Subgroups — #90470⋊C₃ #13057513

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621519	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621516	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#6513⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#1144⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6481⋊C₃	C₂	1	2	yes	maximal, complemented
7	Q₁₆×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
8	#6479⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6478⋊C₃	C₂	1	2	yes	maximal, complemented
10	#6472⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#6470⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#90470	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	#41×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	#66659	—	2	1	no	—
16	#66659	C₂²	1	2	yes	—
17	Q₈×AGL(1,𝔽₇)	—	2	1	no	—
18	Q₈×AGL(1,𝔽₇)	C₂²	1	2	yes	—
19	Q₈×AGL(1,𝔽₇)	C₂²	1	2	yes	—
20	#1159⋊C₃	—	2	2	no	—
21	#1159⋊C₃	C₂²	1	2	yes	—
22	#1157⋊C₃	—	2	2	no	—
23	#1157⋊C₃	—	2	1	no	—
24	#1157⋊C₃	C₂²	1	1	yes	characteristic
25	#1157⋊C₃	C₂²	1	1	yes	characteristic
26	C₂×ΓL(1,𝔽₈)⋊Q₈	—	2	1	no	—
27	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
28	Q₁₆×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
29	SD₁₆×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
30	M₁₆×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
31	#1144⋊C₃	C₂²	1	2	yes	—
32	#1144⋊C₃	C₂²	1	2	yes	—
33	#1144⋊C₃	C₂²	1	1	yes	characteristic
34	#1143⋊C₃	C₂²	1	1	yes	characteristic
35	#1142⋊C₃	C₂²	1	1	yes	characteristic
36	#1141⋊C₃	C₂²	1	1	yes	characteristic
37	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₂₈⋊C₃	C₂²	1	2	yes	—
39	#1132⋊C₃	C₂²	1	2	yes	—
40	#1131⋊C₃	C₂²	1	2	yes	—
41	C₈×AGL(1,𝔽₇)	C₂²	1	2	yes	—
42	#6554	C₂×C₃	1	1	yes	characteristic, complemented
43	#6551	C₂×C₃	1	1	yes	characteristic, complemented
44	C₇×#137	C₂×C₃	1	1	yes	characteristic, complemented
45	#6513	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#1144	C₂×C₃	1	1	yes	characteristic, complemented
47	#6481	C₂×C₃	1	2	yes	complemented
48	D₇×Q₁₆	C₂×C₃	1	2	yes	complemented
49	#6479	C₂×C₃	1	2	yes	complemented
50	#6478	C₂×C₃	1	2	yes	complemented
51	#6472	C₂×C₃	1	1	yes	characteristic, complemented
52	#6470	C₂×C₃	1	1	yes	characteristic, complemented
53	C₃×#578	—	7	1	no	maximal, Hall
54	Q₈×ΓL(1,𝔽₈)	—	2	1	no	—
55	Q₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
56	Q₈×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
57	D₄×ΓL(1,𝔽₈)	—	2	1	no	—
58	D₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
59	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
60	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
61	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
62	(C₂²⋊D₇)⋊C₃	—	2	1	no	—
63	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
64	(C₂²⋊D₇)⋊C₃	D₄	1	2	yes	—
65	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
66	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
67	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
68	C₄⋊AGL(1,𝔽₇)	D₄	1	1	yes	characteristic
69	C₄⋊AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
