Subgroups — C₄×#17985821 #31912837

Order 18176.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985821	C₂	1	2	yes	maximal
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
4	#22586438	C₂	1	2	yes	maximal
5	C₄×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	C₂×#14556068	C₂²	1	1	yes	characteristic
8	C₂×#14556064	C₂²	1	2	yes	—
9	C₄×#11765849	—	2	1	no	—
10	#17985821	C₄	1	8	yes	direct product factor
11	#17985775	C₄	1	2	yes	complemented
12	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
13	#17985764	—	2	1	no	—
14	C₂×#106×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
17	#340×C₇₁	—	2	1	no	—
18	C₂×#11765849	—	2	2	no	—
19	#14556068	—	2	2	no	—
20	#14556068	C₂×C₄	1	4	yes	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
23	#14556064	—	2	2	no	—
24	#14556064	C₂×C₄	1	4	yes	—
25	#14556039	—	4	1	no	—
26	C₄×C₇₁⋊C₈	—	4	1	no	—
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₄²×C₇₁	—	4	1	no	—
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	#106×C₇₁	—	2	2	no	—
33	#106×C₇₁	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
38	#95×C₇₁	—	4	1	no	—
39	#11765849	—	2	4	no	—
40	#11765849	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₇₁⋊C₁₆	—	2	2	no	—
43	C₇₁⋊C₁₆	C₄²	1	4	yes	complemented
44	C₂²×C₄×C₇₁	—	4	2	no	—
45	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	#41	1	2	yes	—
55	C₄⋊C₄×C₇₁	—	2	4	no	—
56	C₄⋊C₄×C₇₁	—	4	2	no	—
57	C₄²×C₇₁	—	8	2	no	—
58	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
61	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
62	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	4	1	no	—
65	C₂×C₄×C₇₁	—	4	4	no	—
66	C₂×C₄×C₇₁	—	8	2	no	—
67	C₂×C₄×C₇₁	—	8	2	no	—
68	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#135	1	2	yes	—
76	C₈×C₇₁	—	2	1	no	—
77	C₈×C₇₁	—	2	1	no	—
78	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
79	C₈×C₇₁	#117	1	1	yes	characteristic
80	C₈×C₇₁	C₄×Q₈	1	2	yes	—
81	C₇₁⋊C₈	—	4	4	no	—
82	C₂²×C₇₁	—	2	1	no	—
83	C₂²×C₇₁	—	2	1	no	—
84	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
85	C₂²×C₇₁	C₄×SD₁₆	1	2	yes	—
86	C₄×C₇₁	—	2	1	no	—
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	—	2	1	no	—
89	C₄×C₇₁	—	8	4	no	—
90	C₄×C₇₁	#365	1	2	yes	complemented
91	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₇₁	#427	1	1	yes	characteristic
93	C₄×C₇₁	C₄×SD₁₆	1	2	yes	—
94	C₄×#365	—	71	1	no	maximal, Sylow
95	C₂×C₇₁	—	2	2	no	—
96	C₂×C₇₁	C₄×#106	1	1	yes	characteristic
97	C₂×C₇₁	C₂×#365	1	1	yes	characteristic
98	C₂×C₇₁	#2155	1	1	yes	characteristic
99	C₂×#365	—	71	2	no	—
100	C₄×M₃₂	—	71	1	no	—
101	C₄×#108	—	71	1	no	—
102	C₄×#106	D₇₁	1	1	yes	characteristic
103	#1383	—	71	2	no	—
104	C₇₁	C₄×#365	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	—	71	1	no	—
106	C₂×M₃₂	—	71	1	no	—
107	C₂×#108	—	71	2	no	—
108	C₂×#106	C₂×D₇₁	1	2	yes	—
109	C₄×M₁₆	—	142	1	no	—
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	1	no	—
112	#365	—	71	8	no	—
113	#346	—	71	2	no	—
114	C₄×C₁₆	—	71	2	no	—
115	#341	—	142	1	no	—
116	#340	—	2	1	no	—
117	C₂×M₁₆	—	142	2	no	—
118	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
123	M₃₂	—	71	4	no	—
124	M₃₂	—	142	2	no	—
125	C₂×C₁₆	—	71	2	no	—
126	C₂×C₁₆	—	71	2	no	—
127	#108	—	71	4	no	—
128	#108	—	142	2	no	—
129	#106	—	2	2	no	—
130	#106	C₄×D₇₁	1	4	yes	—
131	#97	—	284	1	no	—
132	C₄×C₈	—	284	1	no	—
133	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
134	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
137	#95	—	4	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	#11765853	1	1	yes	characteristic
140	C₂²×C₄	#11765853	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
142	M₁₆	—	284	2	no	—
143	M₁₆	—	142	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	284	2	no	—
146	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
147	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
150	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
151	C₂×C₈	#11765865	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	8	2	no	—
155	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
156	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
157	C₄²	#11765852	1	1	yes	characteristic
158	C₄²	#11765852	1	1	yes	characteristic
159	C₁₆	—	71	4	no	—
160	C₁₆	—	142	2	no	—
161	C₂³	C₂×#11765853	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	#14556059	1	1	yes	characteristic
168	C₂×C₄	#14556059	1	1	yes	characteristic
169	C₂×C₄	#14556059	1	1	yes	characteristic
170	C₂×C₄	#14556059	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
173	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
174	C₂×C₄	#14556148	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	284	4	no	—
178	C₈	C₄²×D₇₁	1	1	yes	characteristic
179	C₈	#14556085	1	1	yes	characteristic
180	C₈	C₄×Dic₁₄₂	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#14556059	1	1	yes	characteristic
184	C₂²	C₄×#11765853	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	#17985821	1	2	yes	direct product factor
190	C₄	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
191	C₄	#17986035	1	1	yes	characteristic
192	C₄	C₄×#11765852	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#14556059	1	1	yes	characteristic
195	C₂	C₂×#17985821	1	1	yes	characteristic
196	C₂	#22590112	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556059
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄×#106×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#106
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄×#106	D₇₁
hypocenter	C₇₁	C₄×#365
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
10. #17985821	C₄	direct (N × H)
11. #17985775	C₄	semidirect
12. C₄×C₇₁⋊C₁₆	C₄	semidirect
43. C₇₁⋊C₁₆	C₄²	semidirect
90. C₄×C₇₁	#365	semidirect
104. C₇₁	C₄×#365	semidirect
189. C₄	#17985821	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×#17985821	2
C₄×#14556068	2
C₄×#14556064	2
#22586438	2
C₄×#106×C₇₁	2
C₄×#365	71

Subgroups — C4×#17985821 #31912837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17985821 #31912837