Subgroups — C₄×#17989533 #34014622

Order 18688.

2094 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989533	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊#64408	C₂	1	6	yes	maximal
3	#22625109	C₂	1	6	yes	maximal
4	C₄³×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#64408	C₂²	1	12	yes	—
6	C₂²×C₄×#64408	C₂²	1	4	yes	—
7	#17989533	C₄	1	16	yes	direct product factor
8	C₄×C₄⋊#64408	—	2	24	no	—
9	C₂×(C₂×#64408)⋊C₄	C₂²	1	12	yes	—
10	C₂×C₄²×D₇₃	C₂²	1	3	yes	—
11	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₇₃	C₂²	1	3	yes	—
13	C₄²×Dic₇₃	C₄	1	3	yes	complemented
14	C₄²×Dic₇₃	C₄	1	1	yes	characteristic, complemented
15	C₄²×Dic₇₃	C₄	1	3	yes	complemented
16	C₄³×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂³×#64408	C₂³	1	8	yes	—
18	C₂×C₄⋊#64408	—	2	24	no	—
19	C₂×C₄⋊#64408	—	2	24	no	—
20	C₂×C₄⋊#64408	C₂×C₄	1	48	yes	—
21	C₂×C₄×#64408	—	2	24	no	—
22	(C₂×#64408)⋊C₄	—	2	24	no	—
23	C₂²×C₄×D₇₃	C₂³	1	3	yes	—
24	C₂²×C₄×D₇₃	C₂³	1	3	yes	—
25	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
27	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
32	C₄²×D₇₃	—	2	12	no	—
33	C₄²×D₇₃	C₂×C₄	1	8	yes	—
34	C₄²×D₇₃	D₄	1	12	yes	—
35	C₄²×D₇₃	Q₈	1	12	yes	—
36	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
39	C₂²×#64408	—	2	24	no	—
40	C₂²×#64408	—	2	24	no	—
41	C₂²×#64408	C₂²×C₄	1	16	yes	—
42	C₄⋊#64408	—	2	96	no	—
43	C₄×#64408	—	4	16	no	—
44	C₂³×D₇₃	C₂⁴	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×D₇₃	—	2	12	no	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
50	C₂×C₄×D₇₃	C₂×D₄	1	6	yes	—
51	C₂×C₄×D₇₃	C₂×D₄	1	6	yes	—
52	C₂×C₄×D₇₃	C₂×D₄	1	6	yes	—
53	C₂×C₄×D₇₃	C₂×Q₈	1	6	yes	—
54	C₂×C₄×D₇₃	C₂×Q₈	1	6	yes	—
55	C₂×C₄×D₇₃	C₂×Q₈	1	6	yes	—
56	C₂×C₄×D₇₃	#41	1	12	yes	—
57	C₄×Dic₇₃	—	2	6	no	—
58	C₄×Dic₇₃	—	2	12	no	—
59	C₄×Dic₇₃	C₄²	1	12	yes	complemented
60	C₄×Dic₇₃	C₄⋊C₄	1	12	yes	complemented
61	C₄×Dic₇₃	C₄⋊C₄	1	12	yes	complemented
62	C₄×Dic₇₃	C₄⋊C₄	1	12	yes	complemented
63	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₇₃	—	2	6	no	—
67	C₄²×C₇₃	C₄²	1	4	yes	complemented
68	C₄²×C₇₃	C₄⋊C₄	1	12	yes	complemented
69	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
70	C₂²×D₇₃	C₂³×C₄	1	2	yes	—
71	C₂²×D₇₃	C₂²×D₄	1	3	yes	—
72	C₂²×D₇₃	C₂²×Q₈	1	3	yes	—
73	C₂²×D₇₃	C₂×#41	1	6	yes	—
74	C₂×#64408	—	2	48	no	—
75	C₂×#64408	—	4	8	no	—
76	C₂×#64408	—	4	24	no	—
77	C₂×C₄×C₇₃	—	2	6	no	—
78	C₂×C₄×C₇₃	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
80	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
81	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₇₃	#117	1	6	yes	—
83	C₂×Dic₇₃	—	2	6	no	—
84	C₂×Dic₇₃	C₂×C₄²	1	6	yes	—
