Subgroups — #90293⋊C₃ #13057338

Order 1344.

436 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#121	C₂	1	1	yes	maximal, characteristic
2	#6498⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6498⋊C₃	C₂	1	2	yes	maximal, complemented
4	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	2	yes	maximal, complemented
5	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#6452⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6441⋊C₃	C₂	1	2	yes	maximal, complemented
10	#90293	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
15	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
16	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
21	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
24	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
25	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
26	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
27	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
31	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
32	C₇×#121	C₂×C₃	1	1	yes	characteristic
33	#6498	C₂×C₃	1	2	yes	complemented
34	#6498	C₂×C₃	1	2	yes	complemented
35	D₄×Dic₇	C₂×C₃	1	2	yes	complemented
36	#6487	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×Dic₇⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
38	#6452	C₂×C₃	1	2	yes	complemented
39	#6442	C₂×C₃	1	2	yes	complemented
40	#6441	C₂×C₃	1	2	yes	complemented
41	C₃×#539	—	7	1	no	maximal, Hall
42	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
43	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
45	D₄×ΓL(1,𝔽₈)	—	4	1	no	—
46	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
48	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
51	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
52	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	1	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	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
57	ΓL(1,𝔽₈)⋊Q₈	—	4	1	no	—
58	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
59	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
61	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
62	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
63	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
64	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
65	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
68	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
69	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
70	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
71	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
72	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
73	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
76	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
77	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄×Dic₇	C₂²×C₃	1	2	yes	—
79	C₃×#126	—	7	2	no	—
80	C₃×#125	—	7	2	no	—
81	C₃×#123	—	7	2	no	—
82	C₃×#123	—	7	2	no	—
83	C₃×C₂²⋊Q₈	—	7	2	no	—
84	C₃×C₂²⋊Q₈	—	7	1	no	—
85	C₃×#121	—	7	1	no	—
86	C₃×C₄×D₄	—	7	2	no	—
87	C₂×C₃×C₄⋊C₄	—	7	1	no	—
88	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
89	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
90	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
91	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
93	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
94	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
95	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
96	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
97	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
98	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
99	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
100	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
101	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
102	#539	—	7	1	no	Sylow
103	C₂³×C₇	C₂³×C₃	1	2	yes	—
104	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
105	C₇×D₄	—	2	4	no	—
106	C₇×D₄	—	4	1	no	—
107	C₂×C₄×C₇	—	4	1	no	—
108	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
109	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
111	C₂×Dic₇	—	2	2	no	—
112	C₂×Dic₇	—	4	2	no	—
113	C₂×Dic₇	—	4	1	no	—
114	C₂×Dic₇	C₂³×C₃	1	2	yes	—
115	C₂×Dic₇	C₂³×C₃	1	2	yes	—
116	C₂×Dic₇	C₂³×C₃	1	2	yes	—
117	C₂×Dic₇	C₂³×C₃	1	2	yes	—
118	Dic₁₄	—	4	1	no	—
119	C₂×C₃×Q₈	—	7	1	no	—
120	C₂×C₃×D₄	—	7	2	no	—
121	C₂×C₃×D₄	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	1	no	—
123	C₂²×C₃×C₄	—	7	1	no	—
124	C₂²×C₃×C₄	—	7	2	no	—
125	C₂²×C₃×C₄	—	7	1	no	—
126	C₃×C₄⋊C₄	—	7	2	no	—
127	C₃×C₄⋊C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	2	no	—
