Subgroups — C₄×#17988443 #33007192

Order 18688.

6734 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988443	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊Dic₇₃	C₂	1	6	yes	maximal, complemented
3	#22608154	C₂	1	6	yes	maximal, complemented
4	C₂⁵×C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂²⋊Dic₇₃	C₂²	1	12	yes	—
6	C₂²×C₄×Dic₇₃	C₂²	1	4	yes	complemented
7	#17988443	C₄	1	16	yes	direct product factor
8	C₄×C₂²⋊Dic₇₃	—	2	24	no	—
9	C₂×#14556498	C₂²	1	12	yes	complemented
10	C₂⁶×C₇₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₇₃	—	2	24	no	—
12	C₂⁴×C₄×C₇₃	C₄	1	8	yes	complemented
13	C₂⁴×C₄×C₇₃	C₂²	1	3	yes	—
14	C₂⁴×C₄×C₇₃	C₂²	1	3	yes	—
15	C₂³×Dic₇₃	C₂³	1	8	yes	—
16	C₂×C₂²⋊Dic₇₃	—	2	24	no	—
17	C₂×C₂²⋊Dic₇₃	—	2	24	no	—
18	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	48	yes	complemented
19	C₂×C₄×Dic₇₃	—	2	24	no	—
20	#14556498	—	2	24	no	—
21	C₂⁵×C₇₃	—	2	12	no	—
22	C₂⁵×C₇₃	—	2	12	no	—
23	C₂⁵×C₇₃	C₂×C₄	1	8	yes	—
24	C₂⁵×C₇₃	C₂×C₄	1	4	yes	—
25	C₂⁵×C₇₃	C₂³	1	3	yes	—
26	C₂³×C₄×C₇₃	—	2	128	no	—
27	C₂³×C₄×C₇₃	—	2	12	no	—
28	C₂³×C₄×C₇₃	—	2	96	no	—
29	C₂³×C₄×C₇₃	—	2	24	no	—
30	C₂³×C₄×C₇₃	—	2	24	no	—
31	C₂³×C₄×C₇₃	C₂×C₄	1	12	yes	—
32	C₂³×C₄×C₇₃	C₂×C₄	1	12	yes	—
33	C₂³×C₄×C₇₃	D₄	1	24	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
35	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₇₃	—	2	24	no	—
37	C₂²×Dic₇₃	—	2	24	no	—
38	C₂²×Dic₇₃	C₂²×C₄	1	16	yes	complemented
39	C₂²⋊Dic₇₃	—	2	96	no	—
40	C₄×Dic₇₃	—	4	16	no	—
41	C₂⁴×C₇₃	—	2	32	no	—
42	C₂⁴×C₇₃	—	2	96	no	—
43	C₂⁴×C₇₃	—	2	24	no	—
44	C₂⁴×C₇₃	—	2	24	no	—
45	C₂⁴×C₇₃	—	2	96	no	—
46	C₂⁴×C₇₃	—	2	12	no	—
47	C₂⁴×C₇₃	—	2	12	no	—
48	C₂⁴×C₇₃	C₄²	1	16	yes	complemented
49	C₂⁴×C₇₃	C₂²×C₄	1	12	yes	—
50	C₂⁴×C₇₃	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₇₃	C₂×D₄	1	12	yes	—
52	C₂⁴×C₇₃	#41	1	12	yes	—
53	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	—	2	96	no	—
55	C₂²×C₄×C₇₃	—	2	256	no	—
56	C₂²×C₄×C₇₃	—	2	24	no	—
57	C₂²×C₄×C₇₃	—	2	12	no	—
58	C₂²×C₄×C₇₃	—	2	48	no	—
59	C₂²×C₄×C₇₃	—	2	48	no	—
60	C₂²×C₄×C₇₃	—	2	96	no	—
61	C₂²×C₄×C₇₃	C₂²⋊C₄	1	48	yes	complemented
62	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₇₃	C₂×D₄	1	12	yes	—
66	C₂³×C₇₃	—	2	32	no	—
67	C₂³×C₇₃	—	2	96	no	—
68	C₂³×C₇₃	—	2	256	no	—
69	C₂³×C₇₃	—	2	12	no	—
70	C₂³×C₇₃	—	2	12	no	—
71	C₂³×C₇₃	—	2	48	no	—
72	C₂³×C₇₃	—	2	24	no	—
73	C₂³×C₇₃	—	2	24	no	—
74	C₂³×C₇₃	—	2	48	no	—
75	C₂³×C₇₃	—	2	96	no	—
76	C₂³×C₇₃	C₂×C₄²	1	12	yes	—
77	C₂³×C₇₃	C₂×C₂²⋊C₄	1	12	yes	—
78	C₂³×C₇₃	#117	1	12	yes	—
79	C₂³×C₇₃	C₄×D₄	1	48	yes	—
80	C₂³×C₇₃	C₂³×C₄	1	2	yes	—
81	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
82	C₂³×C₇₃	C₂²×D₄	1	6	yes	—
83	C₂³×C₇₃	C₂×#41	1	6	yes	—
