Subgroups — #90409⋊C₃ #13057447

Order 1344.

624 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1159⋊C₃	C₂	1	2	yes	maximal, complemented
2	ΓL(1,𝔽₈)×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#6503⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6502⋊C₃	C₂	1	4	yes	maximal, complemented
5	#6501⋊C₃	C₂	1	2	yes	maximal, complemented
6	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#90409	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
11	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
13	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	4	yes	—
14	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	4	yes	—
15	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	4	yes	—
16	#1159⋊C₃	—	2	8	no	—
17	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
18	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
19	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
20	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
21	(C₂²⋊Dic₇)⋊C₃	C₂²	1	4	yes	—
22	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
23	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
24	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
25	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
26	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
27	C₂×#1159	C₂×C₃	1	2	yes	complemented
28	C₇×#127	C₂×C₃	1	1	yes	characteristic, complemented
29	#6503	C₂×C₃	1	2	yes	complemented
30	#6502	C₂×C₃	1	4	yes	complemented
31	#6501	C₂×C₃	1	2	yes	complemented
32	D₄×Dic₇	C₂×C₃	1	2	yes	complemented
33	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×#550	—	7	1	no	maximal, Hall
36	C₂³×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
37	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
38	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
39	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
40	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
42	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
43	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	(C₂²⋊D₇)⋊C₃	—	2	8	no	—
45	(C₂²⋊D₇)⋊C₃	—	4	4	no	—
46	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	8	no	—
47	C₂×ΓL(1,𝔽₈)⋊C₄	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	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
51	C₄×AGL(1,𝔽₇)	—	2	4	no	—
52	ΓL(1,𝔽₈)⋊Q₈	D₄	1	4	yes	complemented
53	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
54	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
55	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
56	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
57	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
58	C₂²×Dic₇	C₂²×C₃	1	4	yes	—
59	#1159	—	2	8	no	—
60	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
62	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
63	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
64	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
65	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
66	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
67	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
68	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
69	C₄×Dic₇	C₂²×C₃	1	2	yes	—
70	C₂×C₃×#41	—	7	2	no	—
71	C₃×#127	—	7	2	no	—
72	C₃×#127	—	7	1	no	—
73	C₃×#121	—	7	2	no	—
74	C₃×#121	—	7	4	no	—
75	C₃×C₄×Q₈	—	7	1	no	—
76	C₃×C₄×D₄	—	7	1	no	—
77	C₃×C₄×D₄	—	7	2	no	—
78	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
79	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
80	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
81	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
82	C₂×AGL(1,𝔽₇)	—	2	2	no	—
83	C₂×AGL(1,𝔽₇)	—	2	2	no	—
84	C₂×AGL(1,𝔽₇)	—	2	2	no	—
85	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
86	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
87	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	complemented
88	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
89	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
90	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	4	yes	—
91	#550	—	7	1	no	Sylow
92	C₂³×C₇	C₂³×C₃	1	4	yes	—
93	C₂²×D₇	C₂³×C₃	1	2	yes	—
94	C₇×D₄	—	2	4	no	—
95	C₇×D₄	—	2	4	no	—
96	C₇×D₄	—	4	2	no	—
97	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
100	C₂²⋊D₇	—	2	8	no	—
101	C₂²⋊D₇	—	4	4	no	—
102	C₂×Dic₇	—	2	8	no	—
103	C₂×Dic₇	C₂³×C₃	1	2	yes	—
104	C₂×Dic₇	C₂³×C₃	1	2	yes	—
105	C₂×Dic₇	C₂³×C₃	1	2	yes	—
106	D₂₈	—	2	2	no	—
107	C₄×D₇	—	2	4	no	—
108	Dic₁₄	C₃×D₄	1	4	yes	complemented
109	C₃×#41	—	14	8	no	—
110	C₂×C₃×Q₈	—	7	1	no	—
111	C₂×C₃×D₄	—	7	1	no	—
112	C₂×C₃×D₄	—	7	4	no	—
113	C₂×C₃×D₄	—	7	4	no	—
114	C₂×C₃×D₄	—	7	2	no	—
115	C₂×C₃×D₄	—	7	2	no	—
116	C₂×C₃×D₄	—	7	2	no	—
117	C₂²×C₃×C₄	—	7	2	no	—
118	C₂²×C₃×C₄	—	7	4	no	—
119	C₃×C₄⋊C₄	—	7	1	no	—
