Subgroups — C₂×#22586415 #31909044

Order 18176.

230 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985820	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22586415	C₂	1	4	yes	maximal, direct product factor
4	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
8	#17985820	—	2	1	no	—
9	#17985820	C₂²	1	2	yes	—
10	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
12	C₂×M₃₂×C₇₁	—	2	1	no	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#364×C₇₁	—	2	1	no	—
15	#364×C₇₁	C₂²	1	2	yes	—
16	#14556068	—	2	2	no	—
17	#14556068	—	4	1	no	—
18	C₂×C₇₁⋊C₁₆	—	2	1	no	—
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₂×C₇₁⋊C₁₆	—	4	1	no	—
21	C₂×C₇₁⋊C₁₆	C₈	1	2	yes	—
22	C₂×C₇₁⋊C₁₆	C₈	1	2	yes	—
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
28	M₃₂×C₇₁	—	2	2	no	—
29	M₃₂×C₇₁	—	4	1	no	—
30	C₂×C₁₆×C₇₁	—	4	1	no	—
31	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₇₁⋊C₁₆	—	2	2	no	—
36	C₇₁⋊C₁₆	—	4	2	no	—
37	C₇₁⋊C₁₆	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₇₁	—	2	1	no	—
39	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇₁	—	2	1	no	—
53	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₁₆×C₇₁	—	4	2	no	—
56	C₂³×C₇₁	#108	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	4	no	—
58	C₂×C₄×C₇₁	—	2	1	no	—
59	C₂×C₄×C₇₁	—	2	1	no	—
60	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
67	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
68	C₈×C₇₁	#107	1	1	yes	characteristic
69	C₈×C₇₁	#107	1	1	yes	characteristic
70	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
73	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₂²×C₇₁	—	2	2	no	—
75	C₂²×C₇₁	#335	1	1	yes	characteristic
76	C₂²×C₇₁	#335	1	1	yes	characteristic
77	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
78	C₄×C₇₁	—	2	2	no	—
79	C₄×C₇₁	—	2	2	no	—
80	C₄×C₇₁	#335	1	1	yes	characteristic
81	C₄×C₇₁	#335	1	1	yes	characteristic
82	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
84	C₂×#1373	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	—	2	1	no	—
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	#1373	1	2	yes	complemented
88	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
89	C₂×#364	—	71	1	no	—
90	C₂×#364	D₇₁	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	71	1	no	—
92	#1373	—	71	4	no	—
93	C₇₁	C₂×#1373	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	—	2	1	no	—
95	C₂×M₃₂	—	142	1	no	—
96	C₂²×C₁₆	—	71	1	no	—
97	C₂²×C₁₆	—	71	1	no	—
98	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
99	#364	—	2	1	no	—
100	#364	—	71	2	no	—
101	#364	—	142	1	no	—
102	#364	C₂×D₇₁	1	2	yes	—
103	C₄×C₁₆	—	71	2	no	—
104	C₄×C₁₆	—	71	2	no	—
105	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
107	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	4	1	no	—
110	M₃₂	—	284	1	no	—
111	M₃₂	—	142	2	no	—
112	C₂×C₁₆	—	4	1	no	—
113	C₂×C₁₆	—	284	1	no	—
114	C₂×C₁₆	—	71	2	no	—
115	C₂×C₁₆	—	71	2	no	—
116	C₂×C₁₆	—	71	2	no	—
117	C₂×C₁₆	—	71	2	no	—
118	C₂×C₁₆	—	142	1	no	—
119	C₂×C₁₆	—	142	1	no	—
120	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
121	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
128	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
129	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
130	C₂×C₈	#11765848	1	1	yes	characteristic
131	C₂×C₈	#11765848	1	1	yes	characteristic
132	C₂×C₈	#11765850	1	1	yes	characteristic
133	C₂×C₈	#11765850	1	1	yes	characteristic
134	C₂×C₈	#11765855	1	1	yes	characteristic
135	C₂×C₈	#11765855	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
138	C₄²	—	2	1	no	—
139	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
141	C₁₆	—	4	2	no	—
142	C₁₆	—	284	2	no	—
143	C₁₆	—	71	4	no	—
144	C₁₆	—	142	2	no	—
145	C₂³	#14556063	1	1	yes	characteristic
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
150	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
151	C₂×C₄	#14556061	1	1	yes	characteristic
152	C₂×C₄	#14556061	1	1	yes	characteristic
153	C₂×C₄	#14556063	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
155	C₈	Dic₇₁⋊C₈	1	1	yes	characteristic
156	C₈	Dic₇₁⋊C₈	1	1	yes	characteristic
157	C₈	#14556061	1	1	yes	characteristic
158	C₈	#14556061	1	1	yes	characteristic
159	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
160	C₈	C₂×#11765848	1	1	yes	characteristic
161	C₈	C₂×#11765850	1	1	yes	characteristic
162	C₈	C₂×#11765855	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#17985751	1	1	yes	characteristic
165	C₂²	#17985751	1	1	yes	characteristic
166	C₂²	C₂×#14556063	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	#17985751	1	1	yes	characteristic
170	C₄	#17985751	1	1	yes	characteristic
171	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
172	C₄	C₂×#14556061	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#22586415	1	2	yes	direct product factor
176	C₂	C₂×#17985751	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11765850
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	#335
prime radical	C₂³×C₇₁	#108
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	C₂×#1373
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #22586415	C₂	direct (N × H)
87. C₂×C₇₁	#1373	semidirect
93. C₇₁	C₂×#1373	semidirect
175. C₂	#22586415	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×#17985820	2
C₂×C₄×C₇₁⋊C₁₆	2
#22586415	2
C₂×#364×C₇₁	2
C₂×#1373	71

Subgroups — C2×#22586415 #31909044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586415 #31909044