Subgroups — #90380⋊C₃ #13057424

Order 1344.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 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	#6460⋊C₃	C₂	1	2	yes	maximal
3	#6459⋊C₃	C₂	1	2	yes	maximal
4	#6455⋊C₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
6	#6453⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6452⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6451⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6451⋊C₃	C₂	1	2	yes	maximal, complemented
10	#6435⋊C₃	C₂	1	1	yes	maximal, characteristic
11	#6432⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#90380	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
14	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
15	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
16	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
17	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
21	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
22	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
23	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
27	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
32	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
33	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
34	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
36	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
37	#6460	C₂×C₃	1	2	yes	—
38	#6459	C₂×C₃	1	2	yes	—
39	#6455	C₂×C₃	1	1	yes	characteristic
40	D₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
41	#6453	C₂×C₃	1	1	yes	characteristic, complemented
42	#6452	C₂×C₃	1	2	yes	complemented
43	#6451	C₂×C₃	1	1	yes	characteristic, complemented
44	#6451	C₂×C₃	1	2	yes	complemented
45	#6435	C₂×C₃	1	1	yes	characteristic
46	#6432	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#544	—	7	1	no	maximal, Hall
48	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
55	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
59	C₄×AGL(1,𝔽₇)	—	4	2	no	—
60	C₄×AGL(1,𝔽₇)	Q₈	1	4	yes	complemented
61	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
62	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
63	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
64	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
65	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
66	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
67	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
76	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
77	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
78	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
79	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
82	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
85	C₄×Dic₇	C₂²×C₃	1	2	yes	—
86	C₃×C₄⋊Q₈	—	7	1	no	—
87	C₃×C₄⋊Q₈	—	7	2	no	—
88	C₃×C₄⋊Q₈	—	7	1	no	—
89	C₃×#125	—	7	2	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×#125	—	7	1	no	—
92	C₃×C₂²⋊Q₈	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	2	no	—
94	C₃×#117	—	7	1	no	—
95	C₃×#117	—	7	1	no	—
96	C₂×C₃×C₄⋊C₄	—	7	1	no	—
97	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
98	C₂×AGL(1,𝔽₇)	—	2	1	no	—
99	C₂×AGL(1,𝔽₇)	—	2	1	no	—
100	C₂×AGL(1,𝔽₇)	C₂×Q₈	1	2	yes	—
101	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
102	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
103	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
104	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
105	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
106	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
107	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
108	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
109	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
110	ΓL(1,𝔽₈)⋊C₄	C₂×Q₈	1	2	yes	—
111	#544	—	7	1	no	Sylow
112	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
114	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×Dic₇	C₂³×C₃	1	2	yes	—
121	C₂×Dic₇	C₂³×C₃	1	2	yes	—
122	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
123	C₄×D₇	—	4	2	no	—
124	C₄×D₇	C₃×Q₈	1	4	yes	complemented
125	Dic₁₄	—	4	2	no	—
126	Dic₁₄	—	4	2	no	—
127	C₂×C₃×Q₈	—	7	2	no	—
128	C₂×C₃×Q₈	—	7	2	no	—
129	C₂²×C₃×C₄	—	7	1	no	—
130	C₂²×C₃×C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	1	no	—
133	C₃×C₄⋊C₄	—	7	1	no	—
134	C₃×C₄⋊C₄	—	7	1	no	—
135	C₃×C₄⋊C₄	—	7	1	no	—
136	C₃×C₄⋊C₄	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	2	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	1	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₂²⋊C₄	—	7	1	no	—
