Subgroups — C₂×#10620846 #13059416

Order 1344.

532 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#66557	C₂	1	2	yes	maximal, complemented
5	#10620846	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₇×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×#6470	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₂²⋊D₇	C₄	1	2	yes	—
10	C₃×#1157	C₄	1	4	yes	—
11	C₃×#1157	C₂²	1	4	yes	—
12	C₂×C₃×D₂₈	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
14	C₂×C₃×Dic₁₄	C₄	1	1	yes	characteristic
15	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
18	C₇×#208	C₂²	1	4	yes	complemented
19	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	—
20	#66557	C₂²	1	8	yes	complemented
21	D₇×C₃⋊C₈	C₂²	1	8	yes	complemented
22	C₂×#1157	D₃	1	1	yes	characteristic
23	C₂×C₇×M₁₆	—	3	1	no	—
24	C₂²×C₇⋊C₈	—	3	1	no	—
25	#6470	—	3	8	no	—
26	C₂×#1131	—	3	2	no	—
27	C₂×C₈×D₇	—	3	2	no	—
28	C₂×#948	—	7	1	no	maximal, Hall
29	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×D₇	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
35	C₃×C₂²⋊D₇	C₂×C₄	1	8	yes	—
36	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
37	C₃×D₂₈	C₂×C₄	1	4	yes	—
38	C₃×C₄×D₇	C₂×C₄	1	4	yes	—
39	C₃×C₄×D₇	C₂³	1	4	yes	—
40	C₃×Dic₁₄	C₂×C₄	1	4	yes	—
41	C₃⋊(C₇⋊C₈)	C₂³	1	2	yes	—
42	C₃⋊(C₇⋊C₈)	C₂³	1	2	yes	—
43	C₇×C₃⋊C₈	C₂³	1	4	yes	—
44	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
45	C₂×C₂²⋊D₇	Dic₃	1	2	yes	—
46	#1157	Dic₃	1	4	yes	—
47	#1157	C₂×D₃	1	4	yes	—
48	C₂×D₂₈	Dic₃	1	1	yes	characteristic
49	C₂×C₄×D₇	C₂×D₃	1	2	yes	—
50	C₂×Dic₁₄	Dic₃	1	1	yes	characteristic
51	C₇×M₁₆	—	3	4	no	—
52	C₂×C₇×C₈	—	3	2	no	—
53	C₂×C₇⋊C₈	—	3	1	no	—
54	C₂×C₇⋊C₈	—	3	1	no	—
55	C₂×C₇⋊C₈	—	3	4	no	—
56	#1131	—	3	8	no	—
57	C₈×D₇	—	3	8	no	—
58	C₂×C₃×#41	—	7	1	no	—
59	#948	—	7	8	no	—
60	C₂×#208	—	7	2	no	—
61	C₂×#208	D₇	1	1	yes	characteristic, complemented
62	C₂²×C₃⋊C₈	—	7	1	no	—
63	C₂²×C₃⋊C₈	—	7	2	no	—
64	C₂²×C₃×C₇	—	2	2	no	—
65	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×D₇	—	2	4	no	—
68	C₂×C₃×D₇	C₂²×C₄	1	4	yes	—
69	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₃×Dic₇	C₂²×C₄	1	4	yes	—
73	C₂×#131	—	21	1	no	Sylow
74	C₂³×C₇	C₂×Dic₃	1	1	yes	characteristic
75	C₂²×D₇	C₂×Dic₃	1	2	yes	—
76	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
78	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
79	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
80	C₂²⋊D₇	C₂×Dic₃	1	8	yes	—
81	C₂×Dic₇	C₂×Dic₃	1	2	yes	—
82	D₂₈	C₂×Dic₃	1	4	yes	—
83	C₄×D₇	C₂×Dic₃	1	4	yes	—
84	C₄×D₇	C₂²×D₃	1	4	yes	—
85	Dic₁₄	C₂×Dic₃	1	4	yes	—
86	C₇×C₈	—	3	4	no	—
87	C₇⋊C₈	—	3	2	no	—
88	C₇⋊C₈	—	3	2	no	—
89	C₃×#41	—	7	4	no	—
90	C₃×#41	—	7	4	no	—
91	C₂×C₃×Q₈	—	7	1	no	—
92	C₂×C₃×D₄	—	7	2	no	—
93	C₂×C₃×D₄	—	7	1	no	—
94	C₂²×C₃×C₄	—	7	2	no	—
95	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
96	#208	—	7	8	no	—
97	#208	C₂×D₇	1	4	yes	complemented
98	C₂×C₃⋊C₈	—	7	1	no	—
99	C₂×C₃⋊C₈	—	7	1	no	—
100	C₂×C₃⋊C₈	—	7	4	no	—
101	C₂×C₃⋊C₈	—	7	8	no	—
102	C₂×C₃⋊C₈	C₂×D₇	1	2	yes	—
103	C₂×C₃×C₇	—	2	2	no	—
104	C₂×C₃×C₇	#131	1	2	yes	complemented
105	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
106	C₃×D₇	—	2	4	no	—
107	C₂×#41	—	7	1	no	—
108	#131	—	21	8	no	—
109	C₂×M₁₆	—	3	1	no	—
110	C₂×M₁₆	—	21	2	no	—
111	C₂²×C₈	—	21	2	no	—
112	C₂²×C₈	—	21	1	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	C₂²×Dic₃	1	2	yes	—
115	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
116	C₂×D₇	—	2	4	no	—
117	C₂×D₇	C₂²×Dic₃	1	4	yes	—
118	C₄×C₇	C₂²×Dic₃	1	2	yes	—
119	C₄×C₇	C₂²×Dic₃	1	1	yes	characteristic
120	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
