Subgroups — #90418⋊C₃ #13057455

Order 1344.

560 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
2	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#6509⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6459⋊C₃	C₂	1	4	yes	maximal, complemented
5	#6456⋊C₃	C₂	1	2	yes	maximal
6	#6451⋊C₃	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
8	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#90418	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
11	Q₈×AGL(1,𝔽₇)	—	2	8	no	—
12	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
13	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
14	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	4	yes	—
15	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	4	yes	—
16	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
21	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
22	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
23	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
24	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
25	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
26	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
27	C₂×Q₈×D₇	C₂×C₃	1	2	yes	complemented
28	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	#6509	C₂×C₃	1	2	yes	complemented
30	#6459	C₂×C₃	1	4	yes	complemented
31	#6456	C₂×C₃	1	2	yes	—
32	#6451	C₂×C₃	1	2	yes	complemented
33	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic
34	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×D₄×Q₈	—	7	1	no	maximal, Hall
36	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
37	Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
38	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
39	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
40	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
42	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
43	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
44	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
45	C₄⋊AGL(1,𝔽₇)	Q₈	1	4	yes	complemented
46	C₄×AGL(1,𝔽₇)	—	2	4	no	—
47	C₄×AGL(1,𝔽₇)	—	2	8	no	—
48	ΓL(1,𝔽₈)⋊Q₈	—	2	8	no	—
49	ΓL(1,𝔽₈)⋊Q₈	D₄	1	4	yes	complemented
50	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
51	Q₈×D₇	—	2	8	no	—
52	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
54	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
55	C₂×Dic₁₄	C₂²×C₃	1	4	yes	—
56	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
57	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
60	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
61	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
62	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
63	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
64	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
65	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
66	C₄×Dic₇	C₂²×C₃	1	2	yes	—
67	C₂²×C₃×Q₈	—	7	2	no	—
68	C₃×C₄⋊Q₈	—	7	2	no	—
69	C₃×C₄⋊Q₈	—	7	1	no	—
70	C₃×C₂²⋊Q₈	—	7	4	no	—
71	C₃×C₂²⋊Q₈	—	7	2	no	—
72	C₃×C₄×Q₈	—	7	1	no	—
73	C₃×C₄×D₄	—	7	1	no	—
74	C₃×C₄×D₄	—	7	2	no	—
75	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
76	C₂×AGL(1,𝔽₇)	—	2	2	no	—
77	C₂×AGL(1,𝔽₇)	—	2	2	no	—
78	C₂×AGL(1,𝔽₇)	C₂×Q₈	1	4	yes	complemented
79	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
80	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
81	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
82	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	complemented
83	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
84	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
85	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	4	yes	—
86	D₄×Q₈	—	7	1	no	Sylow
87	C₂²×D₇	C₂³×C₃	1	2	yes	—
88	C₇×Q₈	—	2	4	no	—
89	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
93	C₂×Dic₇	C₂³×C₃	1	2	yes	—
94	C₂×Dic₇	C₂³×C₃	1	2	yes	—
95	C₂×Dic₇	C₂³×C₃	1	2	yes	—
96	D₂₈	C₃×Q₈	1	4	yes	complemented
97	C₄×D₇	—	2	4	no	—
98	C₄×D₇	—	2	8	no	—
99	Dic₁₄	—	2	8	no	—
100	Dic₁₄	C₃×D₄	1	4	yes	complemented
101	C₂×C₃×Q₈	—	7	1	no	—
102	C₂×C₃×Q₈	—	7	4	no	—
103	C₂×C₃×Q₈	—	7	2	no	—
104	C₂×C₃×Q₈	—	14	8	no	—
105	C₂×C₃×D₄	—	7	1	no	—
106	C₂²×C₃×C₄	—	7	4	no	—
107	C₂²×C₃×C₄	—	7	2	no	—
108	C₃×C₄⋊C₄	—	7	1	no	—
109	C₃×C₄⋊C₄	—	7	2	no	—
110	C₃×C₄⋊C₄	—	7	1	no	—
111	C₃×C₄⋊C₄	—	7	4	no	—
112	C₃×C₄⋊C₄	—	7	2	no	—
113	C₃×C₄⋊C₄	—	7	2	no	—
114	C₃×C₂²⋊C₄	—	7	4	no	—
115	C₃×C₂²⋊C₄	—	7	2	no	—
116	C₃×C₄²	—	7	2	no	—
117	C₃×C₄²	—	7	1	no	—
