Subgroups — #90350⋊C₃ #13057396

Order 1344.

496 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)×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#6500⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#6499⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×(C₂²⋊D₇)⋊C₃	C₂	1	2	yes	maximal, complemented
5	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
6	#6455⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6446⋊C₃	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₄×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
9	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
10	#90350	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂³×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
14	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
15	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
16	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
19	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
21	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	complemented
24	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
27	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	complemented
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
30	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
31	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₇×#123	C₂×C₃	1	1	yes	characteristic, complemented
33	#6500	C₂×C₃	1	1	yes	characteristic
34	#6499	C₂×C₃	1	1	yes	characteristic, complemented
35	C₄×C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
36	#6459	C₂×C₃	1	2	yes	complemented
37	#6455	C₂×C₃	1	2	yes	complemented
38	#6446	C₂×C₃	1	2	yes	complemented
39	D₇×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
40	#6442	C₂×C₃	1	2	yes	complemented
41	C₃×#551	—	7	1	no	maximal, Hall
42	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
43	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	C₂²×AGL(1,𝔽₇)	—	2	4	no	—
45	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
46	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
47	D₄×ΓL(1,𝔽₈)	—	4	1	no	—
48	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
52	(C₂²⋊D₇)⋊C₃	—	2	4	no	—
53	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
54	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
55	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
56	C₄×AGL(1,𝔽₇)	—	2	4	no	—
57	C₄×AGL(1,𝔽₇)	—	4	2	no	—
58	ΓL(1,𝔽₈)⋊Q₈	—	4	1	no	—
59	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
62	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
63	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
64	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
65	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
66	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
67	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
68	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
69	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
70	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
71	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	complemented
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
74	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
76	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
77	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
78	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
79	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₃×#126	—	7	2	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#123	—	7	2	no	—
83	C₃×#123	—	7	1	no	—
84	C₃×C₂²⋊Q₈	—	7	2	no	—
85	C₃×#120	—	7	1	no	—
86	C₃×C₄×D₄	—	7	2	no	—
87	C₃×#117	—	7	2	no	—
88	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
89	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
