Subgroups — #31897281

Order 18176.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986379	C₂	1	1	yes	maximal, characteristic
2	#22589301	C₂	1	2	yes	maximal
3	#22589301	C₂	1	2	yes	maximal
4	#22588468	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₈₄	—	2	2	no	—
7	C₂×#14556128	C₂²	1	2	yes	—
8	#17986379	—	2	2	no	—
9	#17986379	C₄	1	4	yes	complemented
10	#17985869	C₂²	1	2	yes	—
11	#17985762	—	2	2	no	—
12	#17985762	—	2	4	no	—
13	#17985762	—	2	4	no	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#322×C₇₁	—	2	2	no	—
18	C₂²×Dic₁₄₂	—	2	2	no	—
19	C₂×Dic₂₈₄	—	2	8	no	—
20	C₂×Dic₂₈₄	—	4	8	no	—
21	#14556128	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	—	2	2	no	—
24	#14556128	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	—	2	2	no	—
28	C₂×C₄⋊Dic₇₁	—	2	2	no	—
29	C₂×C₄⋊Dic₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	2	yes	—
31	C₂²×C₈×C₇₁	D₄	1	2	yes	—
32	C₂²×C₈×C₇₁	D₄	1	2	yes	—
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₇₁	—	2	1	no	—
35	C₄×C₈×C₇₁	—	2	1	no	—
36	C₄×C₈×C₇₁	—	2	1	no	—
37	C₄×C₈×C₇₁	—	2	1	no	—
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	D₄	1	2	yes	—
40	C₂×Dic₁₄₂	—	2	4	no	—
41	C₂×Dic₁₄₂	—	4	8	no	—
42	C₂×Dic₁₄₂	—	4	2	no	—
43	C₂²×Dic₇₁	—	4	2	no	—
44	C₂²×Dic₇₁	—	4	2	no	—
45	Dic₂₈₄	—	4	16	no	—
46	C₄⋊Dic₇₁	—	2	4	no	—
47	C₄⋊Dic₇₁	—	2	4	no	—
48	C₄⋊Dic₇₁	—	4	2	no	—
49	C₄⋊Dic₇₁	—	4	2	no	—
50	C₄⋊Dic₇₁	—	4	4	no	—
51	C₄⋊Dic₇₁	—	4	4	no	—
52	C₄⋊Dic₇₁	—	4	4	no	—
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	—	2	4	no	—
61	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₇₁	Q₁₆	1	4	yes	complemented
63	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	#41	1	1	yes	characteristic
67	C₂³×C₇₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
75	C₂×C₄×C₇₁	#135	1	2	yes	—
76	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
77	C₂×C₄×C₇₁	#137	1	4	yes	—
78	C₂×Dic₇₁	—	4	4	no	—
79	C₂×Dic₇₁	—	8	4	no	—
80	C₂×Dic₇₁	—	8	2	no	—
81	C₂×Dic₇₁	—	8	2	no	—
82	C₂×Dic₇₁	—	8	2	no	—
83	Dic₁₄₂	—	4	8	no	—
84	Dic₁₄₂	—	8	4	no	—
85	C₈×C₇₁	—	2	4	no	—
86	C₈×C₇₁	—	2	4	no	—
87	C₈×C₇₁	—	4	2	no	—
88	C₂²×C₇₁	#390	1	1	yes	characteristic
89	C₂²×C₇₁	#494	1	1	yes	characteristic
90	C₂²×C₇₁	#495	1	1	yes	characteristic
91	C₂²×C₇₁	#496	1	2	yes	—
92	C₂²×C₇₁	#497	1	2	yes	—
93	C₄×C₇₁	#418	1	2	yes	—
94	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
95	C₄×C₇₁	#441	1	2	yes	—
96	C₄×C₇₁	#467	1	2	yes	—
97	C₄×C₇₁	#470	1	2	yes	—
98	C₄×C₇₁	#490	1	2	yes	—
99	Dic₇₁	—	8	4	no	—
100	Dic₇₁	—	16	2	no	—
101	#9835	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	#1720	1	2	yes	—
103	C₂×C₇₁	#1723	1	2	yes	—
104	C₂×C₇₁	#1961	1	1	yes	characteristic
105	C₂×C₇₁	#1969	1	1	yes	characteristic
106	C₂×C₇₁	#1971	1	1	yes	characteristic
107	C₂×#494	—	71	1	no	—
108	#1990	—	71	2	no	—
109	#1990	—	71	2	no	—
110	#1834	—	71	1	no	—
111	C₂×#322	D₇₁	1	1	yes	characteristic
112	C₇₁	#9835	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	142	2	no	—
