Subgroups — #13060559

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621663	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#950	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6512	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621084	C₂	1	2	yes	maximal, complemented
6	#10621067	C₂	1	2	yes	maximal, complemented
7	#10621059	C₂	1	2	yes	maximal, complemented
8	#10621044	C₂	1	2	yes	maximal, complemented
9	#10620884	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#66565	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620853	C₂	1	1	yes	maximal, characteristic, complemented
12	#90461	—	3	1	no	maximal, Hall
13	C₃×C₇×#41	C₂²	1	1	yes	characteristic, complemented
14	Q₈×D₂₁	C₂²	1	2	yes	complemented
15	#66730	—	2	2	no	—
16	#66730	C₂²	1	2	yes	complemented
17	#66728	—	2	1	no	—
18	#66728	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₂	—	2	1	no	—
20	C₂×Dic₄₂	C₂²	1	1	yes	characteristic, complemented
21	C₇×#235	C₂²	1	1	yes	characteristic
22	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
23	D₇×#67	—	2	2	no	—
24	#66681	C₂²	1	2	yes	—
25	D₃×D₂₈	—	2	1	no	—
26	#66670	C₂²	1	2	yes	—
27	#66669	—	2	1	no	—
28	C₇×#216	C₂²	1	1	yes	characteristic, complemented
29	C₇×#215	C₂²	1	1	yes	characteristic, complemented
30	C₇×#214	C₂²	1	1	yes	characteristic, complemented
31	C₇×#213	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₃×#1143	C₂²	1	2	yes	complemented
34	C₃×#1141	C₂²	1	2	yes	complemented
35	C₃×#1136	C₂²	1	1	yes	characteristic, complemented
36	#66571	C₂²	1	2	yes	complemented
37	#66567	C₂²	1	2	yes	complemented
38	#66565	C₂²	1	2	yes	complemented
39	#66565	C₂²	1	2	yes	complemented
40	#66559	C₂²	1	2	yes	complemented
41	D₂₁⋊C₈	C₂²	1	2	yes	complemented
42	#6554	—	3	1	no	—
43	#6553	—	3	1	no	—
44	C₇×#135	—	3	1	no	—
45	#6514	—	3	1	no	—
46	#6512	D₃	1	1	yes	characteristic, complemented
47	#6481	—	3	2	no	—
48	#6479	—	3	2	no	—
49	D₇×SD₁₆	—	3	2	no	—
50	#6474	—	3	2	no	—
51	C₂×#1132	—	3	1	no	—
52	#6464	—	3	1	no	—
53	#5775	—	7	1	no	maximal, Hall
54	C₂²×C₃×D₇	—	2	1	no	—
55	C₂×D₃×D₇	—	2	2	no	—
56	C₃×C₇×Q₈	C₂³	1	1	yes	characteristic
57	C₃×C₇×D₄	—	2	1	no	—
58	C₃×C₇×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	—	2	1	no	—
60	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
61	#4268	—	2	2	no	—
62	#4268	D₄	1	2	yes	—
63	C₂×Dic₂₁	—	2	1	no	—
64	C₂×Dic₂₁	—	2	2	no	—
65	D₈₄	D₄	1	1	yes	characteristic
