Subgroups — #90225⋊C₃ #13057268

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#6493⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#6461⋊C₃	C₂	1	2	yes	maximal, complemented
5	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
6	#6448⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6446⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6438⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
10	#6432⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#90225	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
15	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
16	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
24	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
25	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
27	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
28	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
29	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
30	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
31	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
32	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
33	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
34	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
36	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
37	#6493	C₂×C₃	1	1	yes	characteristic
38	#6487	C₂×C₃	1	1	yes	characteristic, complemented
39	#6461	C₂×C₃	1	2	yes	complemented
40	#6459	C₂×C₃	1	2	yes	complemented
41	#6448	C₂×C₃	1	2	yes	complemented
42	#6446	C₂×C₃	1	2	yes	complemented
43	#6438	C₂×C₃	1	1	yes	characteristic, complemented
44	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
45	#6432	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#542	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
50	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
55	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
56	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
57	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
59	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
60	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
61	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
62	C₄×AGL(1,𝔽₇)	—	4	2	no	—
63	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
64	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
65	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
67	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
69	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
76	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
77	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
78	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
79	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
80	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
81	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
82	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
84	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
85	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
86	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
87	C₄×Dic₇	C₂²×C₃	1	2	yes	—
88	C₃×C₄⋊Q₈	—	7	1	no	—
89	C₃×#126	—	7	2	no	—
90	C₃×#124	—	7	1	no	—
91	C₃×#124	—	7	2	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	2	no	—
94	C₃×C₂²⋊Q₈	—	7	1	no	—
95	C₃×#121	—	7	1	no	—
96	C₃×C₄×Q₈	—	7	1	no	—
97	C₃×C₄×D₄	—	7	1	no	—
98	C₃×#117	—	7	1	no	—
99	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
100	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
102	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
103	C₂×AGL(1,𝔽₇)	—	2	2	no	—
104	C₂×AGL(1,𝔽₇)	—	2	2	no	—
105	C₂×AGL(1,𝔽₇)	—	2	2	no	—
106	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
107	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
108	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
109	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
110	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
111	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
112	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
113	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
114	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
115	#542	—	7	1	no	Sylow
116	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
117	C₂²×D₇	C₂³×C₃	1	2	yes	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂²⋊D₇	—	4	2	no	—
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	C₂×Dic₇	C₂³×C₃	1	2	yes	—
129	D₂₈	—	2	2	no	—
130	C₄×D₇	—	4	2	no	—
131	Dic₁₄	—	2	2	no	—
132	Dic₁₄	—	4	2	no	—
133	C₂×C₃×Q₈	—	7	1	no	—
134	C₂×C₃×Q₈	—	7	2	no	—
135	C₂×C₃×D₄	—	7	1	no	—
136	C₂×C₃×D₄	—	7	2	no	—
137	C₂²×C₃×C₄	—	7	2	no	—
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₃×C₄⋊C₄	—	7	1	no	—
140	C₃×C₄⋊C₄	—	7	1	no	—
