Finite Group Databasebeta

Subgroups — #13056384

Order 1344.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621657	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621423	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621391	C₂	1	2	yes	maximal, complemented
4	C₃×C₇×#100	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620664	C₂	1	2	yes	maximal, complemented
6	#89519	—	3	1	no	maximal, Hall
7	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂×#4268	—	2	2	no	—
9	C₂×#4268	—	2	2	no	—
10	#66730	—	2	2	no	—
11	D₄×D₂₁	—	2	2	no	—
12	#66728	C₂²	1	2	yes	—
13	C₂×D₈₄	C₂²	1	1	yes	characteristic, complemented
14	C₃×C₇×M₁₆	C₂²	1	2	yes	complemented
15	C₄×Dic₂₁	C₂²	1	1	yes	characteristic, complemented
16	D₁₆₈	—	2	2	no	—
17	#66623	—	2	2	no	—
18	#6548	—	3	1	no	—
19	#6503	—	3	1	no	—
20	#6471	—	3	2	no	—
21	C₇×#100	D₃	1	1	yes	characteristic, complemented
22	#6399	—	3	2	no	—
23	#4747	—	7	1	no	maximal, Hall
24	C₂³×C₃×C₇	D₄	1	2	yes	—
25	C₂²×D₂₁	—	2	2	no	—
26	C₂²×D₂₁	—	2	1	no	—
27	C₃×C₇×D₄	—	2	2	no	—
28	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
29	#4268	—	2	4	no	—
30	#4268	—	2	2	no	—
31	#4268	—	4	4	no	—
32	C₂×Dic₂₁	—	2	2	no	—
33	C₂×Dic₂₁	—	2	1	no	—
34	D₈₄	—	2	2	no	—
35	D₈₄	D₄	1	2	yes	complemented
36	C₄×D₂₁	—	2	2	no	—
37	Dic₄₂	D₄	1	2	yes	complemented
38	C₃×C₇×C₈	—	2	2	no	—
39	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
40	C₂×C₂²⋊D₇	—	6	2	no	—
41	C₂×C₂²⋊D₇	—	6	2	no	—
42	#1159	—	6	2	no	—
43	D₄×D₇	—	6	2	no	—
44	#1157	—	3	2	no	—
45	C₂×D₂₈	—	3	1	no	—
46	C₇×M₁₆	C₂×D₃	1	2	yes	complemented
47	C₄×Dic₇	—	3	1	no	—
48	D₅₆	—	6	2	no	—
49	#1132	—	6	2	no	—
50	#1004	—	7	1	no	—
51	#940	—	7	1	no	—
52	#908	—	7	2	no	—
53	C₃×#100	D₇	1	1	yes	characteristic, complemented
54	#825	—	7	2	no	—
55	C₂²×C₃×C₇	—	2	2	no	—
56	C₂²×C₃×C₇	—	4	2	no	—
57	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×D₂₁	—	2	2	no	—
59	C₂×D₂₁	—	4	2	no	—
60	C₂×D₂₁	—	4	1	no	—
61	C₂×D₂₁	—	8	2	no	—
62	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
63	Dic₂₁	—	2	2	no	—
64	Dic₂₁	—	2	2	no	—
65	#453	—	21	1	no	Sylow
66	C₂³×C₇	D₁₂	1	2	yes	—
67	C₂²×D₇	—	6	2	no	—
68	C₂²×D₇	—	6	1	no	—
69	C₇×D₄	—	2	2	no	—
70	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
71	C₂²⋊D₇	—	6	4	no	—
72	C₂²⋊D₇	—	6	2	no	—
73	C₂²⋊D₇	—	12	4	no	—
74	C₂×Dic₇	—	6	2	no	—
75	C₂×Dic₇	—	6	1	no	—
76	D₂₈	—	3	2	no	—
77	D₂₈	—	6	2	no	—
78	C₄×D₇	—	6	2	no	—
79	Dic₁₄	—	3	2	no	—
80	C₇×C₈	—	2	2	no	—
81	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
82	C₂×#67	—	14	2	no	—
83	C₂×#67	—	14	2	no	—
84	#237	—	14	2	no	—
85	D₃×D₄	—	14	2	no	—
86	#235	—	7	2	no	—
87	C₂×D₁₂	—	7	1	no	—
88	C₃×M₁₆	C₂×D₇	1	2	yes	complemented
89	C₄×Dic₃	—	7	1	no	—
90	D₂₄	—	14	2	no	—
91	#204	—	14	2	no	—
92	C₂×C₃×C₇	—	2	1	no	—
93	C₂×C₃×C₇	—	4	2	no	—
94	C₂×C₃×C₇	#120	1	1	yes	characteristic
95	D₂₁	—	4	2	no	—
96	D₂₁	—	8	1	no	—
97	H(5,𝔽₂)	—	21	1	no	—
98	AGL(1,ℤ/8)	—	21	2	no	—
99	#127	—	21	1	no	—
100	#104	—	21	2	no	—
101	#100	D₂₁	1	1	yes	characteristic, complemented
102	C₂²×C₇	—	2	2	no	—
103	C₂²×C₇	—	4	2	no	—
104	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
105	C₂×D₇	—	6	2	no	—
