Subgroups — #90310⋊C₃ #13057353

Order 1344.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 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, complemented
2	#6503⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#6502⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6501⋊C₃	C₂	1	2	yes	maximal, complemented
5	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6493⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6489⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6456⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6453⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#6449⋊C₃	C₂	1	2	yes	maximal, complemented
11	#6446⋊C₃	C₂	1	2	yes	maximal, complemented
12	#90310	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	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
17	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
18	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
19	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
21	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×ΓL(1,𝔽₈)	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	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
28	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
29	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
30	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
31	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
32	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
33	(ΓL(1,𝔽₈)⋊C₄)⋊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, complemented
37	#6503	C₂×C₃	1	1	yes	characteristic, complemented
38	#6502	C₂×C₃	1	2	yes	complemented
39	#6501	C₂×C₃	1	2	yes	complemented
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6493	C₂×C₃	1	1	yes	characteristic, complemented
42	#6489	C₂×C₃	1	1	yes	characteristic, complemented
43	#6456	C₂×C₃	1	1	yes	characteristic, complemented
44	#6453	C₂×C₃	1	1	yes	characteristic, complemented
45	#6449	C₂×C₃	1	2	yes	complemented
46	#6446	C₂×C₃	1	2	yes	complemented
47	C₃×#540	—	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	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
51	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
52	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
53	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
55	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
56	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
57	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
58	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
59	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
60	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
61	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
62	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
63	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
64	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
65	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
66	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
67	C₄×AGL(1,𝔽₇)	—	4	2	no	—
68	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
70	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
72	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
73	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
74	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
75	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
77	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
80	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
81	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
82	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
83	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
85	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
86	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
87	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
88	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
