Subgroups — #90406⋊C₃ #13057450

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 288 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	1	yes	maximal, characteristic, complemented
2	C₂×D₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	#6503⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6502⋊C₃	C₂	1	4	yes	maximal, complemented
6	#6500⋊C₃	C₂	1	4	yes	maximal, complemented
7	#6499⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6438⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6435⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#90406	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂³×AGL(1,𝔽₇)	C₂²	1	2	yes	—
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
15	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
16	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
17	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	4	yes	complemented
18	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
19	#1159⋊C₃	—	2	4	no	—
20	D₄×AGL(1,𝔽₇)	—	2	4	no	—
21	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
22	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
23	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
25	(C₂²⋊Dic₇)⋊C₃	C₂²	1	4	yes	complemented
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
27	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	complemented
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
29	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₂×#1159	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×D₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
32	C₇×#127	C₂×C₃	1	1	yes	characteristic, complemented
33	#6503	C₂×C₃	1	1	yes	characteristic, complemented
34	#6502	C₂×C₃	1	4	yes	complemented
35	#6500	C₂×C₃	1	4	yes	complemented
36	#6499	C₂×C₃	1	1	yes	characteristic, complemented
37	#6438	C₂×C₃	1	1	yes	characteristic, complemented
38	#6435	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#535	—	7	1	no	maximal, Hall
40	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
42	C₂²×AGL(1,𝔽₇)	—	2	4	no	—
43	C₂²×AGL(1,𝔽₇)	—	4	4	no	—
44	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
45	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
46	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
47	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
48	D₄×ΓL(1,𝔽₈)	—	4	4	no	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
51	(C₂²⋊D₇)⋊C₃	—	2	4	no	—
52	(C₂²⋊D₇)⋊C₃	—	2	4	no	—
53	(C₂²⋊D₇)⋊C₃	—	4	4	no	—
54	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
55	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
58	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
59	C₄×AGL(1,𝔽₇)	D₄	1	4	yes	complemented
60	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
61	C₂³×D₇	C₂²×C₃	1	2	yes	—
62	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
64	C₂×C₇×D₄	C₂²×C₃	1	4	yes	complemented
65	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
66	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	complemented
68	C₂²×Dic₇	C₂²×C₃	1	2	yes	complemented
69	#1159	—	2	4	no	—
70	D₄×D₇	—	2	4	no	—
71	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
73	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
75	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	complemented
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
78	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
79	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₂×C₃×#41	—	7	1	no	—
81	C₂²×C₃×D₄	—	7	1	no	—
82	C₃×#127	—	7	1	no	—
83	C₃×#127	—	7	1	no	—
84	C₃×#124	—	7	1	no	—
85	C₃×#124	—	7	1	no	—
86	C₃×#121	—	7	4	no	—
87	C₃×#120	—	7	4	no	—
88	C₃×#117	—	7	1	no	—
89	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
