Subgroups — C₄²×C₄⋊#64408 #34014574

Order 18688.

2478 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₄²	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄³
Frattini	C₂³	C₂³×#64408
Fitting	C₄³×C₇₃	C₄
socle	C₂³×C₇₃	C₂³×C₄
prime radical, cosocle	C₂³×D₇₃	C₂⁴
hypercenter	C₄³	#64408
hypocenter	C₇₃	C₄²×C₄⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊#64408	C₄	direct (N × H)
13. C₄²×Dic₇₃	C₄	semidirect
14. C₄²×Dic₇₃	C₄	semidirect
15. C₄²×Dic₇₃	C₄	semidirect
16. C₄³×C₇₃	C₄	semidirect
42. C₄⋊#64408	C₄²	direct (N × H)
57. C₄×Dic₇₃	C₄²	semidirect
58. C₄×Dic₇₃	C₄²	semidirect
59. C₄×Dic₇₃	C₄²	semidirect
60. C₄×Dic₇₃	C₄⋊C₄	semidirect
65. C₄²×C₇₃	C₄²	semidirect
66. C₄²×C₇₃	C₄⋊C₄	semidirect
100. C₄×C₇₃	C₄³	semidirect
101. C₄×C₇₃	C₄×C₄⋊C₄	semidirect
103. Dic₇₃	C₄³	semidirect
104. Dic₇₃	C₄×C₄⋊C₄	semidirect
115. C₇₃	C₄²×C₄⋊C₄	semidirect
127. C₄³	#64408	semidirect
176. C₄²	C₄×#64408	semidirect
177. C₄²	C₄⋊#64408	direct (N × H)
214. C₄	C₄²×#64408	semidirect
215. C₄	C₄×C₄⋊#64408	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×C₄×C₄⋊#64408	2
C₂×C₄²×#64408	2
C₄×(C₂×#64408)⋊C₄	2
C₄³×D₇₃	2
C₄²×C₄⋊C₄	73

Subgroups — C42×C4⋊#64408 #34014574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊#64408 #34014574