118	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
119	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
120	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
121	AGL(1,𝔽₇)	—	2	4	no	—
122	C₂²×Q₈	—	7	2	no	—
123	C₄⋊Q₈	—	7	2	no	—
124	C₄⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
125	C₂²⋊Q₈	—	7	2	no	—
126	C₂²⋊Q₈	—	7	4	no	—
127	C₄×Q₈	—	7	1	no	—
128	C₄×D₄	—	7	2	no	—
129	C₄×D₄	—	7	1	no	—
130	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
131	C₂×D₇	—	2	2	no	—
132	C₂×D₇	—	2	2	no	—
133	C₂×D₇	C₂×C₃×Q₈	1	4	yes	complemented
134	C₄×C₇	—	2	4	no	—
135	C₄×C₇	—	2	1	no	—
136	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
137	C₄×C₇	C₂×C₃×Q₈	1	2	yes	complemented
138	Dic₇	—	2	2	no	—
139	Dic₇	—	2	2	no	—
140	Dic₇	C₂×C₃×D₄	1	4	yes	—
141	C₂³×C₃	—	7	2	no	—
142	C₃×Q₈	—	7	4	no	—
143	C₃×Q₈	—	14	4	no	—
144	C₃×Q₈	—	14	8	no	—
145	C₃×D₄	—	7	4	no	—
146	C₂×C₃×C₄	—	7	2	no	—
147	C₂×C₃×C₄	—	7	2	no	—
148	C₂×C₃×C₄	—	7	2	no	—
149	C₂×C₃×C₄	—	7	1	no	—
150	C₂×C₃×C₄	—	7	1	no	—
151	C₂×C₃×C₄	—	7	1	no	—
152	C₂×C₃×C₄	—	7	4	no	—
153	C₂×C₃×C₄	—	14	8	no	—
154	C₂×C₃×C₄	—	14	4	no	—
155	ΓL(1,𝔽₈)	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
156	C₂×Q₈	—	7	1	no	—
157	C₂×Q₈	—	7	4	no	—
158	C₂×Q₈	—	14	8	no	—
159	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
160	C₂×D₄	—	7	1	no	—
161	C₂²×C₄	—	7	4	no	—
162	C₂²×C₄	—	7	2	no	—
163	C₄⋊C₄	—	7	2	no	—
164	C₄⋊C₄	—	7	4	no	—
165	C₄⋊C₄	—	7	1	no	—
166	C₄⋊C₄	—	7	1	no	—
167	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
168	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
169	C₂²⋊C₄	—	7	4	no	—
170	C₂²⋊C₄	—	7	2	no	—
171	C₄²	—	7	2	no	—
172	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
173	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
175	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
176	D₇	—	2	4	no	—
177	C₂²×C₃	—	7	4	no	—
178	C₂²×C₃	—	7	1	no	—
179	C₂²×C₃	—	14	2	no	—
180	C₂²×C₃	—	14	2	no	—
181	C₃×C₄	—	7	4	no	—
182	C₃×C₄	—	7	2	no	—
183	C₃×C₄	—	7	2	no	—
184	C₃×C₄	—	14	4	no	—
185	C₃×C₄	—	14	1	no	—
186	C₃×C₄	—	14	2	no	—
187	C₃×C₄	—	14	2	no	—
188	C₂³	—	7	2	no	—
189	Q₈	—	2	4	no	—
190	Q₈	—	7	4	no	—
191	Q₈	—	14	8	no	—
192	D₄	—	7	4	no	—
193	C₂×C₄	—	7	2	no	—
194	C₂×C₄	—	7	2	no	—
195	C₂×C₄	—	7	2	no	—
196	C₂×C₄	—	14	8	no	—
197	C₂×C₄	—	14	4	no	—
198	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
199	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	4	yes	—
200	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
201	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
202	C₇	C₃×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
203	C₂×C₃	—	7	1	no	—
204	C₂×C₃	—	7	1	no	—
205	C₂×C₃	—	7	1	no	—
206	C₂×C₃	—	14	4	no	—
207	C₂²	—	7	4	no	—
208	C₂²	—	14	2	no	—
209	C₂²	—	14	2	no	—
210	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	1	no	—
213	C₄	—	7	4	no	—
214	C₄	—	14	2	no	—
215	C₄	—	14	2	no	—
216	C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
217	C₄	Q₈×AGL(1,𝔽₇)	1	2	yes	complemented
218	C₃	—	7	1	no	Sylow
219	C₂	—	14	4	no	—
220	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
221	C₂	C₂×Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
222	C₂	#10621516	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×C₄⋊Q₈	C₂×C₃
prime radical	C₂×C₄⋊AGL(1,𝔽₇)	C₂²
hypercenter	C₄⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×D₄×Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×AGL(1,𝔽₇)	C₂	semidirect
2. ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	semidirect
3. #6509⋊C₃	C₂	semidirect
4. #6459⋊C₃	C₂	semidirect
6. #6451⋊C₃	C₂	semidirect
8. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
9. #90418	C₃	semidirect
27. C₂×Q₈×D₇	C₂×C₃	semidirect
28. C₇×C₄⋊Q₈	C₂×C₃	semidirect
29. #6509	C₂×C₃	semidirect
30. #6459	C₂×C₃	semidirect
32. #6451	C₂×C₃	semidirect
34. C₄×Dic₁₄	C₂×C₃	semidirect
45. C₄⋊AGL(1,𝔽₇)	Q₈	semidirect
49. ΓL(1,𝔽₈)⋊Q₈	D₄	semidirect
78. C₂×AGL(1,𝔽₇)	C₂×Q₈	semidirect
82. C₄×ΓL(1,𝔽₈)	C₂×Q₈	semidirect
96. D₂₈	C₃×Q₈	semidirect
100. Dic₁₄	C₃×D₄	semidirect
124. C₄⋊Q₈	AGL(1,𝔽₇)	semidirect
133. C₂×D₇	C₂×C₃×Q₈	semidirect
137. C₄×C₇	C₂×C₃×Q₈	semidirect
155. ΓL(1,𝔽₈)	D₄×Q₈	semidirect
202. C₇	C₃×D₄×Q₈	semidirect
217. C₄	Q₈×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)×C₄⋊Q₈	2
#6509⋊C₃	2
#6459⋊C₃	2
#6456⋊C₃	2
#6451⋊C₃	2
C₄×C₄⋊AGL(1,𝔽₇)	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90418	3
C₃×D₄×Q₈	7

Subgroups — #90418⋊C3 #13057455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90418⋊C₃ #13057455