90	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
91	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	4	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)	—	2	4	no	—
97	C₂×AGL(1,𝔽₇)	—	4	8	no	—
98	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
99	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
100	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
101	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
102	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
103	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
104	#535	—	7	1	no	Sylow
105	C₂³×C₇	C₂³×C₃	1	2	yes	—
106	C₂³×C₇	C₂³×C₃	1	2	yes	—
107	C₂²×D₇	—	2	4	no	—
108	C₂²×D₇	—	4	4	no	—
109	C₂²×D₇	C₂³×C₃	1	2	yes	—
110	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
111	C₇×D₄	—	2	2	no	—
112	C₇×D₄	—	2	2	no	—
113	C₇×D₄	—	4	4	no	—
114	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₂²⋊D₇	—	2	4	no	—
117	C₂²⋊D₇	—	2	4	no	—
118	C₂²⋊D₇	—	4	4	no	—
119	C₂×Dic₇	—	2	4	no	—
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	D₂₈	—	2	2	no	—
124	C₄×D₇	C₃×D₄	1	4	yes	complemented
125	Dic₁₄	—	2	2	no	—
126	C₂⁴×C₃	—	7	2	no	—
127	C₃×#41	—	14	4	no	—
128	C₂×C₃×Q₈	—	7	1	no	—
129	C₂×C₃×D₄	—	7	1	no	—
130	C₂×C₃×D₄	—	7	2	no	—
131	C₂×C₃×D₄	—	7	2	no	—
132	C₂×C₃×D₄	—	7	4	no	—
133	C₂×C₃×D₄	—	7	1	no	—
134	C₂×C₃×D₄	—	7	1	no	—
135	C₂×C₃×D₄	—	7	4	no	—
136	C₂×C₃×D₄	—	14	4	no	—
137	C₂²×C₃×C₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₂²⋊C₄	—	7	2	no	—
141	C₃×C₂²⋊C₄	—	7	4	no	—
142	C₃×C₂²⋊C₄	—	7	4	no	—
143	C₃×C₄²	—	7	1	no	—
144	C₃×C₄²	—	7	1	no	—
145	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
146	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
147	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
148	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	2	yes	—
149	AGL(1,𝔽₇)	—	2	2	no	—
150	AGL(1,𝔽₇)	—	4	2	no	—
151	C₂×#41	—	7	1	no	—
152	C₂²×D₄	—	7	1	no	—
153	#127	—	7	1	no	—
154	#127	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
155	#124	—	7	1	no	—
156	#124	—	7	1	no	—
157	#121	—	7	4	no	—
158	#120	—	7	4	no	—
159	#117	—	7	1	no	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	2	2	no	—
162	C₂²×C₇	—	2	4	no	—
163	C₂²×C₇	—	2	4	no	—
164	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
165	C₂×D₇	—	2	2	no	—
166	C₂×D₇	—	2	2	no	—
167	C₂×D₇	—	2	4	no	—
168	C₂×D₇	—	4	8	no	—
169	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
172	Dic₇	—	2	2	no	—
173	Dic₇	—	2	2	no	—
174	Dic₇	C₂×C₃×D₄	1	2	yes	—
175	C₂³×C₃	—	7	1	no	—
176	C₂³×C₃	—	7	2	no	—
177	C₂³×C₃	—	7	2	no	—
178	C₂³×C₃	—	7	2	no	—
179	C₂³×C₃	—	14	4	no	—
180	C₂³×C₃	—	28	4	no	—
181	C₃×Q₈	—	14	2	no	—
182	C₃×D₄	—	14	2	no	—
183	C₃×D₄	—	14	2	no	—
184	C₃×D₄	—	14	2	no	—
185	C₃×D₄	—	14	4	no	—
186	C₃×D₄	—	14	4	no	—
187	C₃×D₄	—	28	4	no	—
188	C₃×D₄	—	28	4	no	—
189	C₂×C₃×C₄	—	7	4	no	—
190	C₂×C₃×C₄	—	7	1	no	—
191	C₂×C₃×C₄	—	7	2	no	—
192	C₂×C₃×C₄	—	7	2	no	—
193	C₂×C₃×C₄	—	7	1	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	—	14	4	no	—
196	ΓL(1,𝔽₈)	#535	1	1	yes	characteristic, complemented, Hall
197	C₂⁴	—	7	2	no	—
198	#41	—	14	4	no	—
199	C₂×Q₈	—	7	1	no	—
200	C₂×D₄	—	7	2	no	—
201	C₂×D₄	—	7	2	no	—
202	C₂×D₄	—	7	4	no	—
203	C₂×D₄	—	7	1	no	—
204	C₂×D₄	—	14	4	no	—
205	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
206	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
207	C₂×D₄	C₂×AGL(1,𝔽₇)	1	4	yes	complemented
208	C₂²×C₄	—	7	1	no	—
209	C₂²×C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₂²⋊C₄	—	7	4	no	—
212	C₂²⋊C₄	—	7	2	no	—
213	C₂²⋊C₄	—	7	4	no	—
214	C₄²	—	7	1	no	—
215	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
216	C₂×C₇	—	4	2	no	—
217	C₂×C₇	—	4	2	no	—
218	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
219	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
220	D₇	—	2	2	no	—