90	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
91	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
93	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
94	C₂×AGL(1,𝔽₇)	—	2	2	no	—
95	C₂×AGL(1,𝔽₇)	—	2	2	no	—
96	C₂×AGL(1,𝔽₇)	—	4	4	no	—
97	C₂×AGL(1,𝔽₇)	#41	1	4	yes	—
98	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
99	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
100	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
101	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
102	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
103	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
104	#551	—	7	1	no	Sylow
105	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
106	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
107	C₂²×D₇	—	2	4	no	—
108	C₂²×D₇	—	4	1	no	—
109	C₂²×D₇	C₂³×C₃	1	2	yes	—
110	C₇×D₄	—	4	1	no	—
111	C₂×C₄×C₇	—	4	1	no	—
112	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
113	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
115	C₂²⋊D₇	—	2	4	no	—
116	C₂×Dic₇	C₂³×C₃	1	2	yes	—
117	C₂×Dic₇	C₂³×C₃	1	2	yes	—
118	C₂×Dic₇	C₂³×C₃	1	2	yes	—
119	C₄×D₇	—	2	4	no	—
120	C₄×D₇	—	4	2	no	—
121	Dic₁₄	—	4	1	no	—
122	C₂⁴×C₃	—	7	1	no	—
123	C₂×C₃×Q₈	—	7	1	no	—
124	C₂×C₃×D₄	—	7	2	no	—
125	C₂×C₃×D₄	—	7	1	no	—
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₂²×C₃×C₄	—	7	1	no	—
129	C₃×C₄⋊C₄	—	7	2	no	—
130	C₃×C₄⋊C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	2	no	—
133	C₃×C₂²⋊C₄	—	7	2	no	—
134	C₃×C₂²⋊C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	2	no	—
136	C₃×C₂²⋊C₄	—	7	2	no	—
137	C₃×C₂²⋊C₄	—	7	2	no	—
138	C₃×C₂²⋊C₄	—	7	1	no	—
139	C₃×C₂²⋊C₄	—	7	2	no	—
140	C₃×C₂²⋊C₄	—	7	1	no	—
141	C₃×C₄²	—	7	1	no	—
142	C₃×C₄²	—	7	2	no	—
143	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
144	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
145	C₂×ΓL(1,𝔽₈)	C₂×#41	1	2	yes	—
146	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
147	AGL(1,𝔽₇)	—	2	4	no	—
148	#126	—	7	2	no	—
149	#124	—	7	1	no	—
150	#123	—	7	2	no	—
151	#123	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
152	C₂²⋊Q₈	—	7	2	no	—
153	#120	—	7	1	no	—
154	C₄×D₄	—	7	2	no	—
155	#117	—	7	2	no	—
156	C₂×C₂²⋊C₄	—	7	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	—	2	1	no	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
162	C₂×D₇	—	2	2	no	—
163	C₂×D₇	—	2	2	no	—
164	C₂×D₇	—	4	4	no	—
165	C₂×D₇	C₃×#41	1	4	yes	—
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	2	no	—
169	Dic₇	—	2	2	no	—
170	Dic₇	—	2	2	no	—
171	Dic₇	—	2	2	no	—
172	C₂³×C₃	—	7	2	no	—
173	C₂³×C₃	—	7	1	no	—
174	C₂³×C₃	—	7	1	no	—
175	C₂³×C₃	—	14	4	no	—
176	C₂³×C₃	—	28	1	no	—
177	C₃×Q₈	—	28	1	no	—
178	C₃×D₄	—	14	4	no	—
179	C₃×D₄	—	28	1	no	—
180	C₂×C₃×C₄	—	7	2	no	—
181	C₂×C₃×C₄	—	7	2	no	—
182	C₂×C₃×C₄	—	7	2	no	—
183	C₂×C₃×C₄	—	7	2	no	—
184	C₂×C₃×C₄	—	7	1	no	—
185	C₂×C₃×C₄	—	7	2	no	—
186	C₂×C₃×C₄	—	14	4	no	—
187	C₂×C₃×C₄	—	28	1	no	—
188	C₂×C₃×C₄	—	28	2	no	—
189	ΓL(1,𝔽₈)	#551	1	1	yes	characteristic, complemented, Hall
190	C₂⁴	—	7	1	no	—
191	C₂×Q₈	—	7	1	no	—
192	C₂×D₄	—	7	2	no	—
193	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
194	C₂²×C₄	—	7	2	no	—
195	C₂²×C₄	—	7	2	no	—
196	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	7	2	no	—
198	C₄⋊C₄	—	7	2	no	—
199	C₄⋊C₄	—	7	2	no	—
200	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
201	C₂²⋊C₄	—	7	2	no	—
202	C₂²⋊C₄	—	7	2	no	—
203	C₂²⋊C₄	—	7	1	no	—
204	C₂²⋊C₄	—	7	2	no	—
205	C₂²⋊C₄	—	7	2	no	—
206	C₂²⋊C₄	—	7	2	no	—
207	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
208	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
209	C₄²	—	7	1	no	—
210	C₄²	—	7	2	no	—
211	C₂×C₇	—	4	1	no	—
212	C₂×C₇	—	4	1	no	—
213	C₂×C₇	C₂×C₃×#41	1	2	yes	—
