Subgroups — C₂×#11726799 #13509278

Order 1792.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89455	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#89300	C₂	1	2	yes	maximal
4	C₂×#89299	C₂	1	2	yes	maximal
5	#11726799	C₂	1	8	yes	maximal, direct product factor
6	C₂×#6433	—	2	2	no	—
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6392	C₂²	1	4	yes	—
11	C₂×#6391	—	2	2	no	—
12	#89455	—	2	4	no	—
13	#89455	C₂²	1	8	yes	—
14	C₄×C₇×C₁₆	C₂²	1	4	yes	—
15	#89300	C₂²	1	8	yes	—
16	#89299	C₂²	1	8	yes	—
17	C₂×#2253	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇	—	2	4	no	—
22	C₂×Dic₇⋊C₄	—	4	4	no	—
23	#6433	—	2	8	no	—
24	#6433	—	4	4	no	—
25	C₂×C₇×C₁₆	Q₈	1	4	yes	—
26	C₂×C₇×C₁₆	Q₈	1	4	yes	—
27	C₂×C₇×C₁₆	C₂³	1	2	yes	—
28	C₂×C₇×C₁₆	C₂³	1	2	yes	—
29	C₄×C₇×C₈	D₄	1	2	yes	—
30	C₄×C₇×C₈	C₂³	1	2	yes	—
31	#6392	—	2	4	no	—
32	#6392	C₂³	1	8	yes	—
33	#6391	—	2	8	no	—
34	C₂×#499	—	7	2	no	—
35	#2253	—	7	8	no	—
36	C₂×#367	—	7	2	no	—
37	C₂×#366	—	7	2	no	—
38	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
39	C₂²×C₄×C₇	D₈	1	2	yes	—
40	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₇	—	4	4	no	—
42	C₂×C₇×C₈	Q₁₆	1	4	yes	—
43	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
45	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
46	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
47	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
48	C₄²×C₇	D₈	1	2	yes	—
49	C₄²×C₇	C₂×D₄	1	2	yes	—
50	C₄⋊Dic₇	—	2	8	no	—
51	C₄⋊Dic₇	—	4	4	no	—
52	Dic₇⋊C₄	—	4	16	no	—
53	C₇×C₁₆	C₂×Q₈	1	4	yes	—
54	C₇×C₁₆	C₂×Q₈	1	4	yes	—
55	C₂×#125	—	14	2	no	—
56	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
57	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
58	#499	—	7	8	no	—
59	#499	—	14	4	no	—
60	C₂×#107	—	7	4	no	—
61	C₂×#106	—	14	2	no	—
62	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
63	#367	—	7	8	no	—
64	#366	—	7	8	no	—
65	C₄×C₁₆	C₂×D₇	1	4	yes	—
66	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇	—	2	4	no	—
68	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
69	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
70	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
72	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₇	—	4	8	no	—
74	C₂×Dic₇	—	8	8	no	—
75	C₇×C₈	C₄⋊Q₈	1	2	yes	—
76	C₇×C₈	C₂×Q₁₆	1	4	yes	—
77	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
78	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₈	D₂₈	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
81	#125	—	14	8	no	—
82	#125	—	28	4	no	—
83	C₂×C₄⋊C₄	—	14	4	no	—
84	C₂×C₄⋊C₄	—	28	4	no	—
85	C₂×C₄²	D₂₈	1	1	yes	characteristic
86	C₂×C₁₆	Dic₁₄	1	4	yes	—
87	C₂×C₁₆	Dic₁₄	1	4	yes	—
88	C₂×C₁₆	C₂²×D₇	1	2	yes	—
89	C₂×C₁₆	C₂²×D₇	1	2	yes	—
90	#107	—	7	8	no	—
91	#107	—	14	4	no	—
92	#106	—	14	8	no	—
93	C₄×C₈	D₂₈	1	2	yes	—
94	C₄×C₈	C₂²×D₇	1	2	yes	—
95	C₂²×C₇	#500	1	2	yes	—
96	C₂²×C₇	#508	1	4	yes	—
97	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₇	—	2	4	no	—
99	C₄×C₇	#500	1	2	yes	—
100	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₇	—	8	8	no	—
103	C₂²×C₄	—	28	4	no	—
104	C₂²×C₄	D₅₆	1	2	yes	—
105	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
106	C₂×C₈	Dic₂₈	1	4	yes	—
107	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
108	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
109	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
110	C₂×C₈	C₂×D₂₈	1	2	yes	—
111	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
112	C₄⋊C₄	—	14	8	no	—
113	C₄⋊C₄	—	28	16	no	—
114	C₄⋊C₄	—	28	4	no	—
115	C₄²	D₅₆	1	2	yes	—
116	C₄²	C₂×D₂₈	1	2	yes	—
117	C₁₆	C₂×Dic₁₄	1	4	yes	—
118	C₁₆	C₂×Dic₁₄	1	4	yes	—
119	C₂×C₇	#2253	1	4	yes	complemented
120	C₂×C₇	C₂×#500	1	1	yes	characteristic
121	C₂×C₇	C₂×#508	1	2	yes	—
122	C₂³	C₂×D₅₆	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	28	8	no	—
125	C₂×C₄	—	56	8	no	—
126	C₂×C₄	#6432	1	2	yes	—
127	C₂×C₄	C₂×D₅₆	1	4	yes	—
128	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
130	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
131	C₈	#6432	1	2	yes	—
132	C₈	C₂×Dic₂₈	1	4	yes	—
133	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
134	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
135	C₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
136	C₂²	#89454	1	2	yes	—
137	C₂²	#89676	1	4	yes	—
138	C₂²	C₂²×D₅₆	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	56	8	no	—
141	C₄	#89454	1	2	yes	—
142	C₄	C₂×#6432	1	1	yes	characteristic
143	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
144	C₂	#11726799	1	4	yes	direct product factor
145	C₂	C₂×#89454	1	1	yes	characteristic
146	C₂	C₂×#89676	1	2	yes	—

	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×#2253
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
5. #11726799	C₂	direct (N × H)
119. C₂×C₇	#2253	semidirect
135. C₇	C₂×#2253	semidirect
144. C₂	#11726799	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×#89455	2
C₂×C₄×C₇×C₁₆	2
C₂×#89300	2
C₂×#89299	2
#11726799	2
C₂×#2253	7

Subgroups — C2×#11726799 #13509278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726799 #13509278