114	C₂×C₄⋊Q₈	—	71	2	no	—
115	#494	—	71	4	no	—
116	#494	—	142	2	no	—
117	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	#384	—	71	2	no	—
121	#340	—	142	2	no	—
122	#340	—	142	4	no	—
123	#340	—	142	4	no	—
124	#322	—	2	2	no	—
125	C₂²×Q₈	—	142	2	no	—
126	C₂×Q₁₆	—	284	8	no	—
127	C₂×Q₁₆	—	142	8	no	—
128	C₂²×C₈	D₂₈₄	1	2	yes	—
129	C₂²×C₈	D₂₈₄	1	2	yes	—
130	C₂²×C₈	D₂₈₄	1	2	yes	—
131	C₄⋊Q₈	—	71	4	no	—
132	C₄⋊Q₈	—	142	2	no	—
133	C₄⋊Q₈	—	142	2	no	—
134	C₄⋊Q₈	—	142	2	no	—
135	C₂×C₄⋊C₄	—	142	2	no	—
136	C₂×C₄⋊C₄	—	142	2	no	—
137	C₂×C₄⋊C₄	—	142	2	no	—
138	C₂×C₄⋊C₄	—	142	2	no	—
139	C₂×C₄⋊C₄	—	142	2	no	—
140	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₄×D₇₁	1	2	yes	—
146	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
147	C₂×Q₈	—	284	2	no	—
148	C₂×Q₈	—	284	8	no	—
149	C₂×Q₈	—	142	4	no	—
150	C₂²×C₄	—	284	2	no	—
151	C₂²×C₄	—	284	2	no	—
152	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
155	Q₁₆	—	284	16	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
162	C₂×C₈	Dic₂₈₄	1	4	yes	complemented
163	C₄⋊C₄	—	284	2	no	—
164	C₄⋊C₄	—	284	2	no	—
165	C₄⋊C₄	—	284	4	no	—
166	C₄⋊C₄	—	284	4	no	—
167	C₄⋊C₄	—	284	4	no	—
168	C₄⋊C₄	—	142	4	no	—
169	C₄⋊C₄	—	142	4	no	—
170	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
171	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
172	C₄²	#11765865	1	1	yes	characteristic
173	C₄²	#11765865	1	1	yes	characteristic
174	C₂³	#14556125	1	1	yes	characteristic
175	Q₈	—	284	8	no	—
176	Q₈	—	568	4	no	—
177	C₂×C₄	—	284	4	no	—
178	C₂×C₄	—	568	4	no	—
179	C₂×C₄	—	568	2	no	—
180	C₂×C₄	—	568	2	no	—
181	C₂×C₄	—	568	2	no	—
182	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
184	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
185	C₂×C₄	#14556102	1	1	yes	characteristic
186	C₂×C₄	#14556102	1	1	yes	characteristic
187	C₂×C₄	#14556116	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
189	C₂×C₄	#14556150	1	2	yes	—
190	C₂×C₄	#14556156	1	4	yes	—
191	C₂×C₄	#14556161	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	2	no	—
195	C₂²	#17985901	1	1	yes	characteristic
196	C₂²	#17986379	1	1	yes	characteristic
197	C₂²	#17986383	1	1	yes	characteristic
198	C₂²	#17986388	1	2	yes	—
199	C₂²	#17986393	1	2	yes	—
200	C₄	—	568	4	no	—
201	C₄	—	1136	2	no	—
202	C₄	#17986005	1	2	yes	—
203	C₄	C₄×Dic₂₈₄	1	2	yes	—
204	C₄	#17986107	1	2	yes	—
205	C₄	#17986240	1	2	yes	—
206	C₄	#17986257	1	2	yes	—
207	C₄	#17986364	1	2	yes	—
208	C₂	#22588000	1	2	yes	—
209	C₂	#22588017	1	2	yes	—
210	C₂	#22589164	1	1	yes	characteristic
211	C₂	#22589201	1	1	yes	characteristic
212	C₂	#22589211	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556125
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#9835
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
9. #17986379	C₄	semidirect
62. C₂×C₈×C₇₁	Q₁₆	semidirect
112. C₇₁	#9835	semidirect
162. C₂×C₈	Dic₂₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986379	2
#22589301	2
#22588468	2
C₂×#322×C₇₁	2
#9835	71