Subgroups — #90390⋊C₃ #13057433

Order 1344.

392 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6461⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6461⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6460⋊C₃	C₂	1	2	yes	maximal, complemented
5	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
6	#6457⋊C₃	C₂	1	2	yes	maximal
7	#6451⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6438⋊C₃	C₂	1	1	yes	maximal, characteristic
9	#6433⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#90390	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
13	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
15	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
16	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
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	2	yes	—
22	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
23	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
24	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
25	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
26	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
27	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
31	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
32	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
33	#6461	C₂×C₃	1	2	yes	complemented
34	#6461	C₂×C₃	1	2	yes	complemented
35	#6460	C₂×C₃	1	2	yes	complemented
36	#6459	C₂×C₃	1	2	yes	complemented
37	#6457	C₂×C₃	1	2	yes	—
38	#6451	C₂×C₃	1	2	yes	complemented
39	#6438	C₂×C₃	1	1	yes	characteristic
40	#6433	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#563	—	7	1	no	maximal, Hall
42	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
43	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
45	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
46	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
48	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
49	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
50	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
51	C₄⋊AGL(1,𝔽₇)	—	4	1	no	—
52	C₄×AGL(1,𝔽₇)	—	4	2	no	—
53	ΓL(1,𝔽₈)⋊Q₈	—	4	1	no	—
54	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
55	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
57	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
59	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
64	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
65	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
66	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
67	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
68	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
69	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
70	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
72	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
73	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄×Dic₇	C₂²×C₃	1	2	yes	—
76	C₃×C₄⋊Q₈	—	7	2	no	—
77	C₃×#126	—	7	2	no	—
78	C₃×#126	—	7	2	no	—
79	C₃×#125	—	7	1	no	—
80	C₃×#125	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#123	—	7	2	no	—
83	C₃×C₂²⋊Q₈	—	7	2	no	—
84	C₃×C₂²⋊Q₈	—	7	2	no	—
85	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
86	C₂×AGL(1,𝔽₇)	—	2	2	no	—
87	C₂×AGL(1,𝔽₇)	—	2	2	no	—
88	C₂×AGL(1,𝔽₇)	—	2	2	no	—
89	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
90	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
91	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
92	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
93	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
94	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
95	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
96	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
97	#563	—	7	1	no	Sylow
98	C₂²×D₇	C₂³×C₃	1	2	yes	—
99	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
100	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
103	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
104	C₂×Dic₇	C₂³×C₃	1	2	yes	—
105	C₂×Dic₇	C₂³×C₃	1	2	yes	—
106	C₂×Dic₇	C₂³×C₃	1	2	yes	—
107	D₂₈	—	4	1	no	—
108	C₄×D₇	—	4	2	no	—
109	Dic₁₄	—	4	1	no	—
110	Dic₁₄	—	4	2	no	—
111	C₂×C₃×Q₈	—	7	2	no	—
112	C₂×C₃×Q₈	—	7	1	no	—
113	C₂×C₃×D₄	—	7	1	no	—
114	C₂²×C₃×C₄	—	7	2	no	—
115	C₃×C₄⋊C₄	—	7	1	no	—
116	C₃×C₄⋊C₄	—	7	2	no	—
117	C₃×C₄⋊C₄	—	7	2	no	—
118	C₃×C₄⋊C₄	—	7	2	no	—
119	C₃×C₄⋊C₄	—	7	2	no	—