84	C₂×C₄×C₇₃	—	2	96	no	—
85	C₂×C₄×C₇₃	—	2	12	no	—
86	C₂×C₄×C₇₃	—	2	24	no	—
87	C₂×C₄×C₇₃	—	2	96	no	—
88	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	12	yes	—
89	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	12	yes	—
90	C₂×C₄×C₇₃	#120	1	16	yes	—
91	C₂×Dic₇₃	—	2	48	no	—
92	C₂×Dic₇₃	—	4	8	no	—
93	C₂×Dic₇₃	—	4	24	no	—
94	C₂²×C₇₃	—	2	128	no	—
95	C₂²×C₇₃	—	2	24	no	—
96	C₂²×C₇₃	—	2	12	no	—
97	C₂²×C₇₃	—	2	12	no	—
98	C₂²×C₇₃	—	2	24	no	—
99	C₂²×C₇₃	—	2	96	no	—
100	C₂²×C₇₃	C₄×C₂²⋊C₄	1	24	yes	complemented
101	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	3	yes	—
103	C₂²×C₇₃	C₂×#117	1	3	yes	—
104	C₂²×C₇₃	C₂×C₄×D₄	1	12	yes	—
105	C₂²×C₇₃	C₂×#120	1	4	yes	—
106	C₂²×C₇₃	#525	1	12	yes	—
107	C₄×C₇₃	—	2	12	no	—
108	C₄×C₇₃	#379	1	8	yes	complemented
109	Dic₇₃	—	4	16	no	—
110	C₄×#379	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	—	2	24	no	—
112	C₂×C₇₃	C₂×C₄×C₂²⋊C₄	1	3	yes	—
113	C₂×C₇₃	C₂×#379	1	1	yes	characteristic
114	C₂×C₇₃	#2280	1	3	yes	—
115	C₂×C₇₃	C₄×#120	1	8	yes	—
116	C₂⁵×C₄	D₇₃	1	1	yes	characteristic
117	C₂×#379	—	73	2	no	—
118	C₂×C₄×C₂²⋊C₄	—	73	6	no	—
119	C₂²⋊#95	—	73	6	no	—
120	C₇₃	C₄×#379	1	1	yes	characteristic, complemented, Sylow
121	C₂⁶	C₂×D₇₃	1	1	yes	characteristic
122	C₂⁴×C₄	—	2	24	no	—
123	C₂⁴×C₄	Dic₇₃	1	8	yes	complemented
124	C₂⁴×C₄	C₂×D₇₃	1	3	yes	—
125	C₂⁴×C₄	C₂×D₇₃	1	3	yes	—
126	C₂²×C₂²⋊C₄	—	73	12	no	—
127	C₂²×C₄²	—	73	4	no	—
128	#379	—	73	16	no	—
129	C₄×C₂²⋊C₄	—	146	24	no	—
130	C₂×#95	—	73	12	no	—
131	C₂⁵	—	2	12	no	—
132	C₂⁵	—	2	12	no	—
133	C₂⁵	C₄×D₇₃	1	8	yes	—
134	C₂⁵	C₂×Dic₇₃	1	4	yes	—
135	C₂⁵	C₂²×D₇₃	1	3	yes	—
136	C₂³×C₄	—	2	128	no	—
137	C₂³×C₄	—	2	12	no	—
138	C₂³×C₄	—	2	96	no	—
139	C₂³×C₄	—	2	24	no	—
140	C₂³×C₄	—	2	24	no	—
141	C₂³×C₄	—	73	8	no	—
142	C₂³×C₄	C₂×Dic₇₃	1	12	yes	—
143	C₂³×C₄	C₂×Dic₇₃	1	12	yes	—
144	C₂³×C₄	C₂²⋊D₇₃	1	24	yes	—
145	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
146	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	—	73	48	no	—
148	C₂×C₂²⋊C₄	—	146	24	no	—
149	C₂×C₂²⋊C₄	—	146	24	no	—
150	C₂×C₄²	—	146	24	no	—
151	#95	—	146	24	no	—
152	C₂⁴	—	2	32	no	—
153	C₂⁴	—	2	96	no	—
154	C₂⁴	—	2	24	no	—
155	C₂⁴	—	2	24	no	—
156	C₂⁴	—	2	96	no	—
157	C₂⁴	—	2	12	no	—
158	C₂⁴	—	2	12	no	—
159	C₂⁴	C₄×Dic₇₃	1	16	yes	complemented
160	C₂⁴	C₂×C₄×D₇₃	1	12	yes	—
161	C₂⁴	C₂²×Dic₇₃	1	6	yes	—
162	C₂⁴	C₂×C₂²⋊D₇₃	1	12	yes	—
163	C₂⁴	#11924028	1	12	yes	—
164	C₂⁴	C₂³×D₇₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	256	no	—
167	C₂²×C₄	—	2	24	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	48	no	—
170	C₂²×C₄	—	2	48	no	—
171	C₂²×C₄	—	2	96	no	—
172	C₂²×C₄	—	73	16	no	—
173	C₂²×C₄	—	146	24	no	—
174	C₂²×C₄	—	146	24	no	—
175	C₂²×C₄	C₂²⋊Dic₇₃	1	48	yes	complemented