214	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
215	D₇	—	2	4	no	—
216	C₂²×C₃	—	7	4	no	—
217	C₂²×C₃	—	7	1	no	—
218	C₂²×C₃	—	14	1	no	—
219	C₂²×C₃	—	14	2	no	—
220	C₂²×C₃	—	14	1	no	—
221	C₂²×C₃	—	14	2	no	—
222	C₂²×C₃	—	14	2	no	—
223	C₂²×C₃	—	14	2	no	—
224	C₂²×C₃	—	28	4	no	—
225	C₃×C₄	—	14	2	no	—
226	C₃×C₄	—	14	2	no	—
227	C₃×C₄	—	14	1	no	—
228	C₃×C₄	—	14	2	no	—
229	C₃×C₄	—	14	2	no	—
230	C₃×C₄	—	14	2	no	—
231	C₂³	—	7	2	no	—
232	C₂³	—	14	4	no	—
233	C₂³	—	28	1	no	—
234	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
235	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
236	Q₈	—	28	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	14	4	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	7	2	no	—
241	C₂×C₄	—	7	2	no	—
242	C₂×C₄	—	7	2	no	—
243	C₂×C₄	—	14	4	no	—
244	C₂×C₄	—	28	2	no	—
245	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
246	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
247	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
248	C₇	C₃×#551	1	1	yes	characteristic, complemented, Sylow
249	C₂×C₃	—	7	1	no	—
250	C₂×C₃	—	7	2	no	—
251	C₂×C₃	—	14	4	no	—
252	C₂×C₃	—	28	1	no	—
253	C₂×C₃	—	28	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	7	4	no	—
259	C₂²	—	14	2	no	—
260	C₂²	—	14	2	no	—
261	C₂²	—	28	4	no	—
262	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	2	no	—
266	C₄	—	14	2	no	—
267	C₄	—	14	2	no	—
268	C₄	—	14	2	no	—
269	C₃	—	7	1	no	Sylow
270	C₂	—	4	1	no	—
271	C₂	—	4	1	no	—
272	C₂	—	14	4	no	—
273	C₂	#6548⋊C₃	1	1	yes	characteristic
274	C₂	#41×AGL(1,𝔽₇)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#123	C₂×C₃
prime radical	#6500⋊C₃	C₂
hypercenter	#123	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#551
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#123	C₂	semidirect
3. #6499⋊C₃	C₂	semidirect
4. C₄×(C₂²⋊D₇)⋊C₃	C₂	semidirect
5. #6459⋊C₃	C₂	semidirect
6. #6455⋊C₃	C₂	semidirect
7. #6446⋊C₃	C₂	semidirect
8. C₂²⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
9. #6442⋊C₃	C₂	semidirect
10. #90350	C₃	semidirect
13. C₂²×C₄×ΓL(1,𝔽₈)	C₂²	semidirect
17. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
18. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
20. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
23. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
27. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
28. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
29. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
30. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
31. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
32. C₇×#123	C₂×C₃	semidirect
34. #6499	C₂×C₃	semidirect
35. C₄×C₂²⋊D₇	C₂×C₃	semidirect
36. #6459	C₂×C₃	semidirect
37. #6455	C₂×C₃	semidirect
38. #6446	C₂×C₃	semidirect
39. D₇×C₂²⋊C₄	C₂×C₃	semidirect
40. #6442	C₂×C₃	semidirect
61. C₂²×C₄×C₇	C₂²×C₃	semidirect
65. C₂×Dic₁₄	C₂²×C₃	semidirect
66. C₇×C₄⋊C₄	C₂²×C₃	semidirect
68. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
71. C₂²⋊Dic₇	C₂²×C₃	semidirect
75. C₄⋊Dic₇	C₂²×C₃	semidirect
76. Dic₇⋊C₄	C₂²×C₃	semidirect
77. Dic₇⋊C₄	C₂²×C₃	semidirect
78. C₄×Dic₇	C₂²×C₃	semidirect
79. C₄×Dic₇	C₂²×C₃	semidirect
151. #123	AGL(1,𝔽₇)	semidirect
189. ΓL(1,𝔽₈)	#551	semidirect
196. C₂²×C₄	C₂×AGL(1,𝔽₇)	semidirect
200. C₄⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
208. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
248. C₇	C₃×#551	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)×#123	2
#6500⋊C₃	2
#6499⋊C₃	2
C₄×(C₂²⋊D₇)⋊C₃	2
#6459⋊C₃	2
#6455⋊C₃	2
#6446⋊C₃	2
C₂²⋊C₄×AGL(1,𝔽₇)	2
#6442⋊C₃	2
#90350	3
C₃×#551	7

Subgroups — #90350⋊C3 #13057396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90350⋊C₃ #13057396