120	C₃×C₄⋊C₄	—	7	1	no	—
121	C₃×C₄⋊C₄	—	7	1	no	—
122	C₃×C₄⋊C₄	—	7	2	no	—
123	C₃×C₄⋊C₄	—	7	2	no	—
124	C₃×C₄⋊C₄	—	7	1	no	—
125	C₃×C₂²⋊C₄	—	7	2	no	—
126	C₃×C₂²⋊C₄	—	7	2	no	—
127	C₃×C₂²⋊C₄	—	7	2	no	—
128	C₃×C₂²⋊C₄	—	7	2	no	—
129	C₃×C₂²⋊C₄	—	7	2	no	—
130	C₃×C₄²	—	7	2	no	—
131	C₃×C₄²	—	7	1	no	—
132	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
135	AGL(1,𝔽₇)	—	4	2	no	—
136	C₄⋊Q₈	—	7	2	no	—
137	#126	—	7	2	no	—
138	#126	—	7	2	no	—
139	#125	—	7	1	no	—
140	#125	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
141	#124	—	7	1	no	—
142	#123	—	7	2	no	—
143	C₂²⋊Q₈	—	7	2	no	—
144	C₂²⋊Q₈	—	7	2	no	—
145	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
146	C₂×D₇	—	2	2	no	—
147	C₂×D₇	—	2	2	no	—
148	C₂×D₇	—	2	2	no	—
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	—	2	1	no	—
151	C₄×C₇	—	2	1	no	—
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	1	no	—
154	Dic₇	—	2	2	no	—
155	Dic₇	—	2	2	no	—
156	Dic₇	—	2	2	no	—
157	C₂³×C₃	—	7	2	no	—
158	C₃×Q₈	—	28	2	no	—
159	C₃×Q₈	—	28	1	no	—
160	C₃×D₄	—	28	1	no	—
161	C₂×C₃×C₄	—	7	2	no	—
162	C₂×C₃×C₄	—	7	2	no	—
163	C₂×C₃×C₄	—	7	2	no	—
164	C₂×C₃×C₄	—	7	2	no	—
165	C₂×C₃×C₄	—	7	1	no	—
166	C₂×C₃×C₄	—	7	1	no	—
167	C₂×C₃×C₄	—	7	1	no	—
168	C₂×C₃×C₄	—	7	2	no	—
169	C₂×C₃×C₄	—	28	2	no	—
170	ΓL(1,𝔽₈)	#563	1	1	yes	characteristic, complemented, Hall
171	C₂×Q₈	—	7	2	no	—
172	C₂×Q₈	—	7	1	no	—
173	C₂×D₄	—	7	1	no	—
174	C₂²×C₄	—	7	2	no	—
175	C₄⋊C₄	—	7	2	no	—
176	C₄⋊C₄	—	7	2	no	—
177	C₄⋊C₄	—	7	2	no	—
178	C₄⋊C₄	—	7	1	no	—
179	C₄⋊C₄	—	7	2	no	—
180	C₄⋊C₄	—	7	1	no	—
181	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
182	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
183	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
184	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
185	C₂²⋊C₄	—	7	2	no	—
186	C₂²⋊C₄	—	7	2	no	—
187	C₂²⋊C₄	—	7	2	no	—
188	C₂²⋊C₄	—	7	2	no	—
189	C₂²⋊C₄	—	7	2	no	—
190	C₄²	—	7	2	no	—
191	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
192	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
193	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
194	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
195	D₇	—	4	2	no	—
196	C₂²×C₃	—	7	1	no	—
197	C₂²×C₃	—	14	2	no	—
198	C₂²×C₃	—	14	2	no	—
199	C₂²×C₃	—	14	2	no	—
200	C₃×C₄	—	14	2	no	—
201	C₃×C₄	—	14	1	no	—
202	C₃×C₄	—	14	1	no	—
203	C₃×C₄	—	14	2	no	—
204	C₃×C₄	—	14	1	no	—
205	C₃×C₄	—	14	2	no	—
206	C₃×C₄	—	14	2	no	—
207	C₃×C₄	—	14	2	no	—
208	C₂³	—	7	2	no	—
209	Q₈	—	28	2	no	—
210	Q₈	—	28	1	no	—
211	D₄	—	28	1	no	—
212	C₂×C₄	—	7	2	no	—
213	C₂×C₄	—	7	2	no	—
214	C₂×C₄	—	7	2	no	—
215	C₂×C₄	—	28	2	no	—
216	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
217	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
218	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
219	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
220	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
221	C₇	C₃×#563	1	1	yes	characteristic, complemented, Sylow
222	C₂×C₃	—	7	1	no	—
223	C₂×C₃	—	7	1	no	—
224	C₂×C₃	—	7	1	no	—
225	C₂×C₃	—	28	2	no	—
226	C₂²	—	14	2	no	—
227	C₂²	—	14	2	no	—
228	C₂²	—	14	2	no	—
229	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	14	2	no	—
236	C₄	—	14	2	no	—
237	C₄	—	14	2	no	—
238	C₃	—	7	1	no	Sylow
239	C₂	—	28	2	no	—
240	C₂	#10621516	1	1	yes	characteristic
241	C₂	#10621518	1	1	yes	characteristic
242	C₂	#10621519	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#125	C₂×C₃
prime radical	C₂×C₄⋊AGL(1,𝔽₇)	C₂²
hypercenter	#125	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#563
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#125	C₂	semidirect
2. #6461⋊C₃	C₂	semidirect
3. #6461⋊C₃	C₂	semidirect
4. #6460⋊C₃	C₂	semidirect
5. #6459⋊C₃	C₂	semidirect
7. #6451⋊C₃	C₂	semidirect
9. #6433⋊C₃	C₂	semidirect
10. #90390	C₃	semidirect
32. C₇×#125	C₂×C₃	semidirect
33. #6461	C₂×C₃	semidirect
34. #6461	C₂×C₃	semidirect
35. #6460	C₂×C₃	semidirect
36. #6459	C₂×C₃	semidirect
38. #6451	C₂×C₃	semidirect
40. #6433	C₂×C₃	semidirect
140. #125	AGL(1,𝔽₇)	semidirect
170. ΓL(1,𝔽₈)	#563	semidirect
221. C₇	C₃×#563	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×#125	2
#6461⋊C₃	2
#6460⋊C₃	2
#6459⋊C₃	2
#6457⋊C₃	2
#6451⋊C₃	2
#6438⋊C₃	2
#6433⋊C₃	2
#90390	3
C₃×#563	7

Subgroups — #90390⋊C3 #13057433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90390⋊C₃ #13057433