141	C₃×C₄⋊C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	1	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₂²⋊C₄	—	7	1	no	—
152	C₃×C₂²⋊C₄	—	7	1	no	—
153	C₃×C₄²	—	7	2	no	—
154	C₃×C₄²	—	7	1	no	—
155	C₃×C₄²	—	7	1	no	—
156	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
157	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
158	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
160	AGL(1,𝔽₇)	—	4	2	no	—
161	C₄⋊Q₈	—	7	1	no	—
162	#126	—	7	2	no	—
163	#124	—	7	2	no	—
164	#124	—	7	1	no	—
165	#123	—	7	2	no	—
166	C₂²⋊Q₈	—	7	1	no	—
167	C₂²⋊Q₈	—	7	2	no	—
168	#121	—	7	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	#117	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
176	C₂×D₇	—	2	2	no	—
177	C₂×D₇	—	2	2	no	—
178	C₂×D₇	—	2	2	no	—
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	C₃×#41	1	2	yes	—
185	Dic₇	—	2	2	no	—
186	Dic₇	—	2	2	no	—
187	Dic₇	—	2	2	no	—
188	C₂³×C₃	—	7	2	no	—
189	C₂³×C₃	—	7	1	no	—
190	C₃×Q₈	—	14	2	no	—
191	C₃×Q₈	—	28	2	no	—
192	C₃×D₄	—	14	2	no	—
193	C₃×D₄	—	28	2	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	7	1	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	1	no	—
200	C₂×C₃×C₄	—	7	1	no	—
201	C₂×C₃×C₄	—	7	1	no	—
202	C₂×C₃×C₄	—	7	1	no	—
203	C₂×C₃×C₄	—	14	2	no	—
204	C₂×C₃×C₄	—	28	2	no	—
205	ΓL(1,𝔽₈)	#542	1	1	yes	characteristic, complemented, Hall
206	C₂×Q₈	—	7	1	no	—
207	C₂×Q₈	—	7	2	no	—
208	C₂×D₄	—	7	2	no	—
209	C₂×D₄	—	7	1	no	—
210	C₂²×C₄	—	7	2	no	—
211	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
212	C₄⋊C₄	—	7	2	no	—
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	—	7	1	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
218	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
225	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
226	C₄²	—	7	2	no	—
227	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
228	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
229	C₂×C₇	—	4	1	no	—
230	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
231	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
233	D₇	—	4	2	no	—
234	C₂²×C₃	—	7	1	no	—
235	C₂²×C₃	—	14	1	no	—
236	C₂²×C₃	—	14	1	no	—
237	C₂²×C₃	—	14	1	no	—
238	C₂²×C₃	—	14	2	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₃×C₄	—	7	2	no	—
242	C₃×C₄	—	14	1	no	—
243	C₃×C₄	—	14	1	no	—
244	C₃×C₄	—	14	1	no	—
245	C₃×C₄	—	14	1	no	—
246	C₃×C₄	—	14	1	no	—
247	C₃×C₄	—	14	2	no	—
248	C₃×C₄	—	14	2	no	—
249	C₃×C₄	—	14	2	no	—
250	C₂³	—	7	2	no	—
251	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
252	Q₈	—	14	2	no	—
253	Q₈	—	28	2	no	—
254	D₄	—	14	2	no	—
255	D₄	—	28	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	7	2	no	—
258	C₂×C₄	—	7	2	no	—
259	C₂×C₄	—	7	2	no	—
260	C₂×C₄	—	28	2	no	—
261	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
262	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
263	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
264	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
265	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
266	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
267	C₇	C₃×#542	1	1	yes	characteristic, complemented, Sylow
268	C₂×C₃	—	7	1	no	—
269	C₂×C₃	—	7	1	no	—
270	C₂×C₃	—	7	1	no	—
271	C₂×C₃	—	28	1	no	—
272	C₂×C₃	—	28	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	14	2	no	—
277	C₂²	—	14	2	no	—
278	C₂²	—	14	2	no	—
279	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	14	2	no	—
286	C₄	—	14	2	no	—
287	C₄	—	14	2	no	—
288	C₄	#1157⋊C₃	1	2	yes	—
289	C₃	—	7	1	no	Sylow
290	C₂	—	4	1	no	—
291	C₂	—	28	2	no	—
292	C₂	C₂×#1157⋊C₃	1	1	yes	characteristic
293	C₂	#10621518	1	1	yes	characteristic
294	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×#117	C₂×C₃
prime radical	#6493⋊C₃	C₂
hypercenter	#117	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#542
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#117	C₂	semidirect
3. #6487⋊C₃	C₂	semidirect
4. #6461⋊C₃	C₂	semidirect
5. #6459⋊C₃	C₂	semidirect
6. #6448⋊C₃	C₂	semidirect
7. #6446⋊C₃	C₂	semidirect
8. #6438⋊C₃	C₂	semidirect
9. C₄×C₄⋊AGL(1,𝔽₇)	C₂	semidirect
10. #6432⋊C₃	C₂	semidirect
11. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
12. #90225	C₃	semidirect
36. C₇×#117	C₂×C₃	semidirect
38. #6487	C₂×C₃	semidirect
39. #6461	C₂×C₃	semidirect
40. #6459	C₂×C₃	semidirect
41. #6448	C₂×C₃	semidirect
42. #6446	C₂×C₃	semidirect
43. #6438	C₂×C₃	semidirect
44. C₄×D₂₈	C₂×C₃	semidirect
45. #6432	C₂×C₃	semidirect
46. C₄×Dic₁₄	C₂×C₃	semidirect
171. #117	AGL(1,𝔽₇)	semidirect
205. ΓL(1,𝔽₈)	#542	semidirect
267. C₇	C₃×#542	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)×#117	2
#6493⋊C₃	2
#6487⋊C₃	2
#6461⋊C₃	2
#6459⋊C₃	2
#6448⋊C₃	2
#6446⋊C₃	2
#6438⋊C₃	2
C₄×C₄⋊AGL(1,𝔽₇)	2
#6432⋊C₃	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90225	3
C₃×#542	7

Subgroups — #90225⋊C3 #13057268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90225⋊C₃ #13057268