	—
151	C₂×#95	—	73	12	no	—
152	C₂×#95	C₂×D₇₃	1	12	yes	—
153	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	73	16	no	—
158	C₂³×C₄	—	73	8	no	—
159	C₂³×C₄	—	73	8	no	—
160	C₂³×C₄	—	73	8	no	—
161	C₂³×C₄	—	73	8	no	—
162	C₂³×C₄	—	73	6	no	—
163	C₂³×C₄	—	73	6	no	—
164	C₂³×C₄	—	73	16	no	—
165	C₂³×C₄	—	73	12	no	—
166	C₂³×C₄	—	146	8	no	—
167	C₂³×C₄	—	146	8	no	—
168	C₂³×C₄	C₄×D₇₃	1	16	yes	—
169	C₂³×C₄	C₄×D₇₃	1	8	yes	—
170	C₂³×C₄	C₂²×D₇₃	1	12	yes	—
171	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
172	C₂×C₄⋊C₄	—	73	192	no	—
173	C₂×C₄⋊C₄	—	73	32	no	—
174	C₂×C₄²	—	73	96	no	—
175	C₂×C₄²	—	73	16	no	—
176	C₂×C₄²	—	73	32	no	—
177	C₂×C₄²	—	146	96	no	—
178	#95	—	73	16	no	—
179	#95	C₂²×D₇₃	1	16	yes	—
180	C₂⁴	C₂×C₄×D₇₃	1	4	yes	—
181	C₂⁴	C₂×C₄×D₇₃	1	8	yes	—
182	C₂⁴	D₄×D₇₃	1	4	yes	—
183	C₂⁴	Q₈×D₇₃	1	4	yes	—
184	C₂⁴	#11924027	1	8	yes	—
185	C₂⁴	C₂³×D₇₃	1	3	yes	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	2	64	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	2	24	no	—
190	C₂²×C₄	—	2	24	no	—
191	C₂²×C₄	—	73	64	no	—
192	C₂²×C₄	—	73	64	no	—
193	C₂²×C₄	—	73	24	no	—
194	C₂²×C₄	—	73	24	no	—
195	C₂²×C₄	—	73	48	no	—
196	C₂²×C₄	—	73	24	no	—
197	C₂²×C₄	—	73	24	no	—
198	C₂²×C₄	—	73	4	no	—
199	C₂²×C₄	—	73	4	no	—
200	C₂²×C₄	—	73	48	no	—
201	C₂²×C₄	—	73	8	no	—
202	C₂²×C₄	—	146	64	no	—
203	C₂²×C₄	—	146	24	no	—
204	C₂²×C₄	—	146	24	no	—
205	C₂²×C₄	C₂×C₄×D₇₃	1	24	yes	—
206	C₂²×C₄	C₂×C₄×D₇₃	1	48	yes	—
207	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
208	C₂²×C₄	C₂³×D₇₃	1	8	yes	—
209	C₄⋊C₄	—	73	128	no	—
210	C₄²	—	73	64	no	—
211	C₄²	—	146	64	no	—
212	C₂³	C₄²×D₇₃	1	16	yes	—
213	C₂³	C₄⋊C₄×D₇₃	1	16	yes	—
214	C₂³	#14556547	1	16	yes	—
215	C₂³	#14556551	1	32	yes	—
216	C₂³	#14556556	1	32	yes	—
217	C₂³	C₂²×C₄×D₇₃	1	6	yes	—
218	C₂³	C₂²×C₄×D₇₃	1	12	yes	—
219	C₂³	C₂×D₄×D₇₃	1	6	yes	—
220	C₂³	C₂×Q₈×D₇₃	1	6	yes	—
221	C₂³	C₂×#11924027	1	12	yes	—
222	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	48	no	—
227	C₂×C₄	—	2	96	no	—
228	C₂×C₄	—	73	8	no	—
229	C₂×C₄	—	73	8	no	—
230	C₂×C₄	—	73	16	no	—
231	C₂×C₄	—	73	8	no	—
232	C₂×C₄	—	73	8	no	—
233	C₂×C₄	—	73	96	no	—
234	C₂×C₄	—	73	96	no	—
235	C₂×C₄	—	73	16	no	—
236	C₂×C₄	—	146	8	no	—
237	C₂×C₄	—	146	8	no	—
238	C₂×C₄	—	146	96	no	—
239	C₂×C₄	C₂²×C₄×D₇₃	1	8	yes	—
240	C₂×C₄	C₂²×C₄×D₇₃	1	16	yes	—
241	C₂²	#17988438	1	64	yes	direct product factor
242	C₂²	C₂×C₄²×D₇₃	1	12	yes	—
243	C₂²	C₂×C₄⋊C₄×D₇₃	1	12	yes	—
244	C₂²	C₂×#14556547	1	12	yes	—
245	C₂²	C₂×#14556551	1	24	yes	—
246	C₂²	C₂×#14556556	1	24	yes	—
247	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
248	C₂²	C₂³×C₄×D₇₃	1	2	yes	—
249	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
250	C₂²	C₂²×Q₈×D₇₃	1	1	yes	characteristic
251	C₂²	C₂²×#11924027	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	2	16	no	—
254	C₄	—	73	16	no	—
255	C₄	—	73	16	no	—
256	C₄	—	146	16	no	—
257	C₂	C₂×#17988438	1	24	yes	direct product factor
258	C₂	C₂²×C₄²×D₇₃	1	1	yes	characteristic
259	C₂	C₂²×C₄⋊C₄×D₇₃	1	1	yes	characteristic
260	C₂	C₂²×#14556547	1	1	yes	characteristic
261	C₂	C₂²×#14556551	1	2	yes	—
262	C₂	C₂²×#14556556	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₇₃
commutator	C₂×C₇₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₇₃
Fitting	C₂²×#95×C₇₃	C₂
socle, prime radical	C₂⁵×C₇₃	C₂³
cosocle	C₂³×C₇₃	C₂⁵
hypercenter	C₂²×#95	D₇₃
hypocenter	C₇₃	C₂²×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁵ ◁ C₂²×#95

Complements

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

N	H ≅ G/N	type
4. C₂×#17988438	C₂	direct (N × H)
10. C₂²×#11924007	C₄	semidirect
13. C₂²×C₄×Dic₇₃	C₄	semidirect
16. #17988438	C₂²	direct (N × H)
32. C₂×#11924007	C₂×C₄	semidirect
35. C₂×C₄×Dic₇₃	C₂×C₄	semidirect
51. C₂²×Dic₇₃	C₄²	semidirect
52. C₂²×Dic₇₃	C₄⋊C₄	semidirect
62. #11924007	C₂²×C₄	semidirect
64. C₄×Dic₇₃	C₂²×C₄	semidirect
101. C₂×Dic₇₃	C₂×C₄²	semidirect
102. C₂×Dic₇₃	C₂×C₄⋊C₄	semidirect
109. C₂²×C₇₃	C₄×C₄⋊C₄	semidirect
123. Dic₇₃	C₂²×C₄²	semidirect
124. Dic₇₃	C₂²×C₄⋊C₄	semidirect
126. C₂×C₇₃	C₂×C₄×C₄⋊C₄	semidirect
138. C₇₃	C₂²×C₄×C₄⋊C₄	semidirect
241. C₂²	#17988438	direct (N × H)
257. C₂	C₂×#17988438	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×#11924007	2
C₂³×C₄×Dic₇₃	2
C₂×#17988438	2
C₂²×#14556498	2
C₂²×#95×C₇₃	2
C₂²×C₄×C₄⋊C₄	73

Subgroups — C22×#17988438 #33244161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988438 #33244161