Subgroups — #13509288

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735887	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#11729158	C₂	1	4	yes	maximal
4	#11726800	C₂	1	2	yes	maximal, complemented
5	#11726799	C₂	1	4	yes	maximal, complemented
6	#11726798	C₂	1	2	yes	maximal, complemented
7	#90161	—	2	2	no	—
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	#89878	—	2	2	no	—
11	#89878	C₂²	1	4	yes	—
12	#89455	—	2	2	no	—
13	#89455	C₂²	1	4	yes	—
14	#89454	C₂²	1	4	yes	—
15	#89452	—	2	2	no	—
16	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₇×C₁₆	C₂²	1	2	yes	—
18	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
19	#89300	C₂²	1	8	yes	—
20	#89299	C₂²	1	8	yes	—
21	#33726	—	7	1	no	maximal, Sylow
22	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
25	#6489	—	2	4	no	—
26	#6487	—	4	4	no	—
27	#6433	—	2	4	no	—
28	#6432	—	2	4	no	—
29	C₄×Dic₁₄	—	4	4	no	—
30	C₂×C₇×C₁₆	Q₈	1	8	yes	complemented
31	C₂×C₇×C₁₆	C₂³	1	2	yes	—
32	C₂×C₇×C₁₆	C₂³	1	2	yes	—
33	C₄×C₇×C₈	D₄	1	1	yes	characteristic
34	C₄×C₇×C₈	D₄	1	1	yes	characteristic
35	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
37	#6392	—	2	4	no	—
38	#6392	C₂³	1	8	yes	—
39	#6391	—	2	4	no	—
40	#6391	—	2	4	no	—
41	#3160	—	7	2	no	—
42	#2254	—	7	2	no	—
43	#2253	—	7	4	no	—
44	#2252	—	7	2	no	—
45	#2158	—	7	4	no	—
46	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
47	C₂²×C₄×C₇	D₈	1	2	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₄	—	4	4	no	—
50	C₂×C₇×C₈	Q₁₆	1	4	yes	—
51	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
55	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
56	C₄²×C₇	D₈	1	2	yes	—
57	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊Dic₇	—	4	4	no	—
60	C₄⋊Dic₇	—	2	8	no	—
61	C₄⋊Dic₇	—	4	4	no	—
62	Dic₇⋊C₄	—	4	8	no	—
63	C₄×Dic₇	—	4	4	no	—
64	C₇×C₁₆	C₂×Q₈	1	4	yes	—
65	C₇×C₁₆	C₂×Q₈	1	4	yes	—
66	#533	—	14	2	no	—
67	C₂²×C₁₆	C₂×D₇	1	2	yes	—
68	#500	—	7	4	no	—
69	#499	—	7	4	no	—
70	#499	—	14	2	no	—
71	#498	—	14	2	no	—
72	#427	—	7	4	no	—
73	#427	—	14	2	no	—
74	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
75	#367	—	7	8	no	—
76	#366	—	7	8	no	—
77	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
78	C₄×C₁₆	C₂×D₇	1	2	yes	—
79	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
80	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₇	—	4	8	no	—
89	Dic₁₄	—	8	8	no	—
90	C₇×C₈	C₄⋊Q₈	1	2	yes	—
91	C₇×C₈	C₂×Q₁₆	1	2	yes	—
92	C₇×C₈	C₂×Q₁₆	1	2	yes	—
93	C₇×C₈	C₂²×Q₈	1	2	yes	—
94	C₂²×C₈	D₂₈	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
96	C₄⋊Q₈	—	14	4	no	—
97	#125	—	14	4	no	—
98	C₂²⋊Q₈	—	28	4	no	—
99	C₄×Q₈	—	28	4	no	—
100	#117	—	14	4	no	—
101	C₂×C₄²	D₂₈	1	1	yes	characteristic
102	C₂×C₁₆	Dic₁₄	1	8	yes	complemented
103	C₂×C₁₆	C₂²×D₇	1	2	yes	—
104	C₂×C₁₆	C₂²×D₇	1	2	yes	—
105	#107	—	7	8	no	—
106	#107	—	14	4	no	—
107	#106	—	14	4	no	—
108	#106	—	14	4	no	—
109	C₄×C₈	D₂₈	1	1	yes	characteristic
110	C₄×C₈	D₂₈	1	1	yes	characteristic
111	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	#500	1	2	yes	—
115	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₇	—	2	2	no	—
117	C₄×C₇	#500	1	2	yes	—
118	C₄×C₇	#508	1	4	yes	—
119	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
121	Dic₇	—	8	8	no	—
122	C₂×Q₈	—	28	4	no	—
123	C₂²×C₄	D₅₆	1	2	yes	—
124	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
125	C₂×C₈	Dic₂₈	1	4	yes	—
126	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
127	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
128	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
129	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
130	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
131	C₄⋊C₄	—	14	8	no	—
132	C₄⋊C₄	—	28	8	no	—
133	C₄⋊C₄	—	28	4	no	—
134	C₂²⋊C₄	—	28	4	no	—
135	C₄²	—	28	4	no	—
136	C₄²	D₅₆	1	2	yes	—
137	C₄²	C₂×D₂₈	1	1	yes	characteristic
138	C₄²	C₂×D₂₈	1	1	yes	characteristic
139	C₁₆	C₂×Dic₁₄	1	4	yes	—
140	C₁₆	C₂×Dic₁₄	1	4	yes	—
141	C₂×C₇	—	2	2	no	—
142	C₂×C₇	C₂×#500	1	1	yes	characteristic
143	C₂×C₇	C₂×#508	1	2	yes	—
144	C₂³	C₂×D₅₆	1	1	yes	characteristic
145	Q₈	—	56	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	28	8	no	—
148	C₂×C₄	#6432	1	2	yes	—
149	C₂×C₄	C₂×D₅₆	1	2	yes	—
150	C₂×C₄	C₂×D₅₆	1	2	yes	—
151	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
152	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
153	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
154	C₈	#6432	1	2	yes	—
155	C₈	C₂×Dic₂₈	1	2	yes	—
156	C₈	C₂×Dic₂₈	1	2	yes	—
157	C₈	C₂²×Dic₁₄	1	2	yes	—
158	C₇	#33726	1	1	yes	characteristic, complemented, Sylow
159	C₂²	—	2	2	no	—
160	C₂²	#89454	1	2	yes	—
161	C₂²	C₂²×D₅₆	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	56	8	no	—
164	C₄	#89454	1	2	yes	—
165	C₄	#89676	1	4	yes	—
166	C₄	C₂×#6432	1	1	yes	characteristic
167	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#89454	1	1	yes	characteristic
170	C₂	C₂×#89676	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#33726
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ 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. #11726800	C₂	semidirect
5. #11726799	C₂	semidirect
6. #11726798	C₂	semidirect
30. C₂×C₇×C₁₆	Q₈	semidirect
102. C₂×C₁₆	Dic₁₄	semidirect
158. C₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#11735887	2
C₂×C₄×C₇×C₁₆	2
#11729158	2
#11726800	2
#11726799	2
#11726798	2
#33726	7