66	C₄×D₂₁	—	2	2	no	—
67	C₄×D₂₁	C₂³	1	2	yes	—
68	Dic₄₂	—	2	1	no	—
69	Dic₄₂	—	2	2	no	—
70	Dic₄₂	D₄	1	1	yes	characteristic
71	Dic₄₂	C₂³	1	2	yes	—
72	C₇×#67	—	2	1	no	—
73	C₇×D₁₂	C₂³	1	1	yes	characteristic
74	C₄×D₃×C₇	—	2	1	no	—
75	C₇×Dic₆	C₂³	1	1	yes	characteristic
76	C₃×D₂₈	—	2	1	no	—
77	C₃×D₂₈	C₂³	1	2	yes	—
78	#4246	—	2	2	no	—
79	#4245	—	2	2	no	—
80	Dic₃×D₇	—	2	2	no	—
81	C₃×C₇⋊C₈	C₂³	1	2	yes	—
82	C₇×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₇×C₃⋊C₈	C₂³	1	1	yes	characteristic
84	C₇×#41	—	3	1	no	—
85	C₇×#41	C₂×D₃	1	1	yes	characteristic, complemented
86	#1161	—	3	2	no	—
87	Q₈×D₇	—	3	2	no	—
88	#1159	—	3	2	no	—
89	#1159	—	6	2	no	—
90	D₄×D₇	—	3	2	no	—
91	D₄×D₇	—	6	2	no	—
92	#1157	—	3	1	no	—
93	#1157	—	6	1	no	—
94	#1157	—	6	1	no	—
95	C₂×D₂₈	—	6	1	no	—
96	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
97	C₂×Dic₁₄	—	3	1	no	—
98	C₂×Dic₁₄	—	6	1	no	—
99	C₇×Q₁₆	—	3	1	no	—
100	C₇×SD₁₆	—	3	1	no	—
101	C₇×SD₁₆	—	3	1	no	—
102	C₇×D₈	—	3	1	no	—
103	C₂×C₇×C₈	—	3	1	no	—
104	#1144	—	3	2	no	—
105	#1143	C₂×D₃	1	2	yes	complemented
106	#1142	—	3	2	no	—
107	#1141	C₂×D₃	1	2	yes	complemented
108	#1136	C₂×D₃	1	1	yes	characteristic, complemented
109	#1132	—	3	2	no	—
110	#1132	—	3	2	no	—
111	#1131	—	3	2	no	—
112	C₈×D₇	—	3	2	no	—
113	#1010	—	7	1	no	—
114	#1004	—	7	1	no	—
115	C₃×AGL(1,ℤ/8)	—	7	1	no	—
116	#950	D₇	1	1	yes	characteristic, complemented
117	C₂×#214	—	7	1	no	—
118	#915	—	7	2	no	—
119	D₃×SD₁₆	—	7	2	no	—
120	#912	—	7	2	no	—
121	#911	—	7	2	no	—
122	#909	—	7	1	no	—
123	#907	—	7	1	no	—
124	C₂²×C₃×C₇	—	2	1	no	—
125	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
126	C₂×D₂₁	C₂×D₄	1	2	yes	—
127	C₂×D₃×C₇	—	2	1	no	—
128	C₂×C₃×D₇	—	2	2	no	—
129	C₂×C₃×D₇	—	8	1	no	—
130	D₃×D₇	—	8	2	no	—
131	C₃×C₄×C₇	—	2	1	no	—
132	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
134	Dic₂₁	—	2	2	no	—
135	Dic₂₁	C₂×D₄	1	2	yes	—
136	C₇×Dic₃	—	2	1	no	—
137	#577	—	21	1	no	Sylow
138	C₂²×D₇	—	2	1	no	—
139	C₂²×D₇	—	6	2	no	—
140	C₇×Q₈	—	3	1	no	—
141	C₇×Q₈	C₂²×D₃	1	1	yes	characteristic
142	C₇×D₄	—	2	1	no	—
143	C₇×D₄	—	3	1	no	—
144	C₇×D₄	—	6	1	no	—
145	C₇×D₄	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₄×C₇	—	2	1	no	—
147	C₂×C₄×C₇	—	6	1	no	—
148	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
149	C₂²⋊D₇	—	3	2	no	—
150	C₂²⋊D₇	—	6	2	no	—
151	C₂²⋊D₇	—	6	2	no	—
152	C₂×Dic₇	—	6	2	no	—
153	C₂×Dic₇	—	6	1	no	—
154	D₂₈	—	2	1	no	—
155	D₂₈	—	3	1	no	—
