Subgroups — #31916057

Order 18176.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22588504	C₂	1	1	yes	maximal, characteristic
3	#22586432	C₂	1	4	yes	maximal
4	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	1	no	—
6	#17986404	—	2	2	no	—
7	#17986404	C₄	1	4	yes	complemented
8	C₂×#14556062	—	2	1	no	—
9	C₂×#14556062	C₂²	1	2	yes	—
10	C₂×#14556062	C₂²	1	2	yes	—
11	C₂×#14556060	—	2	1	no	—
12	#17985865	—	2	1	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#346×C₇₁	—	2	2	no	—
16	C₂²×Dic₁₄₂	—	4	1	no	—
17	#14556128	—	2	4	no	—
18	#14556128	—	4	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	4	1	no	—
22	#14556062	—	2	4	no	—
23	#14556062	—	2	2	no	—
24	#14556062	—	2	4	no	—
25	#14556062	C₂×C₄	1	4	yes	—
26	#14556060	—	4	2	no	—
27	C₇₁⋊#95	—	4	1	no	—
28	C₇₁⋊#95	—	4	1	no	—
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	—	2	2	no	—
33	C₂×C₁₆×C₇₁	—	2	2	no	—
34	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
35	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₁	D₄	1	2	yes	—
37	C₂×Dic₁₄₂	—	4	4	no	—
38	C₂×Dic₁₄₂	—	8	4	no	—
39	C₂²×Dic₇₁	—	4	2	no	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	4	4	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
51	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₁	#41	1	2	yes	—
56	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₇₁	SD₁₆	1	2	yes	—
58	C₁₆×C₇₁	—	2	4	no	—
59	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	#102	1	2	yes	—
63	C₂×C₄×C₇₁	#102	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₇₁	—	4	4	no	—
71	C₂×Dic₇₁	—	8	4	no	—
72	C₂×Dic₇₁	—	8	2	no	—
73	C₂×Dic₇₁	—	8	2	no	—
74	C₂×Dic₇₁	—	8	2	no	—
75	Dic₁₄₂	—	8	8	no	—
76	C₈×C₇₁	#103	1	4	yes	—
77	C₈×C₇₁	C₄×Q₈	1	2	yes	—
78	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
80	C₂²×C₇₁	#486	1	1	yes	characteristic
81	C₂²×C₇₁	#500	1	1	yes	characteristic
82	C₂²×C₇₁	#509	1	2	yes	—
83	C₂²×C₇₁	#510	1	2	yes	—
84	C₄×C₇₁	—	2	2	no	—
85	C₄×C₇₁	—	2	2	no	—
86	C₄×C₇₁	#391	1	1	yes	characteristic
87	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
88	C₄×C₇₁	#487	1	1	yes	characteristic
89	C₄×C₇₁	#498	1	1	yes	characteristic
90	Dic₇₁	—	8	4	no	—
91	Dic₇₁	—	16	2	no	—
92	#13283	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	#1990	1	1	yes	characteristic
94	C₂×C₇₁	#2140	1	1	yes	characteristic
95	C₂×C₇₁	#2141	1	1	yes	characteristic
96	C₂×C₇₁	#2243	1	2	yes	—
97	C₂×C₇₁	#2255	1	2	yes	—
98	C₂×#500	—	71	1	no	—
99	C₂×#346	D₇₁	1	1	yes	characteristic
100	#1846	—	71	1	no	—
101	#1380	—	71	4	no	—
102	C₇₁	#13283	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	142	1	no	—
104	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
105	#500	—	71	4	no	—
106	#500	—	142	2	no	—
107	C₂×#107	—	71	2	no	—
108	C₂×#107	—	71	2	no	—
109	C₂×#107	—	142	1	no	—
110	C₂×#106	—	142	1	no	—
111	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
112	#382	—	142	1	no	—
113	#346	—	2	2	no	—
114	C₂²×Q₈	—	284	1	no	—
115	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	C₄⋊Q₈	—	284	2	no	—
118	C₄⋊Q₈	—	142	4	no	—
119	C₂×C₄⋊C₄	—	284	1	no	—
120	C₂×C₄⋊C₄	—	142	2	no	—
121	C₂×C₄⋊C₄	—	142	2	no	—
122	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	Dic₁₄₂	1	4	yes	complemented
126	#107	—	71	4	no	—
127	#107	—	142	4	no	—
128	#107	—	142	4	no	—
129	#107	—	142	2	no	—
130	#106	—	284	2	no	—
131	C₄×C₈	C₄×D₇₁	1	2	yes	—
132	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
133	#95	—	284	1	no	—
134	#95	—	284	1	no	—
135	C₂×Q₈	—	284	4	no	—
136	C₂×Q₈	—	568	4	no	—
137	C₂²×C₄	—	284	2	no	—
138	C₂²×C₄	—	284	2	no	—
139	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
140	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
142	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
143	C₂×C₈	#11765854	1	2	yes	—
144	C₂×C₈	Dic₂₈₄	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
148	C₂×C₈	#11765865	1	2	yes	—
149	C₄⋊C₄	—	284	4	no	—
150	C₄⋊C₄	—	284	2	no	—
151	C₄⋊C₄	—	284	2	no	—
152	C₄⋊C₄	—	284	2	no	—
153	C₄⋊C₄	—	142	4	no	—
154	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
155	C₄²	#11765854	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
158	Q₈	—	568	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	284	4	no	—
162	C₂×C₄	—	568	4	no	—
163	C₂×C₄	—	568	2	no	—
164	C₂×C₄	—	568	2	no	—
165	C₂×C₄	—	568	2	no	—
166	C₂×C₄	#14556050	1	2	yes	—
167	C₂×C₄	#14556050	1	2	yes	—
168	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#14556116	1	1	yes	characteristic
170	C₂×C₄	#14556128	1	1	yes	characteristic
171	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
172	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
174	C₈	#14556053	1	4	yes	—
175	C₈	C₄×Dic₁₄₂	1	2	yes	—
176	C₈	#14556107	1	2	yes	—
177	C₂²	C₂×#14556050	1	1	yes	characteristic
178	C₂²	#17986346	1	1	yes	characteristic
179	C₂²	#17986404	1	1	yes	characteristic
180	C₂²	#17986437	1	2	yes	—
181	C₂²	#17986442	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	568	4	no	—
185	C₄	—	1136	2	no	—
186	C₄	#17985905	1	1	yes	characteristic
187	C₄	C₂×#14556053	1	1	yes	characteristic
188	C₄	#17986350	1	1	yes	characteristic
189	C₄	#17986397	1	1	yes	characteristic
190	C₂	#22589301	1	1	yes	characteristic
191	C₂	#22590057	1	1	yes	characteristic
192	C₂	#22590061	1	1	yes	characteristic
193	C₂	#22590577	1	2	yes	—
194	C₂	#22590626	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#346×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	#13283
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #17986404	C₄	semidirect
34. C₂×C₁₆×C₇₁	Q₈	semidirect
102. C₇₁	#13283	semidirect
125. C₂×C₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986404	2
#22588504	2
#22586432	2
C₂×#346×C₇₁	2
#13283	71