70	C₄×AGL(1,𝔽₇)	—	2	2	no	—
71	C₄×AGL(1,𝔽₇)	—	2	2	no	—
72	C₄×AGL(1,𝔽₇)	—	2	1	no	—
73	C₄×AGL(1,𝔽₇)	—	2	2	no	—
74	C₄×AGL(1,𝔽₇)	C₂³	1	2	yes	—
75	ΓL(1,𝔽₈)⋊Q₈	—	2	1	no	—
76	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
77	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
78	ΓL(1,𝔽₈)⋊Q₈	D₄	1	1	yes	characteristic
79	ΓL(1,𝔽₈)⋊Q₈	C₂³	1	2	yes	—
80	ΓL(1,𝔽₈)⋊Q₈	C₂³	1	1	yes	characteristic
81	C₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
82	ΓL(1,𝔽₈)⋊C₈	C₂³	1	1	yes	characteristic
83	ΓL(1,𝔽₈)⋊C₈	C₂³	1	1	yes	characteristic
84	C₇×#41	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
86	#1161	—	2	1	no	—
87	#1161	C₂²×C₃	1	2	yes	—
88	Q₈×D₇	—	2	1	no	—
89	Q₈×D₇	C₂²×C₃	1	2	yes	—
90	Q₈×D₇	C₂²×C₃	1	2	yes	—
91	#1159	—	2	2	no	—
92	#1159	C₂²×C₃	1	2	yes	—
93	#1157	—	2	2	no	—
94	#1157	—	2	1	no	—
95	#1157	C₂²×C₃	1	1	yes	characteristic
96	#1157	C₂²×C₃	1	1	yes	characteristic
97	C₂×Dic₁₄	—	2	1	no	—
98	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
99	C₇×Q₁₆	C₂²×C₃	1	2	yes	—
100	C₇×SD₁₆	C₂²×C₃	1	2	yes	—
101	C₇×M₁₆	C₂²×C₃	1	1	yes	characteristic
102	#1144	C₂²×C₃	1	2	yes	—
103	#1144	C₂²×C₃	1	2	yes	—
104	#1144	C₂²×C₃	1	1	yes	characteristic
105	#1143	C₂²×C₃	1	1	yes	characteristic
106	#1142	C₂²×C₃	1	1	yes	characteristic
107	#1141	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
109	Dic₂₈	C₂²×C₃	1	2	yes	—
110	#1132	C₂²×C₃	1	2	yes	—
111	#1131	C₂²×C₃	1	2	yes	—
112	C₈×D₇	C₂²×C₃	1	2	yes	—
113	C₃×2₋¹⁺⁴	—	7	1	no	—
114	C₃×2₋¹⁺⁴	—	7	1	no	—
115	C₃×#137	—	7	1	no	—
116	C₃×#137	—	7	2	no	—
117	C₃×#137	—	7	2	no	—
118	C₃×#137	—	7	1	no	—
119	C₃×#135	—	7	2	no	—
120	C₃×#135	—	7	1	no	—
121	C₂×C₃×Q₁₆	—	7	2	no	—
122	C₂×C₃×Q₁₆	—	7	1	no	—
123	C₃×#131	—	7	1	no	—
124	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
125	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
126	C₂×AGL(1,𝔽₇)	—	2	1	no	—
127	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
128	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
129	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
130	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
131	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
132	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
133	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
134	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
135	#578	—	7	1	no	Sylow
136	C₇×Q₈	—	2	1	no	—
137	C₇×Q₈	C₂³×C₃	1	1	yes	characteristic
138	C₇×Q₈	C₂³×C₃	1	2	yes	—
139	C₇×D₄	—	2	1	no	—
140	C₇×D₄	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₄×C₇	—	2	1	no	—
142	C₂×C₄×C₇	—	2	1	no	—
143	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
144	C₂²⋊D₇	—	2	1	no	—
145	C₂²⋊D₇	—	2	2	no	—
146	C₂²⋊D₇	C₃×D₄	1	2	yes	—
147	C₂×Dic₇	—	2	1	no	—
148	C₂×Dic₇	—	2	2	no	—
149	D₂₈	—	2	2	no	—
150	D₂₈	C₃×D₄	1	1	yes	characteristic
151	D₂₈	C₂³×C₃	1	1	yes	characteristic
152	C₄×D₇	—	2	2	no	—
153	C₄×D₇	—	2	2	no	—
154	C₄×D₇	—	2	1	no	—
155	C₄×D₇	—	2	2	no	—
156	C₄×D₇	C₂³×C₃	1	2	yes	—
157	Dic₁₄	—	2	1	no	—
158	Dic₁₄	—	2	2	no	—
159	Dic₁₄	—	2	2	no	—
160	Dic₁₄	C₃×D₄	1	1	yes	characteristic
161	Dic₁₄	C₂³×C₃	1	2	yes	—
162	Dic₁₄	C₂³×C₃	1	1	yes	characteristic