85	C₂×Dic₇₃	C₂×C₄⋊C₄	1	6	yes	—
86	C₂×Dic₇₃	C₂×C₄⋊C₄	1	6	yes	—
87	C₂×Dic₇₃	C₂×C₄⋊C₄	1	6	yes	—
88	C₂×Dic₇₃	#117	1	6	yes	—
89	C₄×D₇₃	—	2	12	no	—
90	C₄×D₇₃	—	2	12	no	—
91	C₄×D₇₃	C₄×D₄	1	24	yes	—
92	C₄×D₇₃	C₄×Q₈	1	24	yes	—
93	C₄×D₇₃	#127	1	4	yes	—
94	C₄×D₇₃	C₄⋊Q₈	1	12	yes	—
95	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₇₃	C₂²×C₄⋊C₄	1	3	yes	—
97	C₂²×C₇₃	C₂×#117	1	3	yes	—
98	C₂×D₇₃	C₂×C₄×D₄	1	6	yes	—
99	C₂×D₇₃	C₂×C₄×Q₈	1	6	yes	—
100	C₂×D₇₃	C₂×#127	1	1	yes	characteristic
101	C₂×D₇₃	C₂×C₄⋊Q₈	1	3	yes	—
102	C₂×D₇₃	#532	1	6	yes	—
103	C₂×D₇₃	#533	1	6	yes	—
104	#64408	—	4	16	no	—
105	C₄×C₇₃	—	2	6	no	—
106	C₄×C₇₃	C₄×C₄⋊C₄	1	12	yes	complemented
107	C₄×C₇₃	#384	1	4	yes	complemented
108	Dic₇₃	—	2	6	no	—
109	Dic₇₃	C₄×C₄⋊C₄	1	12	yes	complemented
110	Dic₇₃	#384	1	4	yes	complemented
111	C₄×#384	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	3	yes	—
113	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
114	C₂×C₇₃	#2285	1	3	yes	—
115	D₇₃	C₄×#127	1	2	yes	—
116	D₇₃	C₄×C₄⋊Q₈	1	6	yes	—
117	C₂×#384	—	73	2	no	—
118	C₂×C₄×C₄⋊C₄	—	73	6	no	—
119	C₂×C₄³	—	73	1	no	—
120	C₄⋊#95	—	73	6	no	—
121	C₇₃	C₄×#384	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄⋊C₄	—	73	12	no	—
123	C₂²×C₄²	—	73	3	no	—
124	C₂²×C₄²	—	73	1	no	—
125	C₂²×C₄²	—	73	3	no	—
126	C₂²×C₄²	—	73	4	no	—
127	#384	—	73	16	no	—
128	C₄×C₄⋊C₄	—	146	24	no	—
129	C₂×#95	—	73	12	no	—
130	C₄³	—	73	3	no	—
131	C₄³	—	73	1	no	—
132	C₄³	—	73	3	no	—
133	C₄³	#64408	1	1	yes	characteristic, complemented
134	C₂³×C₄	—	73	3	no	—
135	C₂³×C₄	—	73	1	no	—
136	C₂³×C₄	—	73	3	no	—
137	C₂³×C₄	—	73	8	no	—
138	C₂×C₄⋊C₄	—	73	48	no	—
139	C₂×C₄⋊C₄	—	146	24	no	—
140	C₂×C₄⋊C₄	—	146	24	no	—
141	C₂×C₄²	—	73	3	no	—
142	C₂×C₄²	—	73	3	no	—
143	C₂×C₄²	—	73	3	no	—
144	C₂×C₄²	—	73	3	no	—
145	C₂×C₄²	—	73	8	no	—
146	C₂×C₄²	—	73	12	no	—
147	C₂×C₄²	—	73	12	no	—
148	C₂×C₄²	—	73	6	no	—
149	C₂×C₄²	—	73	3	no	—
150	C₂×C₄²	—	146	12	no	—
151	C₂×C₄²	—	146	24	no	—
152	C₂×C₄²	C₂×#64408	1	3	yes	—
153	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
154	C₂×C₄²	C₂×#64408	1	3	yes	—
155	#95	—	146	24	no	—
156	C₂⁴	—	73	1	no	—
157	C₂²×C₄	—	73	3	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	12	no	—
160	C₂²×C₄	—	73	6	no	—
161	C₂²×C₄	—	73	6	no	—
162	C₂²×C₄	—	73	6	no	—
163	C₂²×C₄	—	73	6	no	—
164	C₂²×C₄	—	73	12	no	—
165	C₂²×C₄	—	73	3	no	—
166	C₂²×C₄	—	73	6	no	—
167	C₂²×C₄	—	73	6	no	—
168	C₂²×C₄	—	73	16	no	—
169	C₂²×C₄	—	146	12	no	—
170	C₂²×C₄	—	146	24	no	—
171	C₂²×C₄	—	146	24	no	—
172	C₂²×C₄	C₂²×#64408	1	3	yes	—
173	C₂²×C₄	C₂²×#64408	1	3	yes	—
174	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
175	C₄⋊C₄	—	146	96	no	—
176	C₄²	—	2	6	no	—
177	C₄²	—	292	16	no	—
