Subgroups — #90366⋊C₃ #13057410

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1159⋊C₃	C₂	1	1	yes	maximal, characteristic
3	ΓL(1,𝔽₈)×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#6507⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6499⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6446⋊C₃	C₂	1	4	yes	maximal, complemented
7	#6441⋊C₃	C₂	1	4	yes	maximal, complemented
8	#6435⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6432⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#90366	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
14	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
15	Q₈×AGL(1,𝔽₇)	—	2	4	no	—
16	#1159⋊C₃	—	2	4	no	—
17	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
18	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
19	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
22	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
24	(C₂²⋊Dic₇)⋊C₃	C₂²	1	4	yes	—
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
26	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	4	yes	—
27	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
29	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#1159	C₂×C₃	1	1	yes	characteristic
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6507	C₂×C₃	1	1	yes	characteristic, complemented
34	#6499	C₂×C₃	1	1	yes	characteristic, complemented
35	#6446	C₂×C₃	1	4	yes	complemented
36	#6441	C₂×C₃	1	4	yes	complemented
37	#6435	C₂×C₃	1	1	yes	characteristic, complemented
38	#6432	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#536	—	7	1	no	maximal, Hall
40	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
42	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
43	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
44	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
45	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
46	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
47	(C₂²⋊D₇)⋊C₃	—	2	4	no	—
48	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
49	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
50	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
51	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
52	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
53	C₄×AGL(1,𝔽₇)	—	2	4	no	—
54	C₄×AGL(1,𝔽₇)	D₄	1	4	yes	—
55	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
56	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
57	ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
58	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
61	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
62	Q₈×D₇	—	2	4	no	—
63	#1159	—	2	4	no	—
64	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
65	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
69	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
70	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
71	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
73	C₄⋊Dic₇	C₂²×C₃	1	4	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
76	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×#41	—	7	1	no	—
78	C₂²×C₃×Q₈	—	7	1	no	—
79	C₃×C₄⋊Q₈	—	7	1	no	—
80	C₃×C₄⋊Q₈	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#124	—	7	1	no	—
83	C₃×#123	—	7	4	no	—
84	C₃×C₂²⋊Q₈	—	7	4	no	—
85	C₃×#117	—	7	1	no	—
86	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
87	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
88	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
89	C₂×AGL(1,𝔽₇)	—	2	2	no	—
90	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
91	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
92	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
93	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
94	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
95	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
96	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
97	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
98	#536	—	7	1	no	Sylow
99	C₂³×C₇	C₂³×C₃	1	2	yes	—
100	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
101	C₇×Q₈	—	2	2	no	—
102	C₇×D₄	—	2	2	no	—
103	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
104	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
105	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
106	C₂²⋊D₇	—	2	4	no	—
107	C₂×Dic₇	—	2	4	no	—
108	C₂×Dic₇	C₂³×C₃	1	2	yes	—
109	C₂×Dic₇	C₂³×C₃	1	2	yes	—
110	C₂×Dic₇	C₂³×C₃	1	2	yes	—
111	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
112	C₄×D₇	—	2	4	no	—
113	C₄×D₇	C₃×D₄	1	4	yes	—
114	Dic₁₄	—	2	2	no	—
115	Dic₁₄	—	2	2	no	—
116	Dic₁₄	—	2	4	no	—
117	C₃×#41	—	14	4	no	—
