Subgroups — C₂×#22586434 #31912339

Order 18176.

574 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556064	C₂	1	2	yes	maximal
2	#22586434	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11765849	—	2	2	no	—
5	C₂×#14556064	—	2	2	no	—
6	C₂×#14556064	—	2	4	no	—
7	C₂×#14556064	C₄	1	8	yes	—
8	C₂×#14556064	C₂²	1	4	yes	—
9	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₁	—	2	4	no	—
11	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
12	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
13	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×#11765849	—	2	8	no	—
15	C₂×#11765849	—	2	4	no	—
16	C₂×#11765849	—	4	2	no	—
17	C₂×#11765849	—	4	4	no	—
18	C₂²×C₇₁⋊C₈	—	4	2	no	—
19	#14556064	—	2	8	no	—
20	#14556064	—	2	4	no	—
21	#14556064	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	D₄	1	4	yes	—
28	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	—	2	4	no	—
32	C₂×C₁₆×C₇₁	—	2	4	no	—
33	C₂×C₁₆×C₇₁	—	2	4	no	—
34	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
35	#11765849	—	2	8	no	—
36	#11765849	—	4	8	no	—
37	#11765849	—	4	4	no	—
38	C₂×C₇₁⋊C₈	—	4	8	no	—
39	C₂×C₇₁⋊C₈	—	4	4	no	—
40	C₂⁴×C₇₁	Q₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄²	1	2	yes	—
50	C₂×C₈×C₇₁	C₄²	1	2	yes	—
51	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₁₆×C₇₁	—	2	4	no	—
63	C₂³×C₇₁	—	2	4	no	—
64	C₂³×C₇₁	#103	1	4	yes	—
65	C₂³×C₇₁	#107	1	1	yes	characteristic
66	C₂³×C₇₁	#107	1	1	yes	characteristic
67	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#95	1	2	yes	—
69	C₂×C₄×C₇₁	#95	1	2	yes	—
70	C₂×C₄×C₇₁	#102	1	4	yes	—
71	C₂×C₄×C₇₁	#102	1	4	yes	—
72	C₂×C₄×C₇₁	#103	1	4	yes	—
73	C₂×C₄×C₇₁	#106	1	2	yes	—
74	C₂×C₄×C₇₁	#106	1	2	yes	—
75	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
81	C₈×C₇₁	#95	1	2	yes	—
82	C₈×C₇₁	#95	1	2	yes	—
83	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
84	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₇₁⋊C₈	—	4	8	no	—
87	C₂²×C₇₁	—	2	8	no	—
88	C₂²×C₇₁	—	2	2	no	—
89	C₂²×C₇₁	—	2	2	no	—
90	C₂²×C₇₁	#340	1	2	yes	—
91	C₂²×C₇₁	#340	1	2	yes	—
92	C₂²×C₇₁	#368	1	4	yes	—
93	C₂²×C₇₁	C₂×#103	1	2	yes	—
94	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
95	C₄×C₇₁	#340	1	2	yes	—
96	C₄×C₇₁	#340	1	2	yes	—
97	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
98	C₄×C₇₁	C₂×#102	1	2	yes	—
99	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
100	C₂×#1381	—	71	1	no	maximal, Sylow
101	C₂×C₇₁	—	2	2	no	—
102	C₂×C₇₁	—	2	2	no	—
103	C₂×C₇₁	#1381	1	4	yes	complemented
104	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
105	C₂×C₇₁	C₂×#368	1	2	yes	—
106	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
107	C₂²×#108	—	71	2	no	—
108	#1381	—	71	4	no	—
109	C₇₁	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
110	C₂²×M₁₆	—	142	2	no	—
111	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
112	C₂²×C₁₆	—	2	4	no	—
113	C₂²×C₁₆	Dic₇₁	1	2	yes	—
114	C₂²×C₁₆	Dic₇₁	1	2	yes	—
115	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
116	C₂×#108	—	71	8	no	—
117	C₂×#108	—	71	4	no	—
118	C₂×#108	—	142	2	no	—
119	C₂×#108	—	142	4	no	—
120	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
121	C₂×M₁₆	—	284	2	no	—
122	C₂×M₁₆	—	284	4	no	—
123	C₂×M₁₆	—	142	8	no	—
124	C₂×M₁₆	—	142	4	no	—
125	C₂²×C₈	—	284	2	no	—
126	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
127	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₇₁	1	4	yes	—
129	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₇₁	1	4	yes	—
133	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
138	#108	—	71	8	no	—
139	#108	—	142	8	no	—
140	#108	—	142	4	no	—
141	C₂⁴	Dic₂₈₄	1	1	yes	characteristic
142	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	4	yes	—
143	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
145	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
146	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
147	C₂²×C₄	#11765854	1	4	yes	—
148	C₂²×C₄	Dic₂₈₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
150	M₁₆	—	284	8	no	—
151	M₁₆	—	284	4	no	—
152	M₁₆	—	142	8	no	—
153	C₂×C₈	—	284	8	no	—
154	C₂×C₈	—	284	4	no	—
155	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
156	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
157	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
158	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
159	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
160	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
161	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
162	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
163	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
165	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
168	C₁₆	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	#14556053	1	4	yes	—
171	C₂³	#14556062	1	1	yes	characteristic
172	C₂³	#14556062	1	1	yes	characteristic
173	C₂³	C₂×Dic₂₈₄	1	1	yes	characteristic
174	C₂×C₄	C₇₁⋊#95	1	2	yes	—
175	C₂×C₄	C₇₁⋊#95	1	2	yes	—
176	C₂×C₄	#14556050	1	4	yes	—
177	C₂×C₄	#14556050	1	4	yes	—
178	C₂×C₄	#14556053	1	4	yes	—
179	C₂×C₄	#14556060	1	2	yes	—
180	C₂×C₄	#14556060	1	2	yes	—
181	C₂×C₄	#14556062	1	1	yes	characteristic
182	C₂×C₄	#14556062	1	1	yes	characteristic
183	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
185	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
186	C₂×C₄	C₂×#11765854	1	2	yes	—
187	C₈	—	284	8	no	—
188	C₈	C₇₁⋊#95	1	2	yes	—
189	C₈	C₇₁⋊#95	1	2	yes	—
190	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
191	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
192	C₈	C₂×Dic₇₁⋊C₄	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#17985762	1	2	yes	—
197	C₂²	#17985762	1	2	yes	—
198	C₂²	#17985829	1	4	yes	—
199	C₂²	C₂×#14556053	1	2	yes	—
200	C₂²	C₂×#14556062	1	1	yes	characteristic
201	C₄	#17985762	1	2	yes	—
202	C₄	#17985762	1	2	yes	—
203	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
204	C₄	C₂×#14556050	1	2	yes	—
205	C₄	C₂×#14556060	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#22586434	1	4	yes	direct product factor
209	C₂	C₂×#17985762	1	1	yes	characteristic
210	C₂	C₂×#17985829	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₅₆₈
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂³×C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	#107
prime radical	C₂⁴×C₇₁	Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂³×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#1381
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22586434	C₂	direct (N × H)
103. C₂×C₇₁	#1381	semidirect
109. C₇₁	C₂×#1381	semidirect
208. C₂	#22586434	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556064	2
#22586434	2
C₂³×C₁₆×C₇₁	2
C₂×#1381	71

Subgroups — C2×#22586434 #31912339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586434 #31912339