89	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
90	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
91	C₃×#127	—	7	1	no	—
92	C₃×#125	—	7	1	no	—
93	C₃×#123	—	7	2	no	—
94	C₃×#123	—	7	2	no	—
95	C₃×#121	—	7	1	no	—
96	C₃×#121	—	7	2	no	—
97	C₃×#121	—	7	2	no	—
98	C₃×#121	—	7	1	no	—
99	C₃×C₄×D₄	—	7	1	no	—
100	C₃×C₄×D₄	—	7	1	no	—
101	C₃×#117	—	7	1	no	—
102	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
103	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
105	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
106	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
107	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
108	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
109	C₂×AGL(1,𝔽₇)	—	2	2	no	—
110	C₂×AGL(1,𝔽₇)	—	2	2	no	—
111	C₂×AGL(1,𝔽₇)	—	2	2	no	—
112	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
113	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
114	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
115	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
116	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
117	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
118	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
119	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
120	#540	—	7	1	no	Sylow
121	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂³×C₇	C₂³×C₃	1	2	yes	—
123	C₂²×D₇	C₂³×C₃	1	2	yes	—
124	C₇×D₄	—	2	2	no	—
125	C₇×D₄	—	4	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
128	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
130	C₂²⋊D₇	—	4	2	no	—
131	C₂²⋊D₇	—	4	2	no	—
132	C₂²⋊D₇	—	4	2	no	—
133	C₂×Dic₇	—	4	2	no	—
134	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
135	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
136	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
137	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
138	C₂×Dic₇	C₂³×C₃	1	2	yes	—
139	D₂₈	—	2	2	no	—
140	C₄×D₇	—	4	2	no	—
141	C₂×C₃×D₄	—	7	1	no	—
142	C₂×C₃×D₄	—	7	2	no	—
143	C₂×C₃×D₄	—	7	2	no	—
144	C₂×C₃×D₄	—	7	2	no	—
145	C₂×C₃×D₄	—	7	1	no	—
146	C₂×C₃×D₄	—	7	2	no	—
147	C₂²×C₃×C₄	—	7	2	no	—
148	C₂²×C₃×C₄	—	7	2	no	—
149	C₂²×C₃×C₄	—	7	1	no	—
150	C₃×C₄⋊C₄	—	7	1	no	—
151	C₃×C₄⋊C₄	—	7	1	no	—
152	C₃×C₄⋊C₄	—	7	1	no	—
153	C₃×C₄⋊C₄	—	7	2	no	—
154	C₃×C₄⋊C₄	—	7	2	no	—
155	C₃×C₄⋊C₄	—	7	1	no	—
156	C₃×C₂²⋊C₄	—	7	1	no	—
157	C₃×C₂²⋊C₄	—	7	1	no	—
158	C₃×C₂²⋊C₄	—	7	2	no	—
159	C₃×C₂²⋊C₄	—	7	2	no	—
160	C₃×C₂²⋊C₄	—	7	2	no	—
161	C₃×C₂²⋊C₄	—	7	2	no	—
162	C₃×C₄²	—	7	1	no	—
163	C₃×C₄²	—	7	1	no	—
164	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
165	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
166	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
167	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
169	AGL(1,𝔽₇)	—	4	2	no	—
170	#127	—	7	1	no	—
171	#125	—	7	1	no	—
172	#123	—	7	2	no	—
173	#123	—	7	2	no	—
174	#121	—	7	2	no	—
175	#121	—	7	2	no	—
176	#121	—	7	1	no	—
177	#121	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
178	C₄×D₄	—	7	1	no	—
179	C₄×D₄	—	7	1	no	—
180	#117	—	7	1	no	—
181	C₂²×C₇	—	2	1	no	—
182	C₂²×C₇	—	2	2	no	—
183	C₂²×C₇	—	2	2	no	—
184	C₂²×C₇	—	2	1	no	—
185	C₂²×C₇	—	2	2	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
188	C₂×D₇	—	2	2	no	—
189	C₂×D₇	—	2	2	no	—
190	C₂×D₇	—	2	2	no	—
191	C₄×C₇	—	2	1	no	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	C₃×#41	1	2	yes	—
194	Dic₇	—	2	1	no	—
195	Dic₇	—	2	1	no	—
196	Dic₇	—	2	2	no	—
197	Dic₇	—	2	1	no	—
198	Dic₇	—	2	1	no	—
199	C₂³×C₃	—	7	2	no	—
200	C₂³×C₃	—	7	1	no	—
201	C₂³×C₃	—	7	2	no	—
202	C₃×D₄	—	14	2	no	—
203	C₃×D₄	—	14	2	no	—
204	C₃×D₄	—	28	2	no	—
205	C₃×D₄	—	28	2	no	—
206	C₃×D₄	—	28	2	no	—
207	C₃×D₄	—	28	2	no	—
208	C₂×C₃×C₄	—	7	1	no	—
209	C₂×C₃×C₄	—	7	1	no	—
210	C₂×C₃×C₄	—	7	1	no	—
211	C₂×C₃×C₄	—	7	1	no	—
212	C₂×C₃×C₄	—	7	2	no	—
213	C₂×C₃×C₄	—	7	1	no	—
214	C₂×C₃×C₄	—	7	1	no	—
215	C₂×C₃×C₄	—	7	2	no	—
216	C₂×C₃×C₄	—	14	2	no	—
217	C₂×C₃×C₄	—	28	2	no	—