156	D₂₈	—	6	2	no	—
157	D₂₈	C₂²×D₃	1	2	yes	—
158	C₄×D₇	—	3	2	no	—
159	C₄×D₇	—	6	2	no	—
160	C₄×D₇	—	6	2	no	—
161	Dic₁₄	—	3	1	no	—
162	Dic₁₄	—	3	2	no	—
163	Dic₁₄	—	6	2	no	—
164	Dic₁₄	—	6	1	no	—
165	C₇×C₈	—	3	1	no	—
166	C₇×C₈	—	3	1	no	—
167	C₇⋊C₈	C₂²×D₃	1	2	yes	—
168	C₃×#41	C₂×D₇	1	1	yes	characteristic, complemented
169	C₂×C₃×D₄	—	7	1	no	—
170	C₂×#67	—	14	2	no	—
171	D₃×Q₈	—	7	2	no	—
172	#237	—	7	2	no	—
173	#237	—	7	2	no	—
174	#237	—	14	1	no	—
175	#237	—	14	2	no	—
176	D₃×D₄	—	7	2	no	—
177	D₃×D₄	—	14	1	no	—
178	#235	—	7	1	no	—
179	#235	—	14	1	no	—
180	#235	C₂×D₇	1	1	yes	characteristic
181	C₂×Dic₆	—	7	1	no	—
182	C₂×Dic₆	—	14	1	no	—
183	C₃×SD₁₆	—	7	2	no	—
184	C₃×D₈	—	7	2	no	—
185	C₃×M₁₆	—	7	1	no	—
186	#216	C₂×D₇	1	1	yes	characteristic, complemented
187	#215	C₂×D₇	1	1	yes	characteristic, complemented
188	#214	—	7	2	no	—
189	#214	—	7	2	no	—
190	#214	C₂×D₇	1	1	yes	characteristic, complemented
191	#213	C₂×D₇	1	1	yes	characteristic, complemented
192	C₂×C₃⋊C₈	C₂×D₇	1	1	yes	characteristic, complemented
193	Dic₁₂	—	7	2	no	—
194	#204	—	7	2	no	—
195	ΓL(1,𝔽₂₅)	—	7	2	no	—
196	D₃×C₈	—	7	2	no	—
197	C₂×C₃×C₇	—	2	1	no	—
198	C₂×C₃×C₇	—	4	1	no	—
199	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
200	D₂₁	—	2	2	no	—
201	C₃×D₇	—	4	2	no	—
202	D₃×C₇	—	4	1	no	—
203	2₋¹⁺⁴	—	21	1	no	—
204	H(5,𝔽₂)	—	21	1	no	—
205	#137	—	21	1	no	—
206	#137	—	21	2	no	—
207	AGL(1,ℤ/8)	—	7	1	no	—
208	AGL(1,ℤ/8)	—	21	2	no	—
209	#135	—	3	1	no	—
210	#135	—	21	2	no	—
211	C₂×SD₁₆	—	21	1	no	—
212	C₂×SD₁₆	—	21	2	no	—
213	#131	—	21	1	no	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	6	1	no	—
216	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
217	C₂×D₇	—	2	2	no	—
218	C₂×D₇	—	3	2	no	—
219	C₂×D₇	—	8	1	no	—
220	C₂×D₇	—	24	2	no	—
221	C₄×C₇	—	2	1	no	—
222	C₄×C₇	—	6	1	no	—
223	C₄×C₇	D₃×D₄	1	1	yes	characteristic
224	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
225	Dic₇	—	3	2	no	—
226	Dic₇	—	6	2	no	—
227	C₂³×C₃	—	14	1	no	—
228	C₂²×D₃	—	14	2	no	—
229	C₃×Q₈	C₂²×D₇	1	1	yes	characteristic
230	C₃×D₄	—	2	1	no	—
231	C₃×D₄	—	7	2	no	—
232	C₃×D₄	—	14	1	no	—
233	C₃×D₄	C₂²×D₇	1	1	yes	characteristic
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
236	#67	—	2	1	no	—
237	#67	—	7	2	no	—
238	#67	—	14	2	no	—
239	#67	—	14	2	no	—
240	#67	—	14	2	no	—
241	C₂×Dic₃	—	14	2	no	—
242	C₂×Dic₃	—	14	2	no	—
243	C₂×Dic₃	—	14	1	no	—
244	D₁₂	—	7	1	no	—
