Subgroups — #90294⋊C₃ #13057339

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)×#121	C₂	1	1	yes	maximal, characteristic
2	#6499⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#6499⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6498⋊C₃	C₂	1	2	yes	maximal, complemented
5	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6489⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6451⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6450⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
11	#6441⋊C₃	C₂	1	2	yes	maximal, complemented
12	#90294	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
15	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₂²×ΓL(1,𝔽₈)⋊C₄	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	C₂²⋊C₄×ΓL(1,𝔽₈)	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	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
25	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
26	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
27	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
29	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
32	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
33	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
34	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
36	C₇×#121	C₂×C₃	1	1	yes	characteristic
37	#6499	C₂×C₃	1	2	yes	complemented
38	#6499	C₂×C₃	1	1	yes	characteristic, complemented
39	#6498	C₂×C₃	1	2	yes	complemented
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6489	C₂×C₃	1	1	yes	characteristic, complemented
42	#6487	C₂×C₃	1	1	yes	characteristic, complemented
43	#6451	C₂×C₃	1	1	yes	characteristic, complemented
44	#6450	C₂×C₃	1	1	yes	characteristic, complemented
45	#6442	C₂×C₃	1	2	yes	complemented
46	#6441	C₂×C₃	1	2	yes	complemented
47	C₃×#542	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
49	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
51	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
52	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
55	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
56	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
61	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
62	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
63	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
64	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
65	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
67	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂²×Dic₇	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×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
76	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
77	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
78	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
81	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
82	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	C₄×Dic₇	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
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	2	no	—
101	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
102	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
103	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
105	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
106	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
107	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
108	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
109	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
110	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
111	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
112	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
113	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
114	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
115	#542	—	7	1	no	Sylow
116	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
117	C₂³×C₇	C₂³×C₃	1	2	yes	—
118	C₇×D₄	—	2	2	no	—
119	C₇×D₄	—	4	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	—	4	2	no	—
125	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂³×C₃	1	2	yes	—
129	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
130	C₂×Dic₇	C₂³×C₃	1	2	yes	—
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	1	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	2	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₂²⋊C₄	—	7	1	no	—
147	C₃×C₂²⋊C₄	—	7	1	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	2	no	—
152	C₃×C₂²⋊C₄	—	7	2	no	—
153	C₃×C₄²	—	7	1	no	—
154	C₃×C₄²	—	7	1	no	—
155	C₃×C₄²	—	7	2	no	—
156	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
157	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
158	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
159	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
161	C₄⋊Q₈	—	7	1	no	—
162	#126	—	7	2	no	—
163	#124	—	7	1	no	—
164	#124	—	7	2	no	—
165	#123	—	7	2	no	—
166	C₂²⋊Q₈	—	7	1	no	—
167	C₂²⋊Q₈	—	7	2	no	—
168	#121	AGL(1,𝔽₇)	1	1	yes	characteristic
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	#117	—	7	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	—	2	1	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	C₃×#41	1	2	yes	—
182	Dic₇	—	2	2	no	—
183	Dic₇	—	2	1	no	—
184	Dic₇	—	2	2	no	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	2	1	no	—
187	Dic₇	—	2	1	no	—
188	C₂³×C₃	—	7	1	no	—
189	C₂³×C₃	—	7	2	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	1	no	—
195	C₂×C₃×C₄	—	7	1	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	2	no	—
200	C₂×C₃×C₄	—	7	1	no	—
201	C₂×C₃×C₄	—	7	2	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	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
209	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
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	2	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	—	7	1	no	—
217	C₄⋊C₄	—	7	1	no	—
218	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
219	C₂²⋊C₄	—	7	1	no	—
220	C₂²⋊C₄	—	7	1	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
226	C₄²	—	7	1	no	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	2	no	—
229	C₂×C₇	—	4	1	no	—
230	C₂×C₇	—	4	2	no	—
231	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
232	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
234	C₂²×C₃	—	7	1	no	—
235	C₂²×C₃	—	14	1	no	—
236	C₂²×C₃	—	14	2	no	—
237	C₂²×C₃	—	14	2	no	—
238	C₂²×C₃	—	14	1	no	—
239	C₂²×C₃	—	14	1	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₃×C₄	—	7	2	no	—
242	C₃×C₄	—	14	1	no	—
243	C₃×C₄	—	14	2	no	—
244	C₃×C₄	—	14	1	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	—	14	2	no	—
247	C₃×C₄	—	14	1	no	—
248	C₃×C₄	—	14	1	no	—
249	C₃×C₄	—	14	1	no	—
250	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
251	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
252	Q₈	—	14	2	no	—
253	Q₈	—	28	2	no	—
254	D₄	—	2	2	no	—
255	D₄	—	4	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	7	1	no	—
258	C₂×C₄	—	7	1	no	—
259	C₂×C₄	—	7	2	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	28	2	no	—
264	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
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	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	14	2	no	—
283	C₄	—	14	1	no	—
284	C₄	—	14	2	no	—
285	C₄	—	14	1	no	—
286	C₄	—	14	1	no	—
287	C₄	—	14	1	no	—
288	C₄	#1159⋊C₃	1	2	yes	—
289	C₃	—	7	1	no	Sylow
290	C₂	—	4	1	no	—
291	C₂	—	4	2	no	—
292	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
293	C₂	#6548⋊C₃	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×#121	C₂×C₃
prime radical	ΓL(1,𝔽₈)×#121	C₂
hypercenter	#121	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#542
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. #6499⋊C₃	C₂	semidirect
3. #6499⋊C₃	C₂	semidirect
4. #6498⋊C₃	C₂	semidirect
5. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
6. #6489⋊C₃	C₂	semidirect
7. #6487⋊C₃	C₂	semidirect
8. #6451⋊C₃	C₂	semidirect
9. #6450⋊C₃	C₂	semidirect
10. #6442⋊C₃	C₂	semidirect
11. #6441⋊C₃	C₂	semidirect
12. #90294	C₃	semidirect
37. #6499	C₂×C₃	semidirect
38. #6499	C₂×C₃	semidirect
39. #6498	C₂×C₃	semidirect
40. D₄×Dic₇	C₂×C₃	semidirect
41. #6489	C₂×C₃	semidirect
42. #6487	C₂×C₃	semidirect
43. #6451	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
45. #6442	C₂×C₃	semidirect
46. #6441	C₂×C₃	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)×#121	2
#6499⋊C₃	2
#6498⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6489⋊C₃	2
#6487⋊C₃	2
#6451⋊C₃	2
#6450⋊C₃	2
#6442⋊C₃	2
#6441⋊C₃	2
#90294	3
C₃×#542	7

Subgroups — #90294⋊C3 #13057339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90294⋊C₃ #13057339