176	C₂²×C₄	C₂²×Dic₇₃	1	6	yes	—
177	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₇₃	1	12	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₇₃	1	12	yes	—
180	C₂²⋊C₄	—	146	96	no	—
181	C₄²	—	292	16	no	—
182	C₂³	—	2	32	no	—
183	C₂³	—	2	96	no	—
184	C₂³	—	2	256	no	—
185	C₂³	—	2	96	no	—
186	C₂³	—	2	12	no	—
187	C₂³	—	2	12	no	—
188	C₂³	—	2	48	no	—
189	C₂³	—	2	24	no	—
190	C₂³	—	2	24	no	—
191	C₂³	—	2	48	no	—
192	C₂³	C₂×C₄×Dic₇₃	1	12	yes	—
193	C₂³	C₂×C₂²⋊Dic₇₃	1	12	yes	—
194	C₂³	#14556550	1	12	yes	—
195	C₂³	C₄×C₂²⋊D₇₃	1	48	yes	—
196	C₂³	C₂²×C₄×D₇₃	1	2	yes	—
197	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
198	C₂³	C₂²×C₂²⋊D₇₃	1	6	yes	—
199	C₂³	C₂×#11924028	1	6	yes	—
200	C₂×C₄	—	2	96	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	24	no	—
203	C₂×C₄	—	2	96	no	—
204	C₂×C₄	—	292	8	no	—
205	C₂×C₄	—	292	24	no	—
206	C₂×C₄	—	146	48	no	—
207	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	12	yes	—
208	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	12	yes	—
209	C₂×C₄	#14556560	1	16	yes	—
210	C₂²	—	2	128	no	—
211	C₂²	—	2	24	no	—
212	C₂²	—	2	12	no	—
213	C₂²	—	2	12	no	—
214	C₂²	—	2	24	no	—
215	C₂²	—	2	96	no	—
216	C₂²	C₄×C₂²⋊Dic₇₃	1	24	yes	complemented
217	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
218	C₂²	C₂²×C₂²⋊Dic₇₃	1	3	yes	—
219	C₂²	C₂×#14556550	1	3	yes	—
220	C₂²	C₂×C₄×C₂²⋊D₇₃	1	12	yes	—
221	C₂²	C₂×#14556560	1	4	yes	—
222	C₂²	#17989098	1	12	yes	—
223	C₄	—	2	12	no	—
224	C₄	—	292	16	no	—
225	C₄	#17988443	1	8	yes	direct product factor
226	C₂	—	2	24	no	—
227	C₂	C₂×C₄×C₂²⋊Dic₇₃	1	3	yes	—
228	C₂	C₂×#17988443	1	1	yes	characteristic
229	C₂	#22612314	1	3	yes	—
230	C₂	C₄×#14556560	1	8	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Frattini	C₂⁴	C₂³×D₇₃
Fitting	C₂⁵×C₄×C₇₃	C₂
socle, cosocle	C₂⁴×C₇₃	C₂⁴
prime radical	C₂⁶×C₇₃	C₂²
hypercenter	C₂⁵×C₄	D₇₃
hypocenter	C₇₃	C₄×#379
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₂²⋊Dic₇₃	C₂	semidirect
3. #22608154	C₂	semidirect
6. C₂²×C₄×Dic₇₃	C₂²	semidirect
7. #17988443	C₄	direct (N × H)
9. C₂×#14556498	C₂²	semidirect
12. C₂⁴×C₄×C₇₃	C₄	semidirect
18. C₂×C₂²⋊Dic₇₃	C₂×C₄	semidirect
38. C₂²×Dic₇₃	C₂²×C₄	semidirect
48. C₂⁴×C₇₃	C₄²	semidirect
61. C₂²×C₄×C₇₃	C₂²⋊C₄	semidirect
100. C₂²×C₇₃	C₄×C₂²⋊C₄	semidirect
108. C₄×C₇₃	#379	semidirect
120. C₇₃	C₄×#379	semidirect
123. C₂⁴×C₄	Dic₇₃	semidirect
159. C₂⁴	C₄×Dic₇₃	semidirect
175. C₂²×C₄	C₂²⋊Dic₇₃	semidirect
216. C₂²	C₄×C₂²⋊Dic₇₃	semidirect
225. C₄	#17988443	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17988443	2
C₂×C₄×C₂²⋊Dic₇₃	2
#22608154	2
C₂⁵×C₄×C₇₃	2
C₄×#379	73

Subgroups — C4×#17988443 #33007192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17988443 #33007192