245	D₁₂	C₂²×D₇	1	1	yes	characteristic
246	C₄×D₃	—	2	1	no	—
247	C₄×D₃	—	7	2	no	—
248	C₄×D₃	—	14	2	no	—
249	Dic₆	—	7	1	no	—
250	Dic₆	—	7	2	no	—
251	Dic₆	—	14	2	no	—
252	Dic₆	—	14	1	no	—
253	Dic₆	C₂²×D₇	1	1	yes	characteristic
254	C₃×C₈	—	7	2	no	—
255	C₃⋊C₈	C₂²×D₇	1	1	yes	characteristic
256	C₃⋊C₈	C₂²×D₇	1	1	yes	characteristic
257	C₃×C₇	#577	1	1	yes	characteristic, complemented, Hall
258	#41	—	3	1	no	—
259	#41	—	21	1	no	—
260	#41	—	21	2	no	—
261	#41	—	21	2	no	—
262	#41	—	42	1	no	—
263	#41	—	42	1	no	—
264	#41	—	42	2	no	—
265	#41	D₃×D₇	1	1	yes	characteristic, complemented
266	C₂×Q₈	—	21	2	no	—
267	C₂×Q₈	—	21	1	no	—
268	C₂×Q₈	—	42	1	no	—
269	C₂×D₄	—	7	1	no	—
270	C₂×D₄	—	21	2	no	—
271	C₂×D₄	—	42	1	no	—
272	C₂×D₄	—	42	2	no	—
273	Q₁₆	—	3	1	no	—
274	Q₁₆	—	21	2	no	—
275	SD₁₆	—	3	1	no	—
276	SD₁₆	—	3	1	no	—
277	SD₁₆	—	7	2	no	—
278	SD₁₆	—	21	2	no	—
279	SD₁₆	—	21	2	no	—
280	SD₁₆	—	21	2	no	—
281	D₈	—	3	1	no	—
282	D₈	—	7	2	no	—
283	M₁₆	—	7	1	no	—
284	M₁₆	—	21	2	no	—
285	C₂×C₈	—	3	1	no	—
286	C₂×C₈	—	21	2	no	—
287	C₂×C₇	—	2	1	no	—
288	C₂×C₇	—	4	1	no	—
289	C₂×C₇	—	12	1	no	—
290	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
291	D₇	—	4	2	no	—
292	D₇	—	6	2	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	14	2	no	—
295	C₂²×C₃	—	56	1	no	—
296	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
297	C₂×D₃	—	2	1	no	—
298	C₂×D₃	—	7	2	no	—
299	C₂×D₃	—	56	2	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	D₄×D₇	1	1	yes	characteristic
302	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
303	Dic₃	—	2	1	no	—
304	Dic₃	—	7	2	no	—
305	Dic₃	—	14	2	no	—
306	C₂³	—	14	1	no	—
307	C₂³	—	42	2	no	—
308	Q₈	—	3	1	no	—
309	Q₈	—	21	1	no	—
310	Q₈	—	21	2	no	—
311	Q₈	—	42	2	no	—
312	Q₈	—	42	1	no	—
313	Q₈	C₂×D₃×D₇	1	1	yes	characteristic
314	D₄	—	2	1	no	—
315	D₄	—	3	1	no	—
316	D₄	—	6	1	no	—
317	D₄	—	7	2	no	—
318	D₄	—	14	1	no	—
319	D₄	—	21	2	no	—
320	D₄	—	21	1	no	—
321	D₄	—	42	2	no	—
322	D₄	—	42	2	no	—
323	D₄	—	42	2	no	—
324	D₄	C₂×D₃×D₇	1	1	yes	characteristic
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	6	1	no	—
327	C₂×C₄	—	21	2	no	—
328	C₂×C₄	—	42	2	no	—
329	C₂×C₄	—	42	2	no	—
330	C₂×C₄	—	42	2	no	—
331	C₂×C₄	—	42	1	no	—
332	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
333	C₈	—	3	1	no	—
334	C₈	—	3	1	no	—
335	C₈	—	7	2	no	—
336	C₇	#5775	1	1	yes	characteristic, complemented, Sylow
337	C₂×C₃	—	2	1	no	—
338	C₂×C₃	—	4	1	no	—
339	C₂×C₃	—	28	2	no	—