163	C₇×C₈	C₂³×C₃	1	2	yes	—
164	C₇⋊C₈	C₂³×C₃	1	1	yes	characteristic
165	C₇⋊C₈	C₂³×C₃	1	1	yes	characteristic
166	C₃×#41	—	7	1	no	—
167	C₃×#41	—	7	1	no	—
168	C₃×#41	—	7	2	no	—
169	C₃×#41	—	7	2	no	—
170	C₃×#41	—	7	1	no	—
171	C₃×#41	—	14	1	no	—
172	C₃×#41	—	14	2	no	—
173	C₃×#41	—	14	2	no	—
174	C₃×#41	—	14	1	no	—
175	C₂×C₃×Q₈	—	7	1	no	—
176	C₂×C₃×Q₈	—	7	2	no	—
177	C₂×C₃×Q₈	—	7	2	no	—
178	C₂×C₃×Q₈	—	7	1	no	—
179	C₂×C₃×Q₈	—	14	1	no	—
180	C₂×C₃×Q₈	—	14	1	no	—
181	C₃×Q₁₆	—	7	1	no	—
182	C₃×Q₁₆	—	7	2	no	—
183	C₃×Q₁₆	—	7	2	no	—
184	C₃×Q₁₆	—	7	2	no	—
185	C₃×Q₁₆	—	7	2	no	—
186	C₃×SD₁₆	—	7	2	no	—
187	C₃×SD₁₆	—	7	1	no	—
188	C₃×SD₁₆	—	7	1	no	—
189	C₃×SD₁₆	—	7	2	no	—
190	C₃×D₈	—	7	1	no	—
191	C₃×M₁₆	—	7	2	no	—
192	C₃×M₁₆	—	7	1	no	—
193	C₂×C₃×C₈	—	7	2	no	—
194	C₂×C₃×C₈	—	7	1	no	—
195	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
196	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
197	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
198	AGL(1,𝔽₇)	—	2	2	no	—
199	AGL(1,𝔽₇)	—	4	1	no	—
200	2₋¹⁺⁴	—	7	1	no	—
201	2₋¹⁺⁴	—	7	1	no	—
202	#137	—	7	2	no	—
203	#137	—	7	2	no	—
204	#137	—	7	1	no	—
205	#137	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
206	#135	—	7	1	no	—
207	#135	—	7	2	no	—
208	C₂×Q₁₆	—	7	1	no	—
209	C₂×Q₁₆	—	7	2	no	—
210	#131	—	7	1	no	—
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×D₇	—	2	1	no	—
214	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
215	C₄×C₇	—	2	1	no	—
216	C₄×C₇	—	2	2	no	—
217	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
218	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
219	Dic₇	—	2	1	no	—
220	Dic₇	—	2	2	no	—
221	Dic₇	C₂×C₃×D₄	1	2	yes	—
222	C₃×Q₈	—	7	1	no	—
223	C₃×Q₈	—	7	2	no	—
224	C₃×Q₈	—	7	1	no	—
225	C₃×Q₈	—	7	1	no	—
226	C₃×Q₈	—	7	2	no	—
227	C₃×Q₈	—	14	1	no	—
228	C₃×Q₈	—	14	2	no	—
229	C₃×Q₈	—	14	2	no	—
230	C₃×Q₈	—	14	1	no	—
231	C₃×D₄	—	7	1	no	—
232	C₃×D₄	—	7	2	no	—
233	C₃×D₄	—	7	1	no	—
234	C₃×D₄	—	7	1	no	—
235	C₃×D₄	—	14	1	no	—
236	C₃×D₄	—	14	2	no	—
237	C₃×D₄	—	14	2	no	—
238	C₃×D₄	—	14	1	no	—
239	C₂×C₃×C₄	—	7	2	no	—
240	C₂×C₃×C₄	—	7	1	no	—
241	C₂×C₃×C₄	—	14	1	no	—
242	C₂×C₃×C₄	—	14	1	no	—
243	C₂×C₃×C₄	—	14	2	no	—
244	C₂×C₃×C₄	—	14	1	no	—
245	C₂×C₃×C₄	—	14	2	no	—
246	C₂×C₃×C₄	—	14	2	no	—
247	C₂×C₃×C₄	—	14	2	no	—
248	C₂×C₃×C₄	—	14	1	no	—
249	C₃×C₈	—	7	1	no	—
250	C₃×C₈	—	7	1	no	—
251	C₃×C₈	—	7	2	no	—
252	ΓL(1,𝔽₈)	#578	1	1	yes	characteristic, complemented, Hall
253	#41	—	7	1	no	—
254	#41	—	7	2	no	—
255	#41	—	7	2	no	—
256	#41	—	7	1	no	—
257	#41	—	14	2	no	—
258	#41	—	14	1	no	—
259	#41	—	14	2	no	—
260	#41	—	14	1	no	—
261	#41	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
262	C₂×Q₈	—	7	2	no	—
263	C₂×Q₈	—	7	2	no	—
264	C₂×Q₈	—	7	1	no	—
265	C₂×Q₈	—	14	1	no	—
266	C₂×Q₈	—	14	1	no	—
267	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
268	Q₁₆	—	7	1	no	—
269	Q₁₆	—	7	2	no	—
270	Q₁₆	—	7	2	no	—
271	Q₁₆	—	7	2	no	—
272	Q₁₆	C₂×AGL(1,𝔽₇)	1	2	yes	—
273	SD₁₆	—	7	1	no	—
274	SD₁₆	—	7	1	no	—
275	SD₁₆	—	7	2	no	—
276	SD₁₆	C₂×AGL(1,𝔽₇)	1	2	yes	—
277	D₈	—	7	1	no	—
278	M₁₆	—	7	2	no	—
279	M₁₆	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