106	C₂×D₇	—	12	2	no	—
107	C₂×D₇	—	12	1	no	—
108	C₂×D₇	—	24	2	no	—
109	C₄×C₇	D₃×D₄	1	2	yes	—
110	Dic₇	—	6	2	no	—
111	Dic₇	—	6	2	no	—
112	C₂³×C₃	D₂₈	1	2	yes	—
113	C₂²×D₃	—	14	2	no	—
114	C₂²×D₃	—	14	1	no	—
115	C₃×D₄	—	2	2	no	—
116	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
117	#67	—	14	4	no	—
118	#67	—	14	2	no	—
119	#67	—	28	4	no	—
120	C₂×Dic₃	—	14	2	no	—
121	C₂×Dic₃	—	14	1	no	—
122	D₁₂	—	7	2	no	—
123	D₁₂	—	14	2	no	—
124	C₄×D₃	—	14	2	no	—
125	Dic₆	—	7	2	no	—
126	C₃×C₈	—	2	2	no	—
127	C₃×C₇	#453	1	1	yes	characteristic, complemented, Hall
128	#41	—	21	2	no	—
129	#41	—	42	2	no	—
130	C₂×D₄	—	21	1	no	—
131	C₂×D₄	—	42	2	no	—
132	C₂×D₄	—	42	2	no	—
133	C₂×D₄	—	42	2	no	—
134	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
135	SD₁₆	—	42	2	no	—
136	D₈	—	42	2	no	—
137	M₁₆	C₂×D₂₁	1	2	yes	complemented
138	C₄²	—	21	1	no	—
139	C₂×C₇	—	2	1	no	—
140	C₂×C₇	—	4	2	no	—
141	C₂×C₇	#882	1	1	yes	characteristic
142	D₇	—	12	2	no	—
143	D₇	—	24	1	no	—
144	C₂²×C₃	—	2	2	no	—
145	C₂²×C₃	—	4	2	no	—
146	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
147	C₂×D₃	—	14	2	no	—
148	C₂×D₃	—	28	2	no	—
149	C₂×D₃	—	28	1	no	—
150	C₂×D₃	—	56	2	no	—
151	C₃×C₄	D₄×D₇	1	2	yes	—
152	Dic₃	—	14	2	no	—
153	Dic₃	—	14	2	no	—
154	C₂³	—	42	2	no	—
155	C₂³	—	42	1	no	—
156	C₂³	D₈₄	1	2	yes	—
157	Q₈	—	21	2	no	—
158	D₄	—	2	2	no	—
159	D₄	—	21	2	no	—
160	D₄	—	42	4	no	—
161	D₄	—	42	2	no	—
162	D₄	—	42	2	no	—
163	D₄	—	84	4	no	—
164	C₂×C₄	—	42	2	no	—
165	C₂×C₄	—	42	1	no	—
166	C₂×C₄	—	42	2	no	—
167	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₇	#4747	1	1	yes	characteristic, complemented, Sylow
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	—	4	2	no	—
172	C₂×C₃	#6445	1	1	yes	characteristic
173	D₃	—	28	2	no	—
174	D₃	—	56	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	42	2	no	—
178	C₂²	—	84	2	no	—
179	C₂²	—	84	1	no	—
180	C₂²	—	168	2	no	—
181	C₂²	C₂×D₈₄	1	1	yes	characteristic
182	C₄	—	42	2	no	—
183	C₄	—	42	2	no	—
184	C₄	D₄×D₂₁	1	2	yes	—
185	C₃	#89519	1	1	yes	characteristic, complemented, Sylow
186	C₂	—	2	1	no	—
187	C₂	—	4	2	no	—
188	C₂	—	84	2	no	—
189	C₂	—	168	1	no	—
190	C₂	#10621365	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10621365
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₁
Fitting	C₃×C₇×#100	C₂
socle	C₂×C₃×C₇	#120
hypercenter	#100	D₂₁
hypocenter	C₃×C₇	#453
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
1. #10621657	C₂	semidirect
2. #10621423	C₂	semidirect
3. #10621391	C₂	semidirect
4. C₃×C₇×#100	C₂	semidirect
5. #10620664	C₂	semidirect
13. C₂×D₈₄	C₂²	semidirect
14. C₃×C₇×M₁₆	C₂²	semidirect
15. C₄×Dic₂₁	C₂²	semidirect
21. C₇×#100	D₃	semidirect
35. D₈₄	D₄	semidirect
37. Dic₄₂	D₄	semidirect
46. C₇×M₁₆	C₂×D₃	semidirect
53. C₃×#100	D₇	semidirect
88. C₃×M₁₆	C₂×D₇	semidirect
101. #100	D₂₁	semidirect
127. C₃×C₇	#453	semidirect
137. M₁₆	C₂×D₂₁	semidirect
169. C₇	#4747	semidirect
185. C₃	#89519	semidirect

Maximal subgroups

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

subgroup	index
#10621657	2
#10621423	2
#10621391	2
C₃×C₇×#100	2
#10620664	2
#89519	3
#4747	7