Subgroups — #90519⋊C₃ #13057561

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	#6509⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6501⋊C₃	C₂	1	2	yes	maximal, complemented
5	#6500⋊C₃	C₂	1	2	yes	maximal, complemented
6	#6498⋊C₃	C₂	1	2	yes	maximal, complemented
7	C₄×(C₂²⋊D₇)⋊C₃	C₂	1	4	yes	maximal, complemented
8	#6489⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#90519	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂³×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
11	#41×ΓL(1,𝔽₈)	—	2	4	no	—
12	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
17	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	4	yes	—
18	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄×AGL(1,𝔽₇)	—	2	4	no	—
20	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
21	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	4	yes	complemented
24	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
25	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	complemented
26	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	complemented
27	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
28	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
30	#6509	C₂×C₃	1	2	yes	complemented
31	#6501	C₂×C₃	1	2	yes	complemented
32	#6500	C₂×C₃	1	2	yes	complemented
33	#6498	C₂×C₃	1	2	yes	complemented
34	C₄×C₂²⋊D₇	C₂×C₃	1	4	yes	complemented
35	#6489	C₂×C₃	1	1	yes	characteristic, complemented
36	C₃×#525	—	7	1	no	maximal, Hall
37	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
38	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
39	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
40	C₂²×AGL(1,𝔽₇)	—	2	4	no	—
41	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
42	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
43	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
44	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
45	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
46	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	4	yes	complemented
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
48	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	(C₂²⋊D₇)⋊C₃	—	2	8	no	—
51	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
52	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
53	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
54	C₄×AGL(1,𝔽₇)	—	2	4	no	—
55	C₄×AGL(1,𝔽₇)	—	2	4	no	—
56	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
57	C₇×#41	—	2	4	no	—
58	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
59	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
60	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	complemented
63	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
64	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄×D₇	—	2	4	no	—
66	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
68	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
69	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	complemented
70	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
71	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
72	Dic₇⋊C₄	C₂²×C₃	1	4	yes	complemented
73	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
74	C₂×C₃×#41	—	7	1	no	—
75	C₂³×C₃×C₄	—	7	1	no	—
76	C₃×#123	—	7	2	no	—
77	C₃×C₂²⋊Q₈	—	7	2	no	—
78	C₃×#121	—	7	2	no	—
79	C₃×#120	—	7	2	no	—
80	C₃×C₄×D₄	—	7	4	no	—
81	C₃×#117	—	7	1	no	—
82	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
83	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
84	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