118	C₂×C₃×Q₈	—	7	2	no	—
119	C₂×C₃×Q₈	—	7	1	no	—
120	C₂×C₃×Q₈	—	7	1	no	—
121	C₂×C₃×Q₈	—	7	1	no	—
122	C₂×C₃×Q₈	—	14	4	no	—
123	C₂×C₃×D₄	—	7	2	no	—
124	C₂×C₃×D₄	—	7	1	no	—
125	C₂²×C₃×C₄	—	7	1	no	—
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	4	no	—
130	C₃×C₄⋊C₄	—	7	4	no	—
131	C₃×C₂²⋊C₄	—	7	2	no	—
132	C₃×C₂²⋊C₄	—	7	4	no	—
133	C₃×C₂²⋊C₄	—	7	4	no	—
134	C₃×C₄²	—	7	1	no	—
135	C₃×C₄²	—	7	1	no	—
136	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
137	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
138	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	2	yes	—
139	AGL(1,𝔽₇)	—	2	2	no	—
140	C₂×#41	—	7	1	no	—
141	C₂²×Q₈	—	7	1	no	—
142	C₄⋊Q₈	—	7	1	no	—
143	C₄⋊Q₈	—	7	1	no	—
144	#124	—	7	1	no	—
145	#124	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
146	#123	—	7	4	no	—
147	C₂²⋊Q₈	—	7	4	no	—
148	#117	—	7	1	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	2	4	no	—
151	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
152	C₂×D₇	—	2	2	no	—
153	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
157	Dic₇	—	2	2	no	—
158	Dic₇	—	2	2	no	—
159	Dic₇	—	2	2	no	—
160	Dic₇	C₂×C₃×D₄	1	2	yes	—
161	C₂³×C₃	—	7	1	no	—
162	C₂³×C₃	—	7	2	no	—
163	C₃×Q₈	—	14	2	no	—
164	C₃×Q₈	—	14	4	no	—
165	C₃×Q₈	—	14	2	no	—
166	C₃×Q₈	—	14	2	no	—
167	C₃×D₄	—	14	2	no	—
168	C₃×D₄	—	14	4	no	—
169	C₂×C₃×C₄	—	7	4	no	—
170	C₂×C₃×C₄	—	7	1	no	—
171	C₂×C₃×C₄	—	7	2	no	—
172	C₂×C₃×C₄	—	7	2	no	—
173	C₂×C₃×C₄	—	7	2	no	—
174	C₂×C₃×C₄	—	7	1	no	—
175	C₂×C₃×C₄	—	7	2	no	—
176	C₂×C₃×C₄	—	7	2	no	—
177	C₂×C₃×C₄	—	14	4	no	—
178	C₂×C₃×C₄	—	14	4	no	—
179	ΓL(1,𝔽₈)	#536	1	1	yes	characteristic, complemented, Hall
180	#41	—	14	4	no	—
181	C₂×Q₈	—	7	2	no	—
182	C₂×Q₈	—	7	1	no	—
183	C₂×Q₈	—	7	1	no	—
184	C₂×Q₈	—	14	4	no	—
185	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
186	C₂×D₄	—	7	2	no	—
187	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₄⋊C₄	—	7	2	no	—
192	C₄⋊C₄	—	7	4	no	—
193	C₄⋊C₄	—	7	4	no	—
194	C₂²⋊C₄	—	7	4	no	—
195	C₂²⋊C₄	—	7	2	no	—
196	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	4	yes	—
197	C₄²	—	7	1	no	—
198	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
199	C₂×C₇	—	4	2	no	—
200	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
202	D₇	—	2	2	no	—
203	C₂²×C₃	—	7	2	no	—
204	C₂²×C₃	—	7	1	no	—
205	C₂²×C₃	—	14	2	no	—
206	C₂²×C₃	—	14	4	no	—
207	C₂²×C₃	—	14	2	no	—
208	C₃×C₄	—	7	2	no	—
209	C₃×C₄	—	7	2	no	—
210	C₃×C₄	—	14	2	no	—
211	C₃×C₄	—	14	2	no	—
212	C₃×C₄	—	14	2	no	—
213	C₃×C₄	—	14	2	no	—
214	C₃×C₄	—	14	2	no	—
215	C₂³	—	7	1	no	—
216	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
217	Q₈	—	2	2	no	—
218	Q₈	—	14	4	no	—
219	Q₈	—	14	2	no	—
220	Q₈	—	14	2	no	—
221	D₄	—	2	2	no	—
222	D₄	—	14	4	no	—
223	C₂×C₄	—	7	1	no	—
224	C₂×C₄	—	7	4	no	—
225	C₂×C₄	—	7	2	no	—
226	C₂×C₄	—	7	2	no	—
227	C₂×C₄	—	7	2	no	—
228	C₂×C₄	—	14	4	no	—
229	C₂×C₄	—	14	4	no	—
230	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
231	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
232	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
233	C₇	C₃×#536	1	1	yes	characteristic, complemented, Sylow
234	C₂×C₃	—	7	1	no	—
235	C₂×C₃	—	7	2	no	—
236	C₂×C₃	—	14	2	no	—
237	C₂×C₃	—	28	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	7	2	no	—
241	C₂²	—	14	2	no	—
242	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	7	2	no	—
246	C₄	—	14	2	no	—
247	C₄	—	14	2	no	—
248	C₄	—	14	2	no	—
249	C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
250	C₃	—	7	1	no	Sylow
251	C₂	—	4	2	no	—
252	C₂	—	14	2	no	—
253	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
254	C₂	#10621519	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#124	C₂×C₃
prime radical	C₂×#1159⋊C₃	C₂
hypercenter	#124	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#536
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×AGL(1,𝔽₇)	C₂	semidirect
3. ΓL(1,𝔽₈)×#124	C₂	semidirect
4. #6507⋊C₃	C₂	semidirect
5. #6499⋊C₃	C₂	semidirect
6. #6446⋊C₃	C₂	semidirect
7. #6441⋊C₃	C₂	semidirect
8. #6435⋊C₃	C₂	semidirect
9. #6432⋊C₃	C₂	semidirect
10. #90366	C₃	semidirect
30. C₂×Q₈×D₇	C₂×C₃	semidirect
32. C₇×#124	C₂×C₃	semidirect
33. #6507	C₂×C₃	semidirect
34. #6499	C₂×C₃	semidirect
35. #6446	C₂×C₃	semidirect
36. #6441	C₂×C₃	semidirect
37. #6435	C₂×C₃	semidirect
38. #6432	C₂×C₃	semidirect
145. #124	AGL(1,𝔽₇)	semidirect
179. ΓL(1,𝔽₈)	#536	semidirect
233. C₇	C₃×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×AGL(1,𝔽₇)	2
C₂×#1159⋊C₃	2
ΓL(1,𝔽₈)×#124	2
#6507⋊C₃	2
#6499⋊C₃	2
#6446⋊C₃	2
#6441⋊C₃	2
#6435⋊C₃	2
#6432⋊C₃	2
#90366	3
C₃×#536	7

Subgroups — #90366⋊C3 #13057410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90366⋊C₃ #13057410