221	D₇	—	4	2	no	—
222	C₂²×C₃	—	7	2	no	—
223	C₂²×C₃	—	7	1	no	—
224	C₂²×C₃	—	14	2	no	—
225	C₂²×C₃	—	14	2	no	—
226	C₂²×C₃	—	14	4	no	—
227	C₂²×C₃	—	14	4	no	—
228	C₂²×C₃	—	14	2	no	—
229	C₂²×C₃	—	14	2	no	—
230	C₂²×C₃	—	14	4	no	—
231	C₂²×C₃	—	28	8	no	—
232	C₃×C₄	—	7	2	no	—
233	C₃×C₄	—	7	2	no	—
234	C₃×C₄	—	14	2	no	—
235	C₃×C₄	—	14	2	no	—
236	C₃×C₄	—	14	2	no	—
237	C₂³	—	7	1	no	—
238	C₂³	—	7	2	no	—
239	C₂³	—	14	4	no	—
240	C₂³	—	28	4	no	—
241	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
242	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
243	Q₈	—	14	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	4	no	—
247	D₄	—	14	4	no	—
248	D₄	—	14	4	no	—
249	D₄	—	14	2	no	—
250	D₄	—	28	4	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	7	4	no	—
253	C₂×C₄	—	7	2	no	—
254	C₂×C₄	—	7	2	no	—
255	C₂×C₄	—	14	4	no	—
256	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
257	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
258	C₇	C₃×#535	1	1	yes	characteristic, complemented, Sylow
259	C₂×C₃	—	7	1	no	—
260	C₂×C₃	—	7	2	no	—
261	C₂×C₃	—	14	2	no	—
262	C₂×C₃	—	28	2	no	—
263	C₂×C₃	—	28	2	no	—
264	C₂×C₃	—	28	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	7	2	no	—
270	C₂²	—	14	2	no	—
271	C₂²	—	14	2	no	—
272	C₂²	—	14	4	no	—
273	C₂²	—	28	8	no	—
274	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	7	2	no	—
277	C₄	—	14	2	no	—
278	C₄	—	14	2	no	—
279	C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
280	C₃	—	7	1	no	Sylow
281	C₂	—	4	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	14	2	no	—
284	C₂	—	28	2	no	—
285	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
286	C₂	#6548⋊C₃	1	2	yes	—

	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×#535
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. C₂×D₄×AGL(1,𝔽₇)	C₂	semidirect
3. ΓL(1,𝔽₈)×#127	C₂	semidirect
4. #6503⋊C₃	C₂	semidirect
5. #6502⋊C₃	C₂	semidirect
6. #6500⋊C₃	C₂	semidirect
7. #6499⋊C₃	C₂	semidirect
8. #6438⋊C₃	C₂	semidirect
9. #6435⋊C₃	C₂	semidirect
10. #90406	C₃	semidirect
13. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
14. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
17. C₂×(C₂²⋊D₇)⋊C₃	C₂²	semidirect
18. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
23. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
24. C₄²×ΓL(1,𝔽₈)	C₂²	semidirect
25. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
27. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
28. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
29. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
30. C₂×#1159	C₂×C₃	semidirect
31. C₂×D₄×D₇	C₂×C₃	semidirect
32. C₇×#127	C₂×C₃	semidirect
33. #6503	C₂×C₃	semidirect
34. #6502	C₂×C₃	semidirect
35. #6500	C₂×C₃	semidirect
36. #6499	C₂×C₃	semidirect
37. #6438	C₂×C₃	semidirect
38. #6435	C₂×C₃	semidirect
59. C₄×AGL(1,𝔽₇)	D₄	semidirect
63. C₂×C₇×D₄	C₂²×C₃	semidirect
64. C₂×C₇×D₄	C₂²×C₃	semidirect
67. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
68. C₂²×Dic₇	C₂²×C₃	semidirect
73. C₂×Dic₁₄	C₂²×C₃	semidirect
74. C₄²×C₇	C₂²×C₃	semidirect
75. C₂²⋊Dic₇	C₂²×C₃	semidirect
77. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
78. Dic₇⋊C₄	C₂²×C₃	semidirect
79. C₄×Dic₇	C₂²×C₃	semidirect
124. C₄×D₇	C₃×D₄	semidirect
154. #127	AGL(1,𝔽₇)	semidirect
196. ΓL(1,𝔽₈)	#535	semidirect
206. C₂×D₄	C₂×AGL(1,𝔽₇)	semidirect
207. C₂×D₄	C₂×AGL(1,𝔽₇)	semidirect
215. C₄²	C₂×AGL(1,𝔽₇)	semidirect
258. C₇	C₃×#535	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
C₂×D₄×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)×#127	2
#6503⋊C₃	2
#6502⋊C₃	2
#6500⋊C₃	2
#6499⋊C₃	2
#6438⋊C₃	2
#6435⋊C₃	2
#90406	3
C₃×#535	7

Subgroups — #90406⋊C3 #13057450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90406⋊C₃ #13057450