85	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
86	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
87	C₂×AGL(1,𝔽₇)	—	2	8	no	—
88	C₂×AGL(1,𝔽₇)	—	2	2	no	—
89	C₂×AGL(1,𝔽₇)	—	2	2	no	—
90	C₂×AGL(1,𝔽₇)	#41	1	4	yes	complemented
91	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
92	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
93	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
94	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
95	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
96	#525	—	7	1	no	Sylow
97	C₂³×C₇	C₂³×C₃	1	2	yes	—
98	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
99	C₂²×D₇	—	2	2	no	—
100	C₂²×D₇	—	2	4	no	—
101	C₂²×D₇	C₂³×C₃	1	2	yes	—
102	C₇×Q₈	—	2	2	no	—
103	C₇×D₄	—	2	4	no	—
104	C₇×D₄	—	2	2	no	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
107	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
108	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
110	C₂²⋊D₇	—	2	8	no	—
111	C₂×Dic₇	—	2	2	no	—
112	C₂×Dic₇	C₂³×C₃	1	4	yes	—
113	C₂×Dic₇	C₂³×C₃	1	2	yes	—
114	C₄×D₇	—	2	4	no	—
115	C₄×D₇	—	2	4	no	—
116	C₂⁴×C₃	—	7	1	no	—
117	C₃×#41	—	14	4	no	—
118	C₂×C₃×Q₈	—	7	1	no	—
119	C₂×C₃×D₄	—	7	4	no	—
120	C₂×C₃×D₄	—	7	2	no	—
121	C₂×C₃×D₄	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	2	no	—
123	C₂²×C₃×C₄	—	7	2	no	—
124	C₂²×C₃×C₄	—	7	1	no	—
125	C₂²×C₃×C₄	—	7	1	no	—
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	14	4	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	4	no	—
130	C₃×C₂²⋊C₄	—	7	2	no	—
131	C₃×C₂²⋊C₄	—	7	4	no	—
132	C₃×C₂²⋊C₄	—	7	4	no	—
133	C₃×C₄²	—	7	2	no	—
134	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
135	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
136	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
137	C₂×ΓL(1,𝔽₈)	C₂×#41	1	2	yes	—
138	AGL(1,𝔽₇)	—	2	4	no	—
139	C₂×#41	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
140	C₂³×C₄	—	7	1	no	—
141	#123	—	7	2	no	—
142	C₂²⋊Q₈	—	7	2	no	—
143	#121	—	7	2	no	—
144	#120	—	7	2	no	—
145	C₄×D₄	—	7	4	no	—
146	#117	—	7	1	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	2	2	no	—
149	C₂²×C₇	—	2	4	no	—
150	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
151	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
152	C₂×D₇	—	2	8	no	—
153	C₂×D₇	—	2	2	no	—
154	C₂×D₇	—	2	2	no	—
155	C₂×D₇	C₃×#41	1	4	yes	complemented
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
158	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
159	Dic₇	—	2	2	no	—
160	Dic₇	—	2	4	no	—
161	C₂³×C₃	—	7	2	no	—
162	C₂³×C₃	—	7	1	no	—
163	C₂³×C₃	—	7	2	no	—
164	C₂³×C₃	—	14	2	no	—
165	C₂³×C₃	—	14	4	no	—
166	C₃×Q₈	—	14	2	no	—
167	C₃×D₄	—	14	2	no	—
168	C₃×D₄	—	14	4	no	—
169	C₃×D₄	—	14	8	no	—
170	C₂×C₃×C₄	—	7	4	no	—
171	C₂×C₃×C₄	—	7	2	no	—
172	C₂×C₃×C₄	—	7	4	no	—
173	C₂×C₃×C₄	—	7	1	no	—
174	C₂×C₃×C₄	—	7	2	no	—
175	C₂×C₃×C₄	—	7	1	no	—
176	C₂×C₃×C₄	—	14	4	no	—
177	C₂×C₃×C₄	—	14	4	no	—
178	C₂×C₃×C₄	—	14	4	no	—
179	C₂×C₃×C₄	—	14	2	no	—
180	ΓL(1,𝔽₈)	#525	1	1	yes	characteristic, complemented, Hall
181	C₂⁴	—	7	1	no	—
182	#41	—	2	4	no	—
183	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
184	C₂×D₄	—	7	4	no	—
185	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
186	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
187	C₂²×C₄	—	7	1	no	—
188	C₂²×C₄	—	7	2	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	14	4	no	—
191	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
192	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
193	C₄⋊C₄	—	7	4	no	—
194	C₄⋊C₄	—	7	2	no	—
195	C₂²⋊C₄	—	7	2	no	—
196	C₂²⋊C₄	—	7	4	no	—
197	C₂²⋊C₄	—	7	4	no	—