280	C₂×C₈	—	7	1	no	—
281	C₂×C₈	—	7	2	no	—
282	C₂×C₇	—	2	1	no	—
283	C₂×C₇	—	4	1	no	—
284	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
285	D₇	—	2	2	no	—
286	D₇	—	4	1	no	—
287	C₂²×C₃	—	7	2	no	—
288	C₂²×C₃	—	7	1	no	—
289	C₂²×C₃	—	14	1	no	—
290	C₂²×C₃	—	14	1	no	—
291	C₃×C₄	—	7	2	no	—
292	C₃×C₄	—	7	1	no	—
293	C₃×C₄	—	7	1	no	—
294	C₃×C₄	—	14	1	no	—
295	C₃×C₄	—	14	2	no	—
296	C₃×C₄	—	14	1	no	—
297	C₃×C₄	—	14	2	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	7	1	no	—
300	Q₈	—	7	2	no	—
301	Q₈	—	7	1	no	—
302	Q₈	—	14	2	no	—
303	Q₈	—	14	2	no	—
304	Q₈	—	14	1	no	—
305	Q₈	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
306	Q₈	C₂²×AGL(1,𝔽₇)	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	7	2	no	—
309	D₄	—	7	1	no	—
310	D₄	—	7	1	no	—
311	D₄	—	14	2	no	—
312	D₄	—	14	2	no	—
313	D₄	—	14	1	no	—
314	D₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	7	2	no	—
318	C₂×C₄	—	14	2	no	—
319	C₂×C₄	—	14	2	no	—
320	C₂×C₄	—	14	2	no	—
321	C₂×C₄	—	14	2	no	—
322	C₂×C₄	—	14	1	no	—
323	C₂×C₄	—	14	1	no	—
324	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
325	C₈	—	7	1	no	—
326	C₈	—	7	1	no	—
327	C₈	C₂²×AGL(1,𝔽₇)	1	2	yes	—
328	C₇	C₃×#578	1	1	yes	characteristic, complemented, Sylow
329	C₂×C₃	—	7	1	no	—
330	C₂×C₃	—	14	1	no	—
331	C₂×C₃	—	14	2	no	—
332	C₂×C₃	—	28	1	no	—
333	C₂×C₃	—	28	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	7	2	no	—
336	C₂²	—	14	1	no	—
337	C₂²	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	7	2	no	—
341	C₄	—	14	1	no	—
342	C₄	—	14	2	no	—
343	C₄	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
344	C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
345	C₃	—	7	1	no	Sylow
346	C₂	—	2	1	no	—
347	C₂	—	4	1	no	—
348	C₂	—	14	2	no	—
349	C₂	—	28	1	no	—
350	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×AGL(1,𝔽₇)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#137	C₂×C₃
socle	C₂×C₇	C₂²×C₃×D₄
hypercenter	#137	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#578
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #10621519	C₂	semidirect
2. #10621516	C₂	semidirect
3. ΓL(1,𝔽₈)×#137	C₂	semidirect
4. #6513⋊C₃	C₂	semidirect
5. C₂×#1144⋊C₃	C₂	semidirect
6. #6481⋊C₃	C₂	semidirect
7. Q₁₆×AGL(1,𝔽₇)	C₂	semidirect
8. #6479⋊C₃	C₂	semidirect
9. #6478⋊C₃	C₂	semidirect
10. #6472⋊C₃	C₂	semidirect
11. #6470⋊C₃	C₂	semidirect
12. #90470	C₃	semidirect
42. #6554	C₂×C₃	semidirect
43. #6551	C₂×C₃	semidirect
44. C₇×#137	C₂×C₃	semidirect
45. #6513	C₂×C₃	semidirect
46. C₂×#1144	C₂×C₃	semidirect
47. #6481	C₂×C₃	semidirect
48. D₇×Q₁₆	C₂×C₃	semidirect
49. #6479	C₂×C₃	semidirect
50. #6478	C₂×C₃	semidirect
51. #6472	C₂×C₃	semidirect
52. #6470	C₂×C₃	semidirect
205. #137	AGL(1,𝔽₇)	semidirect
252. ΓL(1,𝔽₈)	#578	semidirect
328. C₇	C₃×#578	semidirect

Maximal subgroups

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

subgroup	index
#10621519	2
#10621516	2
ΓL(1,𝔽₈)×#137	2
#6513⋊C₃	2
C₂×#1144⋊C₃	2
#6481⋊C₃	2
Q₁₆×AGL(1,𝔽₇)	2
#6479⋊C₃	2
#6478⋊C₃	2
#6472⋊C₃	2
#6470⋊C₃	2
#90470	3
C₃×#578	7

Subgroups — #90470⋊C3 #13057513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90470⋊C₃ #13057513