145	C₃×C₂²⋊C₄	—	7	1	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₄²	—	7	2	no	—
148	C₃×C₄²	—	7	1	no	—
149	C₃×C₄²	—	7	1	no	—
150	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
151	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
153	AGL(1,𝔽₇)	—	2	2	no	—
154	C₄⋊Q₈	—	7	2	no	—
155	C₄⋊Q₈	—	7	1	no	—
156	C₄⋊Q₈	—	7	1	no	—
157	#125	—	7	2	no	—
158	#125	—	7	1	no	—
159	#125	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
160	C₂²⋊Q₈	—	7	2	no	—
161	C₂²⋊Q₈	—	7	2	no	—
162	#117	—	7	1	no	—
163	#117	—	7	1	no	—
164	C₂×C₄⋊C₄	—	7	1	no	—
165	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
166	C₂×D₇	—	2	1	no	—
167	C₂×D₇	—	2	1	no	—
168	C₂×D₇	C₂×C₃×Q₈	1	2	yes	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	C₂×C₃×Q₈	1	2	yes	—
174	Dic₇	—	2	1	no	—
175	Dic₇	—	2	2	no	—
176	Dic₇	—	2	2	no	—
177	Dic₇	—	2	1	no	—
178	Dic₇	C₂×C₃×Q₈	1	2	yes	—
179	C₂³×C₃	—	7	1	no	—
180	C₃×Q₈	—	28	2	no	—
181	C₃×Q₈	—	28	2	no	—
182	C₂×C₃×C₄	—	7	4	no	—
183	C₂×C₃×C₄	—	7	1	no	—
184	C₂×C₃×C₄	—	7	1	no	—
185	C₂×C₃×C₄	—	7	1	no	—
186	C₂×C₃×C₄	—	7	2	no	—
187	C₂×C₃×C₄	—	7	2	no	—
188	C₂×C₃×C₄	—	7	1	no	—
189	C₂×C₃×C₄	—	7	2	no	—
190	C₂×C₃×C₄	—	7	1	no	—
191	C₂×C₃×C₄	—	7	1	no	—
192	C₂×C₃×C₄	—	7	2	no	—
193	C₂×C₃×C₄	—	28	2	no	—
194	ΓL(1,𝔽₈)	#544	1	1	yes	characteristic, complemented, Hall
195	C₂×Q₈	—	7	2	no	—
196	C₂×Q₈	—	7	2	no	—
197	C₂²×C₄	—	7	1	no	—
198	C₂²×C₄	—	7	2	no	—
199	C₄⋊C₄	—	7	2	no	—
200	C₄⋊C₄	—	7	1	no	—
201	C₄⋊C₄	—	7	1	no	—
202	C₄⋊C₄	—	7	2	no	—
203	C₄⋊C₄	—	7	2	no	—
204	C₄⋊C₄	—	7	1	no	—
205	C₄⋊C₄	—	7	1	no	—
206	C₄⋊C₄	—	7	2	no	—
207	C₄⋊C₄	—	7	2	no	—
208	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
209	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
210	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
211	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
212	C₂²⋊C₄	—	7	1	no	—
213	C₂²⋊C₄	—	7	2	no	—
214	C₂²⋊C₄	—	7	1	no	—
215	C₄²	—	7	2	no	—
216	C₄²	—	7	1	no	—
217	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
218	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
219	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
221	D₇	—	2	2	no	—
222	C₂²×C₃	—	7	2	no	—
223	C₂²×C₃	—	7	1	no	—
224	C₂²×C₃	—	14	1	no	—
225	C₂²×C₃	—	14	1	no	—
226	C₃×C₄	—	7	2	no	—
227	C₃×C₄	—	7	2	no	—
228	C₃×C₄	—	14	2	no	—
229	C₃×C₄	—	14	1	no	—
230	C₃×C₄	—	14	2	no	—
231	C₃×C₄	—	14	1	no	—
232	C₃×C₄	—	14	2	no	—
233	C₃×C₄	—	14	1	no	—
234	C₃×C₄	—	14	2	no	—
235	C₃×C₄	—	14	1	no	—
236	C₂³	—	7	1	no	—
237	Q₈	—	28	2	no	—
238	Q₈	—	28	2	no	—
239	C₂×C₄	—	7	1	no	—
240	C₂×C₄	—	7	4	no	—
241	C₂×C₄	—	7	1	no	—
242	C₂×C₄	—	7	1	no	—
243	C₂×C₄	—	7	2	no	—
244	C₂×C₄	—	7	2	no	—
245	C₂×C₄	—	28	2	no	—
246	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
247	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
248	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
249	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
250	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
251	C₇	C₃×#544	1	1	yes	characteristic, complemented, Sylow
252	C₂×C₃	—	7	1	no	—
253	C₂×C₃	—	7	1	no	—
254	C₂×C₃	—	7	1	no	—
255	C₂×C₃	—	14	2	no	—
256	C₂²	—	7	2	no	—
257	C₂²	—	14	1	no	—
258	C₂²	—	14	1	no	—
259	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	7	2	no	—
265	C₄	—	14	2	no	—
266	C₄	—	14	1	no	—
267	C₄	—	14	2	no	—
268	C₄	—	14	1	no	—
269	C₄	Q₈×AGL(1,𝔽₇)	1	2	yes	—
270	C₃	—	7	1	no	Sylow
271	C₂	—	14	2	no	—
272	C₂	C₂×Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
273	C₂	#10621518	1	1	yes	characteristic
274	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	C₂²×AGL(1,𝔽₇)	C₂³
hypercenter	#125	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#544
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
6. #6453⋊C₃	C₂	semidirect
7. #6452⋊C₃	C₂	semidirect
8. #6451⋊C₃	C₂	semidirect
9. #6451⋊C₃	C₂	semidirect
11. #6432⋊C₃	C₂	semidirect
12. #90380	C₃	semidirect
36. C₇×#125	C₂×C₃	semidirect
41. #6453	C₂×C₃	semidirect
42. #6452	C₂×C₃	semidirect
43. #6451	C₂×C₃	semidirect
44. #6451	C₂×C₃	semidirect
46. #6432	C₂×C₃	semidirect
60. C₄×AGL(1,𝔽₇)	Q₈	semidirect
124. C₄×D₇	C₃×Q₈	semidirect
159. #125	AGL(1,𝔽₇)	semidirect
194. ΓL(1,𝔽₈)	#544	semidirect
251. C₇	C₃×#544	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
#6460⋊C₃	2
#6459⋊C₃	2
#6455⋊C₃	2
C₄⋊C₄×AGL(1,𝔽₇)	2
#6453⋊C₃	2
#6452⋊C₃	2
#6451⋊C₃	2
#6435⋊C₃	2
#6432⋊C₃	2
#90380	3
C₃×#544	7

Subgroups — #90380⋊C3 #13057424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90380⋊C₃ #13057424