178	C₄²	—	73	12	no	—
179	C₄²	—	73	12	no	—
180	C₄²	—	73	12	no	—
181	C₄²	—	73	12	no	—
182	C₄²	—	146	12	no	—
183	C₄²	—	146	6	no	—
184	C₄²	C₄×#64408	1	4	yes	complemented
185	C₄²	C₄⋊#64408	1	12	yes	complemented
186	C₂³	—	73	2	no	—
187	C₂³	—	73	3	no	—
188	C₂³	—	73	3	no	—
189	C₂³	—	73	6	no	—
190	C₂³	C₂³×#64408	1	1	yes	characteristic
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	292	8	no	—
193	C₂×C₄	—	292	24	no	—
194	C₂×C₄	—	73	6	no	—
195	C₂×C₄	—	73	6	no	—
196	C₂×C₄	—	73	6	no	—
197	C₂×C₄	—	73	6	no	—
198	C₂×C₄	—	73	24	no	—
199	C₂×C₄	—	73	24	no	—
200	C₂×C₄	—	73	4	no	—
201	C₂×C₄	—	73	12	no	—
202	C₂×C₄	—	73	6	no	—
203	C₂×C₄	—	146	12	no	—
204	C₂×C₄	—	146	12	no	—
205	C₂×C₄	—	146	48	no	—
206	C₂×C₄	—	146	6	no	—
207	C₂×C₄	C₂×C₄×#64408	1	6	yes	—
208	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
209	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
210	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
211	C₂×C₄	#14556670	1	6	yes	—
212	C₂²	—	73	6	no	—
213	C₂²	—	73	6	no	—
214	C₂²	—	73	1	no	—
215	C₂²	—	73	3	no	—
216	C₂²	—	73	6	no	—
217	C₂²	—	73	6	no	—
218	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
219	C₂²	C₂²×C₄⋊#64408	1	3	yes	—
220	C₂²	C₂×#14556670	1	3	yes	—
221	C₄	—	2	6	no	—
222	C₄	—	292	16	no	—
223	C₄	—	73	12	no	—
224	C₄	—	73	4	no	—
225	C₄	—	146	6	no	—
226	C₄	C₄×C₄⋊#64408	1	12	yes	complemented
227	C₄	#17989533	1	4	yes	direct product factor
228	C₂	—	73	2	no	—
229	C₂	—	73	6	no	—
230	C₂	C₂×C₄×C₄⋊#64408	1	3	yes	—
231	C₂	C₂×#17989533	1	1	yes	characteristic
232	C₂	#22626380	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₄²
Frattini	C₂³	C₂³×#64408
Fitting	C₄³×C₇₃	C₄
socle	C₂³×C₇₃	C₂³×C₄
prime radical, cosocle	C₂³×D₇₃	C₂⁴
hypercenter	C₄³	#64408
hypocenter	C₇₃	C₄×#384
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
7. #17989533	C₄	direct (N × H)
13. C₄²×Dic₇₃	C₄	semidirect
14. C₄²×Dic₇₃	C₄	semidirect
15. C₄²×Dic₇₃	C₄	semidirect
16. C₄³×C₇₃	C₄	semidirect
59. C₄×Dic₇₃	C₄²	semidirect
60. C₄×Dic₇₃	C₄⋊C₄	semidirect
61. C₄×Dic₇₃	C₄⋊C₄	semidirect
62. C₄×Dic₇₃	C₄⋊C₄	semidirect
67. C₄²×C₇₃	C₄²	semidirect
68. C₄²×C₇₃	C₄⋊C₄	semidirect
106. C₄×C₇₃	C₄×C₄⋊C₄	semidirect
107. C₄×C₇₃	#384	semidirect
109. Dic₇₃	C₄×C₄⋊C₄	semidirect
110. Dic₇₃	#384	semidirect
121. C₇₃	C₄×#384	semidirect
133. C₄³	#64408	semidirect
184. C₄²	C₄×#64408	semidirect
185. C₄²	C₄⋊#64408	semidirect
226. C₄	C₄×C₄⋊#64408	semidirect
227. C₄	#17989533	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×#17989533	2
C₂×C₄×C₄⋊#64408	2
#22625109	2
C₄³×D₇₃	2
C₄×#384	73

Subgroups — C4×#17989533 #34014622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989533 #34014622