340	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
341	D₃	—	4	1	no	—
342	D₃	—	14	2	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	6	1	no	—
345	C₂²	—	14	2	no	—
346	C₂²	—	21	2	no	—
347	C₂²	—	56	1	no	—
348	C₂²	—	168	2	no	—
349	C₂²	#66694	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	6	1	no	—
352	C₄	—	21	2	no	—
353	C₄	—	42	2	no	—
354	C₄	#66694	1	1	yes	characteristic
355	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
356	C₃	#90461	1	1	yes	characteristic, complemented, Sylow
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	12	1	no	—
360	C₂	—	28	2	no	—
361	C₂	—	42	2	no	—
362	C₂	C₂×#66694	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#66694
commutator, cosocle	C₃×C₄×C₇	C₂⁴
Frattini	C₄	C₂²×D₃×D₇
Fitting	C₃×C₇×#41	C₂²
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	#41	D₃×D₇
hypocenter	C₃×C₇	#577
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #10621663	C₂	semidirect
2. #10621564	C₂	semidirect
3. C₇×#950	C₂	semidirect
4. C₃×#6512	C₂	semidirect
5. #10621084	C₂	semidirect
6. #10621067	C₂	semidirect
7. #10621059	C₂	semidirect
8. #10621044	C₂	semidirect
9. #10620884	C₂	semidirect
10. C₂×#66565	C₂	semidirect
11. #10620853	C₂	semidirect
13. C₃×C₇×#41	C₂²	semidirect
14. Q₈×D₂₁	C₂²	semidirect
16. #66730	C₂²	semidirect
20. C₂×Dic₄₂	C₂²	semidirect
28. C₇×#216	C₂²	semidirect
29. C₇×#215	C₂²	semidirect
30. C₇×#214	C₂²	semidirect
31. C₇×#213	C₂²	semidirect
32. C₂×C₇×C₃⋊C₈	C₂²	semidirect
33. C₃×#1143	C₂²	semidirect
34. C₃×#1141	C₂²	semidirect
35. C₃×#1136	C₂²	semidirect
36. #66571	C₂²	semidirect
37. #66567	C₂²	semidirect
38. #66565	C₂²	semidirect
39. #66565	C₂²	semidirect
40. #66559	C₂²	semidirect
41. D₂₁⋊C₈	C₂²	semidirect
46. #6512	D₃	semidirect
85. C₇×#41	C₂×D₃	semidirect
105. #1143	C₂×D₃	semidirect
107. #1141	C₂×D₃	semidirect
108. #1136	C₂×D₃	semidirect
116. #950	D₇	semidirect
168. C₃×#41	C₂×D₇	semidirect
186. #216	C₂×D₇	semidirect
187. #215	C₂×D₇	semidirect
190. #214	C₂×D₇	semidirect
191. #213	C₂×D₇	semidirect
192. C₂×C₃⋊C₈	C₂×D₇	semidirect
257. C₃×C₇	#577	semidirect
265. #41	D₃×D₇	semidirect
336. C₇	#5775	semidirect
356. C₃	#90461	semidirect

Maximal subgroups

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

subgroup	index
#10621663	2
#10621564	2
C₇×#950	2
C₃×#6512	2
#10621084	2
#10621067	2
#10621059	2
#10621044	2
#10620884	2
C₂×#66565	2
#10620853	2
#90461	3
#5775	7