121	Dic₇	C₂²×Dic₃	1	4	yes	—
122	C₂³×C₃	—	7	2	no	—
123	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
124	C₃×Q₈	—	7	4	no	—
125	C₃×D₄	—	7	8	no	—
126	C₃×D₄	—	7	4	no	—
127	C₂×C₃×C₄	—	7	2	no	—
128	C₂×C₃×C₄	—	7	4	no	—
129	C₂×C₃×C₄	—	7	4	no	—
130	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
131	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
132	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
133	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
134	C₃⋊C₈	—	7	2	no	—
135	C₃⋊C₈	—	7	2	no	—
136	C₃⋊C₈	C₂²×D₇	1	4	yes	—
137	C₃×C₇	C₂×#131	1	1	yes	characteristic, complemented, Hall
138	#41	—	7	4	no	—
139	#41	—	7	4	no	—
140	C₂×Q₈	—	7	1	no	—
141	C₂×D₄	—	7	2	no	—
142	C₂×D₄	—	7	1	no	—
143	C₂²×C₄	—	7	2	no	—
144	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
145	M₁₆	—	3	4	no	—
146	M₁₆	—	21	8	no	—
147	C₂×C₈	—	3	2	no	—
148	C₂×C₈	—	21	8	no	—
149	C₂×C₈	—	21	4	no	—
150	C₂×C₈	—	21	1	no	—
151	C₂×C₈	—	21	1	no	—
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	#948	1	2	yes	complemented
154	C₂×C₇	C₂³×Dic₃	1	1	yes	characteristic
155	D₇	—	2	4	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	7	4	no	—
158	C₂²×C₃	—	14	4	no	—
159	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
160	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
161	C₃×C₄	—	7	4	no	—
162	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
163	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
164	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
165	C₂³	—	7	2	no	—
166	C₂³	Dic₃×D₇	1	1	yes	characteristic
167	Q₈	—	7	4	no	—
168	D₄	—	7	8	no	—
169	D₄	—	7	4	no	—
170	C₂×C₄	—	7	2	no	—
171	C₂×C₄	—	7	4	no	—
172	C₂×C₄	—	7	4	no	—
173	C₂×C₄	Dic₃×D₇	1	1	yes	characteristic
174	C₂×C₄	Dic₃×D₇	1	2	yes	—
175	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
176	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
177	C₈	—	3	4	no	—
178	C₈	—	21	2	no	—
179	C₈	—	21	2	no	—
180	C₇	C₂×#948	1	1	yes	characteristic, complemented, Sylow
181	C₂×C₃	—	2	2	no	—
182	C₂×C₃	—	14	4	no	—
183	C₂×C₃	#6470	1	2	yes	complemented
184	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	—	7	4	no	—
187	C₂²	—	14	4	no	—
188	C₂²	C₂×Dic₃×D₇	1	2	yes	—
189	C₂²	C₂×Dic₃×D₇	1	1	yes	characteristic
190	C₄	—	7	4	no	—
191	C₄	C₂×Dic₃×D₇	1	1	yes	characteristic
192	C₄	C₂×Dic₃×D₇	1	2	yes	—
193	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
194	C₃	C₂×#6470	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	14	4	no	—
197	C₂	#10620846	1	2	yes	direct product factor
198	C₂	C₂²×Dic₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₄	C₂²×D₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×#1157	C₂
cosocle	C₃×C₄×C₇	C₂⁴
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×#131
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₇⋊C₈)	C₂	semidirect
3. C₂×C₇×#208	C₂	semidirect
4. C₂×#66557	C₂	semidirect
5. #10620846	C₂	direct (N × H)
6. C₂×D₇×C₃⋊C₈	C₂	semidirect
17. C₂×C₃⋊(C₇⋊C₈)	C₂²	semidirect
18. C₇×#208	C₂²	semidirect
20. #66557	C₂²	semidirect
21. D₇×C₃⋊C₈	C₂²	semidirect
61. C₂×#208	D₇	semidirect
97. #208	C₂×D₇	semidirect
104. C₂×C₃×C₇	#131	semidirect
137. C₃×C₇	C₂×#131	semidirect
153. C₂×C₇	#948	semidirect
180. C₇	C₂×#948	semidirect
183. C₂×C₃	#6470	semidirect
194. C₃	C₂×#6470	semidirect
197. C₂	#10620846	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×C₃×#1157	2
C₂²×C₃⋊(C₇⋊C₈)	2
C₂×C₇×#208	2
C₂×#66557	2
#10620846	2
C₂×D₇×C₃⋊C₈	2
C₂×#6470	3
C₂×#948	7

Subgroups — C2×#10620846 #13059416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10620846 #13059416