130	C₃×C₄⋊C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	1	no	—
132	C₃×C₄⋊C₄	—	7	2	no	—
133	C₃×C₄⋊C₄	—	7	1	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	2	no	—
139	C₃×C₄²	—	7	2	no	—
140	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
141	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
142	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
143	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
145	#126	—	7	2	no	—
146	#125	—	7	2	no	—
147	#123	—	7	2	no	—
148	#123	—	7	2	no	—
149	C₂²⋊Q₈	—	7	1	no	—
150	C₂²⋊Q₈	—	7	2	no	—
151	#121	AGL(1,𝔽₇)	1	1	yes	characteristic
152	C₄×D₄	—	7	2	no	—
153	C₂×C₄⋊C₄	—	7	1	no	—
154	C₂²×C₇	—	2	1	no	—
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	C₃×#41	1	2	yes	—
160	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	Dic₇	—	2	2	no	—
165	Dic₇	—	2	2	no	—
166	Dic₇	—	2	2	no	—
167	Dic₇	—	2	2	no	—
168	C₂³×C₃	—	7	1	no	—
169	C₂³×C₃	—	7	2	no	—
170	C₃×Q₈	—	28	1	no	—
171	C₃×D₄	—	14	4	no	—
172	C₃×D₄	—	28	1	no	—
173	C₂×C₃×C₄	—	7	2	no	—
174	C₂×C₃×C₄	—	7	2	no	—
175	C₂×C₃×C₄	—	7	2	no	—
176	C₂×C₃×C₄	—	7	2	no	—
177	C₂×C₃×C₄	—	7	2	no	—
178	C₂×C₃×C₄	—	7	1	no	—
179	C₂×C₃×C₄	—	7	1	no	—
180	C₂×C₃×C₄	—	14	2	no	—
181	C₂×C₃×C₄	—	28	1	no	—
182	C₂×C₃×C₄	—	28	1	no	—
183	C₂×C₃×C₄	—	28	2	no	—
184	ΓL(1,𝔽₈)	#539	1	1	yes	characteristic, complemented, Hall
185	C₂×Q₈	—	7	1	no	—
186	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
187	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	7	1	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
192	C₄⋊C₄	—	7	2	no	—
193	C₄⋊C₄	—	7	2	no	—
194	C₄⋊C₄	—	7	2	no	—
195	C₄⋊C₄	—	7	2	no	—
196	C₄⋊C₄	—	7	2	no	—
197	C₄⋊C₄	—	7	2	no	—
198	C₄⋊C₄	—	7	1	no	—
199	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
200	C₂²⋊C₄	—	7	2	no	—
201	C₂²⋊C₄	—	7	2	no	—
202	C₂²⋊C₄	—	7	2	no	—
203	C₂²⋊C₄	—	7	2	no	—
204	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
205	C₄²	—	7	2	no	—
206	C₂×C₇	—	2	2	no	—
207	C₂×C₇	—	4	2	no	—
208	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
209	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂²×C₃	—	7	2	no	—
212	C₂²×C₃	—	7	1	no	—
213	C₂²×C₃	—	14	1	no	—
214	C₂²×C₃	—	14	1	no	—
215	C₂²×C₃	—	14	2	no	—
216	C₂²×C₃	—	14	2	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₃×C₄	—	14	1	no	—
219	C₃×C₄	—	14	1	no	—
220	C₃×C₄	—	14	2	no	—
221	C₃×C₄	—	14	2	no	—
222	C₃×C₄	—	14	2	no	—
223	C₃×C₄	—	14	2	no	—
224	C₃×C₄	—	14	2	no	—
225	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
226	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
227	Q₈	—	28	1	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	7	2	no	—
232	C₂×C₄	—	7	2	no	—
233	C₂×C₄	—	7	2	no	—
234	C₂×C₄	—	7	2	no	—
235	C₂×C₄	—	14	2	no	—
236	C₂×C₄	—	28	1	no	—
237	C₂×C₄	—	28	2	no	—
238	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
239	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
240	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
241	C₇	C₃×#539	1	1	yes	characteristic, complemented, Sylow
242	C₂×C₃	—	7	1	no	—
243	C₂×C₃	—	7	1	no	—
244	C₂×C₃	—	7	1	no	—
245	C₂×C₃	—	14	2	no	—
246	C₂×C₃	—	28	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#1159⋊C₃	1	2	yes	—
253	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	14	2	no	—
258	C₄	—	14	2	no	—
259	C₄	—	14	2	no	—
260	C₄	—	14	2	no	—
261	C₃	—	7	1	no	Sylow
262	C₂	—	2	2	no	—
263	C₂	—	4	2	no	—
264	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
265	C₂	#6548⋊C₃	1	1	yes	characteristic
266	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×#121	C₂×C₃
prime radical	ΓL(1,𝔽₈)×#121	C₂
hypercenter	#121	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#539
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
2. #6498⋊C₃	C₂	semidirect
3. #6498⋊C₃	C₂	semidirect
4. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
5. #6487⋊C₃	C₂	semidirect
6. C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	semidirect
7. #6452⋊C₃	C₂	semidirect
8. #6442⋊C₃	C₂	semidirect
9. #6441⋊C₃	C₂	semidirect
10. #90293	C₃	semidirect
33. #6498	C₂×C₃	semidirect
34. #6498	C₂×C₃	semidirect
35. D₄×Dic₇	C₂×C₃	semidirect
36. #6487	C₂×C₃	semidirect
37. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
38. #6452	C₂×C₃	semidirect
39. #6442	C₂×C₃	semidirect
40. #6441	C₂×C₃	semidirect
184. ΓL(1,𝔽₈)	#539	semidirect
241. C₇	C₃×#539	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)×#121	2
#6498⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6487⋊C₃	2
C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	2
#6452⋊C₃	2
#6442⋊C₃	2
#6441⋊C₃	2
#90293	3
C₃×#539	7

Subgroups — #90293⋊C3 #13057338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90293⋊C₃ #13057338