Subgroups — C₂×#11726798 #13509273

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89454	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#89299	C₂	1	4	yes	maximal
4	#11726798	C₂	1	8	yes	maximal, direct product factor
5	C₂×#6432	—	2	2	no	—
6	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6392	—	2	2	no	—
9	C₂×#6392	C₂²	1	4	yes	—
10	#89454	—	2	4	no	—
11	#89454	C₂²	1	8	yes	—
12	C₄×C₇×C₁₆	C₂²	1	4	yes	—
13	#89299	C₂²	1	16	yes	—
14	C₂×#2252	—	7	1	no	maximal, Sylow
15	C₂²×Dic₁₄	—	4	2	no	—
16	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	—	2	4	no	—
20	C₂×C₄⋊Dic₇	—	4	2	no	—
21	#6432	—	2	8	no	—
22	#6432	—	4	4	no	—
23	C₂×C₇×C₁₆	Q₈	1	8	yes	complemented
24	C₂×C₇×C₁₆	C₂³	1	4	yes	—
25	C₄×C₇×C₈	D₄	1	2	yes	—
26	C₄×C₇×C₈	C₂³	1	2	yes	—
27	#6392	—	2	8	no	—
28	#6392	—	2	4	no	—
29	#6392	C₂³	1	8	yes	—
30	C₂×#500	—	7	2	no	—
31	#2252	—	7	8	no	—
32	C₂×#366	—	7	4	no	—
33	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
34	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₇	—	4	4	no	—
38	C₂×Dic₁₄	—	4	8	no	—
39	C₂×Dic₁₄	—	8	8	no	—
40	C₂×C₇×C₈	Q₁₆	1	4	yes	—
41	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
43	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
44	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
45	C₄²×C₇	D₈	1	2	yes	—
46	C₄²×C₇	C₂×D₄	1	2	yes	—
47	C₄⋊Dic₇	—	2	8	no	—
48	C₄⋊Dic₇	—	4	8	no	—
49	C₄⋊Dic₇	—	4	4	no	—
50	C₇×C₁₆	C₂×Q₈	1	8	yes	complemented
51	C₂×C₄⋊Q₈	—	14	2	no	—
52	C₂²×C₁₆	C₂×D₇	1	2	yes	—
53	#500	—	7	8	no	—
54	#500	—	14	4	no	—
55	C₂×#107	—	7	4	no	—
56	C₂×#107	—	14	2	no	—
57	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
58	#366	—	7	16	no	—
59	C₄×C₁₆	C₂×D₇	1	4	yes	—
60	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇	D₁₆	1	4	yes	—
62	C₂×C₄×C₇	Q₃₂	1	4	yes	—
63	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
69	C₂×Dic₇	—	4	8	no	—
70	C₂×Dic₇	—	8	8	no	—
71	Dic₁₄	—	8	16	no	—
72	C₇×C₈	C₄⋊Q₈	1	2	yes	—
73	C₇×C₈	C₂×Q₁₆	1	4	yes	—
74	C₇×C₈	C₂²×Q₈	1	2	yes	—
75	C₂²×Q₈	—	28	2	no	—
76	C₂²×C₈	D₂₈	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
78	C₄⋊Q₈	—	14	8	no	—
79	C₄⋊Q₈	—	28	4	no	—
80	C₂×C₄⋊C₄	—	14	4	no	—
81	C₂×C₄⋊C₄	—	28	2	no	—
82	C₂×C₄²	D₂₈	1	1	yes	characteristic
83	C₂×C₁₆	Dic₁₄	1	8	yes	complemented
84	C₂×C₁₆	C₂²×D₇	1	4	yes	—
85	#107	—	7	8	no	—
86	#107	—	14	4	no	—
87	#107	—	14	8	no	—
88	C₄×C₈	D₂₈	1	2	yes	—
89	C₄×C₈	C₂²×D₇	1	2	yes	—
90	C₂²×C₇	#500	1	2	yes	—
91	C₂²×C₇	C₂×D₁₆	1	2	yes	—
92	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
93	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇	#500	1	2	yes	—
95	C₄×C₇	C₂×D₁₆	1	4	yes	—
96	C₄×C₇	C₂×Q₃₂	1	4	yes	—
97	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
99	Dic₇	—	8	8	no	—
100	C₂×Q₈	—	28	8	no	—
101	C₂×Q₈	—	56	8	no	—
102	C₂²×C₄	—	28	4	no	—
103	C₂²×C₄	D₅₆	1	1	yes	characteristic
104	C₂²×C₄	D₅₆	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
106	C₂×C₈	Dic₂₈	1	4	yes	—
107	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
108	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
109	C₂×C₈	C₂×D₂₈	1	2	yes	—
110	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
111	C₄⋊C₄	—	14	8	no	—
112	C₄⋊C₄	—	28	4	no	—
113	C₄⋊C₄	—	28	8	no	—
114	C₄²	D₅₆	1	2	yes	—
115	C₄²	C₂×D₂₈	1	2	yes	—
116	C₁₆	C₂×Dic₁₄	1	8	yes	complemented
117	C₂×C₇	#2252	1	4	yes	complemented
118	C₂×C₇	C₂×#500	1	1	yes	characteristic
119	C₂×C₇	C₂²×D₁₆	1	1	yes	characteristic
120	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
121	C₂³	C₂×D₅₆	1	1	yes	characteristic
122	Q₈	—	56	16	no	—
123	C₂×C₄	—	28	8	no	—
124	C₂×C₄	—	56	8	no	—
125	C₂×C₄	D₁₁₂	1	4	yes	—
126	C₂×C₄	Dic₅₆	1	4	yes	—
127	C₂×C₄	#6432	1	2	yes	—
128	C₂×C₄	C₂×D₅₆	1	2	yes	—
129	C₂×C₄	C₂×D₅₆	1	2	yes	—
130	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
132	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
133	C₈	#6432	1	2	yes	—
134	C₈	C₂×Dic₂₈	1	4	yes	—
135	C₈	C₂²×Dic₁₄	1	2	yes	—
136	C₇	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
137	C₂²	#89454	1	2	yes	—
138	C₂²	C₂×D₁₁₂	1	2	yes	—
139	C₂²	C₂×Dic₅₆	1	2	yes	—
140	C₂²	C₂²×D₅₆	1	1	yes	characteristic
141	C₄	—	56	8	no	—
142	C₄	#89454	1	2	yes	—
143	C₄	C₂×D₁₁₂	1	4	yes	—
144	C₄	C₂×Dic₅₆	1	4	yes	—
145	C₄	C₂×#6432	1	1	yes	characteristic
146	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
147	C₂	#11726798	1	4	yes	direct product factor
148	C₂	C₂×#89454	1	1	yes	characteristic
149	C₂	C₂²×D₁₁₂	1	1	yes	characteristic
150	C₂	C₂²×Dic₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	C₂×#2252
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. #11726798	C₂	direct (N × H)
23. C₂×C₇×C₁₆	Q₈	semidirect
50. C₇×C₁₆	C₂×Q₈	semidirect
83. C₂×C₁₆	Dic₁₄	semidirect
116. C₁₆	C₂×Dic₁₄	semidirect
117. C₂×C₇	#2252	semidirect
136. C₇	C₂×#2252	semidirect
147. C₂	#11726798	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×#89454	2
C₂×C₄×C₇×C₁₆	2
C₂×#89299	2
#11726798	2
C₂×#2252	7

Subgroups — C2×#11726798 #13509273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726798 #13509273