198	C₄²	—	7	2	no	—
199	C₂×C₇	—	2	2	no	—
200	C₂×C₇	—	4	2	no	—
201	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
202	C₂×C₇	C₂×C₃×#41	1	2	yes	—
203	D₇	—	2	4	no	—
204	C₂²×C₃	—	7	4	no	—
205	C₂²×C₃	—	7	2	no	—
206	C₂²×C₃	—	7	1	no	—
207	C₂²×C₃	—	14	2	no	—
208	C₂²×C₃	—	14	2	no	—
209	C₂²×C₃	—	14	4	no	—
210	C₂²×C₃	—	14	2	no	—
211	C₂²×C₃	—	14	2	no	—
212	C₂²×C₃	—	14	8	no	—
213	C₃×C₄	—	7	2	no	—
214	C₃×C₄	—	7	2	no	—
215	C₃×C₄	—	14	2	no	—
216	C₃×C₄	—	14	2	no	—
217	C₃×C₄	—	14	4	no	—
218	C₂³	—	7	2	no	—
219	C₂³	—	14	4	no	—
220	C₂³	—	14	2	no	—
221	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
222	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	14	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	7	4	no	—
229	C₂×C₄	—	7	2	no	—
230	C₂×C₄	—	14	2	no	—
231	C₂×C₄	—	14	4	no	—
232	C₂×C₄	—	14	4	no	—
233	C₂×C₄	(C₂²⋊D₇)⋊C₃	1	4	yes	complemented
234	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
235	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
236	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
237	C₇	C₃×#525	1	1	yes	characteristic, complemented, Sylow
238	C₂×C₃	—	7	2	no	—
239	C₂×C₃	—	7	1	no	—
240	C₂×C₃	—	14	2	no	—
241	C₂×C₃	—	14	4	no	—
242	C₂×C₃	—	28	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	7	4	no	—
247	C₂²	—	14	8	no	—
248	C₂²	—	14	2	no	—
249	C₂²	—	14	2	no	—
250	C₂²	C₂×(C₂²⋊D₇)⋊C₃	1	2	yes	—
251	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	14	4	no	—
254	C₄	—	14	2	no	—
255	C₄	C₂×(C₂²⋊D₇)⋊C₃	1	2	yes	—
256	C₄	C₂×(C₂²⋊D₇)⋊C₃	1	2	yes	—
257	C₃	—	7	1	no	Sylow
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	—	14	4	no	—
261	C₂	#41×AGL(1,𝔽₇)	1	2	yes	—
262	C₂	C₂²×(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×AGL(1,𝔽₇)
Fitting	C₂×C₇×#41	C₂×C₃
hypercenter	C₂×#41	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#525
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂²×C₄×AGL(1,𝔽₇)	C₂	semidirect
3. #6509⋊C₃	C₂	semidirect
4. #6501⋊C₃	C₂	semidirect
5. #6500⋊C₃	C₂	semidirect
6. #6498⋊C₃	C₂	semidirect
7. C₄×(C₂²⋊D₇)⋊C₃	C₂	semidirect
8. #6489⋊C₃	C₂	semidirect
9. #90519	C₃	semidirect
12. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
14. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
16. C₂²×C₄×ΓL(1,𝔽₈)	C₂²	semidirect
23. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
25. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
26. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
27. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
28. C₂×C₇×#41	C₂×C₃	semidirect
29. C₂²×C₄×D₇	C₂×C₃	semidirect
30. #6509	C₂×C₃	semidirect
31. #6501	C₂×C₃	semidirect
32. #6500	C₂×C₃	semidirect
33. #6498	C₂×C₃	semidirect
34. C₄×C₂²⋊D₇	C₂×C₃	semidirect
35. #6489	C₂×C₃	semidirect
46. C₂×C₄×ΓL(1,𝔽₈)	D₄	semidirect
58. C₂×C₇×Q₈	C₂²×C₃	semidirect
60. C₂×C₇×D₄	C₂²×C₃	semidirect
62. C₂²×C₄×C₇	C₂²×C₃	semidirect
69. C₂²⋊Dic₇	C₂²×C₃	semidirect
71. C₄⋊Dic₇	C₂²×C₃	semidirect
72. Dic₇⋊C₄	C₂²×C₃	semidirect
73. C₄×Dic₇	C₂²×C₃	semidirect
90. C₂×AGL(1,𝔽₇)	#41	semidirect
106. C₂×C₄×C₇	C₃×D₄	semidirect
139. C₂×#41	AGL(1,𝔽₇)	semidirect
155. C₂×D₇	C₃×#41	semidirect
180. ΓL(1,𝔽₈)	#525	semidirect
183. C₂×Q₈	C₂×AGL(1,𝔽₇)	semidirect
186. C₂×D₄	C₂×AGL(1,𝔽₇)	semidirect
192. C₂²×C₄	C₂×AGL(1,𝔽₇)	semidirect
233. C₂×C₄	(C₂²⋊D₇)⋊C₃	semidirect
237. C₇	C₃×#525	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×ΓL(1,𝔽₈)	2
C₂²×C₄×AGL(1,𝔽₇)	2
#6509⋊C₃	2
#6501⋊C₃	2
#6500⋊C₃	2
#6498⋊C₃	2
C₄×(C₂²⋊D₇)⋊C₃	2
#6489⋊C₃	2
#90519	3
C₃×#525	7

Subgroups — #90519⋊C3 #13057561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90519⋊C₃ #13057561