218	C₂×C₃×C₄	—	28	2	no	—
219	ΓL(1,𝔽₈)	#540	1	1	yes	characteristic, complemented, Hall
220	C₂×D₄	—	7	2	no	—
221	C₂×D₄	—	7	2	no	—
222	C₂×D₄	—	7	2	no	—
223	C₂×D₄	—	7	1	no	—
224	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
225	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
226	C₂²×C₄	—	7	2	no	—
227	C₂²×C₄	—	7	2	no	—
228	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
229	C₄⋊C₄	—	7	2	no	—
230	C₄⋊C₄	—	7	1	no	—
231	C₄⋊C₄	—	7	1	no	—
232	C₄⋊C₄	—	7	1	no	—
233	C₄⋊C₄	—	7	2	no	—
234	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
235	C₂²⋊C₄	—	7	1	no	—
236	C₂²⋊C₄	—	7	1	no	—
237	C₂²⋊C₄	—	7	2	no	—
238	C₂²⋊C₄	—	7	2	no	—
239	C₂²⋊C₄	—	7	2	no	—
240	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
241	C₄²	—	7	1	no	—
242	C₄²	—	7	1	no	—
243	C₂×C₇	—	4	1	no	—
244	C₂×C₇	—	4	2	no	—
245	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
246	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
248	D₇	—	4	2	no	—
249	C₂²×C₃	—	7	1	no	—
250	C₂²×C₃	—	14	1	no	—
251	C₂²×C₃	—	14	2	no	—
252	C₂²×C₃	—	14	2	no	—
253	C₂²×C₃	—	14	1	no	—
254	C₂²×C₃	—	14	2	no	—
255	C₂²×C₃	—	14	1	no	—
256	C₂²×C₃	—	14	2	no	—
257	C₂²×C₃	—	14	2	no	—
258	C₂²×C₃	—	14	2	no	—
259	C₃×C₄	—	7	2	no	—
260	C₃×C₄	—	14	1	no	—
261	C₃×C₄	—	14	2	no	—
262	C₃×C₄	—	14	1	no	—
263	C₃×C₄	—	14	1	no	—
264	C₃×C₄	—	14	2	no	—
265	C₃×C₄	—	14	1	no	—
266	C₃×C₄	—	14	1	no	—
267	C₂³	—	7	2	no	—
268	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
269	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
270	D₄	—	2	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	14	2	no	—
273	D₄	—	28	2	no	—
274	D₄	—	28	2	no	—
275	D₄	—	28	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	7	1	no	—
278	C₂×C₄	—	7	1	no	—
279	C₂×C₄	—	7	1	no	—
280	C₂×C₄	—	7	1	no	—
281	C₂×C₄	—	7	2	no	—
282	C₂×C₄	—	28	2	no	—
283	C₂×C₄	—	28	2	no	—
284	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
285	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
286	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
287	C₇	C₃×#540	1	1	yes	characteristic, complemented, Sylow
288	C₂×C₃	—	7	1	no	—
289	C₂×C₃	—	7	1	no	—
290	C₂×C₃	—	7	1	no	—
291	C₂×C₃	—	28	1	no	—
292	C₂×C₃	—	28	2	no	—
293	C₂×C₃	—	28	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	14	2	no	—
301	C₂²	—	14	2	no	—
302	C₂²	—	14	2	no	—
303	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	14	1	no	—
307	C₄	—	14	1	no	—
308	C₄	—	14	1	no	—
309	C₄	—	14	2	no	—
310	C₄	—	14	1	no	—
311	C₄	#1159⋊C₃	1	2	yes	—
312	C₃	—	7	1	no	Sylow
313	C₂	—	4	1	no	—
314	C₂	—	4	2	no	—
315	C₂	—	28	2	no	—
316	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
317	C₂	#6548⋊C₃	1	1	yes	characteristic
318	C₂	#10621518	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
hypercenter	#121	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#540
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
1. ΓL(1,𝔽₈)×#121	C₂	semidirect
2. #6503⋊C₃	C₂	semidirect
3. #6502⋊C₃	C₂	semidirect
4. #6501⋊C₃	C₂	semidirect
5. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
6. #6493⋊C₃	C₂	semidirect
7. #6489⋊C₃	C₂	semidirect
8. #6456⋊C₃	C₂	semidirect
9. #6453⋊C₃	C₂	semidirect
10. #6449⋊C₃	C₂	semidirect
11. #6446⋊C₃	C₂	semidirect
12. #90310	C₃	semidirect
36. C₇×#121	C₂×C₃	semidirect
37. #6503	C₂×C₃	semidirect
38. #6502	C₂×C₃	semidirect
39. #6501	C₂×C₃	semidirect
40. D₄×Dic₇	C₂×C₃	semidirect
41. #6493	C₂×C₃	semidirect
42. #6489	C₂×C₃	semidirect
43. #6456	C₂×C₃	semidirect
44. #6453	C₂×C₃	semidirect
45. #6449	C₂×C₃	semidirect
46. #6446	C₂×C₃	semidirect
177. #121	AGL(1,𝔽₇)	semidirect
219. ΓL(1,𝔽₈)	#540	semidirect
287. C₇	C₃×#540	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
#6503⋊C₃	2
#6502⋊C₃	2
#6501⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6493⋊C₃	2
#6489⋊C₃	2
#6456⋊C₃	2
#6453⋊C₃	2
#6449⋊C₃	2
#6446⋊C₃	2
#90310	3
C₃×#540	7

Subgroups — #90310⋊C3 #13057353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90310⋊C₃ #13057353