120	C₃×C₄⋊C₄	—	7	2	no	—
121	C₃×C₄⋊C₄	—	7	1	no	—
122	C₃×C₂²⋊C₄	—	7	2	no	—
123	C₃×C₂²⋊C₄	—	7	4	no	—
124	C₃×C₄²	—	7	2	no	—
125	C₃×C₄²	—	7	1	no	—
126	C₂×ΓL(1,𝔽₈)	—	4	4	no	—
127	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
128	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
129	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
130	AGL(1,𝔽₇)	—	4	2	no	—
131	C₂×#41	—	7	2	no	—
132	#127	—	7	2	no	—
133	#127	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
134	#121	—	7	2	no	—
135	#121	—	7	4	no	—
136	C₄×Q₈	—	7	1	no	—
137	C₄×D₄	—	7	2	no	—
138	C₄×D₄	—	7	1	no	—
139	C₂²×C₇	—	2	4	no	—
140	C₂²×C₇	—	2	4	no	—
141	C₂²×C₇	—	2	4	no	—
142	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
143	C₂×D₇	—	2	2	no	—
144	C₂×D₇	—	2	2	no	—
145	C₂×D₇	—	2	2	no	—
146	C₄×C₇	—	2	1	no	—
147	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
148	C₄×C₇	C₃×#41	1	2	yes	complemented
149	Dic₇	—	2	2	no	—
150	Dic₇	—	2	2	no	—
151	Dic₇	C₂×C₃×D₄	1	4	yes	—
152	C₂³×C₃	—	7	2	no	—
153	C₂³×C₃	—	7	4	no	—
154	C₃×Q₈	—	7	4	no	—
155	C₃×D₄	—	14	4	no	—
156	C₃×D₄	—	14	4	no	—
157	C₃×D₄	—	14	2	no	—
158	C₃×D₄	—	14	8	no	—
159	C₃×D₄	—	28	2	no	—
160	C₃×D₄	—	28	4	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	1	no	—
165	C₂×C₃×C₄	—	7	1	no	—
166	C₂×C₃×C₄	—	7	1	no	—
167	C₂×C₃×C₄	—	14	4	no	—
168	C₂×C₃×C₄	—	14	8	no	—
169	ΓL(1,𝔽₈)	#550	1	1	yes	characteristic, complemented, Hall
170	#41	—	14	8	no	—
171	C₂×Q₈	—	7	1	no	—
172	C₂×D₄	—	7	4	no	—
173	C₂×D₄	—	7	4	no	—
174	C₂×D₄	—	7	1	no	—
175	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
176	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
177	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
178	C₂²×C₄	—	7	4	no	—
179	C₂²×C₄	—	7	2	no	—
180	C₄⋊C₄	—	7	2	no	—
181	C₄⋊C₄	—	7	1	no	—
182	C₄⋊C₄	—	7	1	no	—
183	C₂²⋊C₄	—	7	4	no	—
184	C₂²⋊C₄	—	7	2	no	—
185	C₄²	—	7	2	no	—
186	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
187	C₂×C₇	—	4	4	no	—
188	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
190	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
191	D₇	—	4	2	no	—
192	C₂²×C₃	—	7	1	no	—
193	C₂²×C₃	—	14	4	no	—
194	C₂²×C₃	—	14	4	no	—
195	C₂²×C₃	—	14	4	no	—
196	C₂²×C₃	—	14	2	no	—
197	C₂²×C₃	—	14	2	no	—
198	C₂²×C₃	—	14	2	no	—
199	C₃×C₄	—	7	4	no	—
200	C₃×C₄	—	7	2	no	—
201	C₃×C₄	—	7	2	no	—
202	C₃×C₄	—	14	1	no	—
203	C₃×C₄	—	14	2	no	—
204	C₃×C₄	—	14	2	no	—
205	C₂³	—	7	2	no	—
206	C₂³	C₂²×AGL(1,𝔽₇)	1	4	yes	—
207	Q₈	—	7	4	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	14	8	no	—
212	D₄	—	14	2	no	—
213	D₄	—	28	4	no	—
214	C₂×C₄	—	7	2	no	—
215	C₂×C₄	—	7	2	no	—
216	C₂×C₄	—	7	2	no	—
217	C₂×C₄	—	14	8	no	—
218	C₂×C₄	—	14	4	no	—
219	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
220	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
221	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
222	C₇	C₃×#550	1	1	yes	characteristic, complemented, Sylow
223	C₂×C₃	—	7	1	no	—
224	C₂×C₃	—	7	1	no	—
225	C₂×C₃	—	7	1	no	—
226	C₂×C₃	—	28	4	no	—
227	C₂×C₃	—	28	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	14	2	no	—
232	C₂²	—	14	2	no	—
233	C₂²	—	14	2	no	—
234	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	7	4	no	—
237	C₄	—	14	2	no	—
238	C₄	—	14	2	no	—
239	C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
240	C₄	#1159⋊C₃	1	2	yes	complemented
241	C₃	—	7	1	no	Sylow
242	C₂	—	4	4	no	—
243	C₂	—	28	2	no	—
244	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
245	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
246	C₂	#6548⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#127	C₂×C₃
hypercenter	#127	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#550
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#1159⋊C₃	C₂	semidirect
2. ΓL(1,𝔽₈)×#127	C₂	semidirect
3. #6503⋊C₃	C₂	semidirect
4. #6502⋊C₃	C₂	semidirect
5. #6501⋊C₃	C₂	semidirect
6. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
7. C₄×C₄⋊AGL(1,𝔽₇)	C₂	semidirect
8. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
9. #90409	C₃	semidirect
27. C₂×#1159	C₂×C₃	semidirect
28. C₇×#127	C₂×C₃	semidirect
29. #6503	C₂×C₃	semidirect
30. #6502	C₂×C₃	semidirect
31. #6501	C₂×C₃	semidirect
32. D₄×Dic₇	C₂×C₃	semidirect
33. C₄×D₂₈	C₂×C₃	semidirect
34. C₄×Dic₁₄	C₂×C₃	semidirect
52. ΓL(1,𝔽₈)⋊Q₈	D₄	semidirect
87. C₄×ΓL(1,𝔽₈)	#41	semidirect
108. Dic₁₄	C₃×D₄	semidirect
133. #127	AGL(1,𝔽₇)	semidirect
148. C₄×C₇	C₃×#41	semidirect
169. ΓL(1,𝔽₈)	#550	semidirect
222. C₇	C₃×#550	semidirect
240. C₄	#1159⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1159⋊C₃	2
ΓL(1,𝔽₈)×#127	2
#6503⋊C₃	2
#6502⋊C₃	2
#6501⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
C₄×C₄⋊AGL(1,𝔽₇)	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90409	3
C₃×#550	7

Subgroups — #90409⋊C3 #13057447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90409⋊C₃ #13057447