Subgroups — #13059451

Order 1344.

524 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66728	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620877	C₂	1	2	yes	maximal, complemented
5	#10620875	C₂	1	2	yes	maximal, complemented
6	#10620874	C₂	1	2	yes	maximal, complemented
7	#10620873	C₂	1	4	yes	maximal, complemented
8	#10620863	C₂	1	2	yes	maximal, complemented
9	#90439	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×#4268	—	2	1	no	—
12	#66728	—	2	1	no	—
13	#66728	—	2	1	no	—
14	#66728	C₂²	1	2	yes	—
15	#66728	C₂²	1	2	yes	complemented
16	C₂×D₈₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×D₂₁	—	2	1	no	—
18	C₂×Dic₄₂	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃×C₂²⋊D₇	—	2	1	no	—
20	C₃×#1157	—	2	1	no	—
21	C₃×#1157	—	2	1	no	—
22	C₃×#1157	C₂²	1	2	yes	—
23	C₃×#1157	C₂²	1	2	yes	complemented
24	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃×C₄×D₇	—	2	1	no	—
26	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
27	C₇×#208	C₂²	1	2	yes	complemented
28	C₇×#208	C₂²	1	2	yes	complemented
29	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
30	#66570	C₂²	1	4	yes	complemented
31	#66566	C₂²	1	4	yes	complemented
32	#66565	C₂²	1	4	yes	complemented
33	#66561	C₂²	1	4	yes	complemented
34	C₂×#1157	—	3	1	no	—
35	C₂×#1157	D₃	1	1	yes	characteristic, complemented
36	C₂×C₇×M₁₆	—	3	1	no	—
37	#6472	—	3	2	no	—
38	#6472	—	3	2	no	—
39	#6471	—	3	2	no	—
40	#6471	—	3	2	no	—
41	#6467	—	3	4	no	—
42	#5822	—	7	1	no	maximal, Hall
43	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
44	C₂²×D₂₁	—	2	1	no	—
45	C₂²×C₃×D₇	—	2	1	no	—
46	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
48	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	#4268	—	2	2	no	—
53	#4268	—	2	2	no	—
54	C₂×Dic₂₁	—	2	1	no	—
55	D₈₄	—	2	1	no	—
56	D₈₄	C₂³	1	2	yes	—
57	C₄×D₂₁	—	2	2	no	—
58	C₄×D₂₁	—	2	2	no	—
59	Dic₄₂	—	2	1	no	—
60	Dic₄₂	C₂³	1	2	yes	—
61	C₃×C₂²⋊D₇	—	2	2	no	—
62	C₃×C₂²⋊D₇	—	2	2	no	—
63	C₂×C₃×Dic₇	—	2	1	no	—
64	C₃×D₂₈	—	2	1	no	—
65	C₃×D₂₈	C₂³	1	2	yes	—
66	C₃×C₄×D₇	—	2	2	no	—
67	C₃×C₄×D₇	—	2	2	no	—
68	C₃×Dic₁₄	—	2	1	no	—
69	C₃×Dic₁₄	C₂³	1	2	yes	—
70	C₇×C₃⋊C₈	C₂³	1	4	yes	—
71	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
72	C₂×C₂²⋊D₇	—	2	1	no	—
73	C₂×C₂²⋊D₇	—	6	1	no	—
74	#1157	—	2	1	no	—
75	#1157	—	2	1	no	—
76	#1157	—	3	2	no	—
77	#1157	—	3	2	no	—
78	#1157	—	6	1	no	—
79	#1157	—	6	1	no	—
80	#1157	C₂×D₃	1	2	yes	—
81	#1157	C₂×D₃	1	2	yes	complemented
82	C₂×D₂₈	—	3	1	no	—
83	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic, complemented
84	C₂×C₄×D₇	—	2	1	no	—
85	C₂×C₄×D₇	—	6	1	no	—
86	C₂×Dic₁₄	—	3	1	no	—
87	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic, complemented
88	C₇×M₁₆	—	3	2	no	—
89	C₇×M₁₆	—	3	2	no	—
90	C₂×C₇×C₈	—	3	2	no	—
91	Dic₂₈	—	3	4	no	—
92	D₅₆	—	3	4	no	—
93	#1132	—	3	4	no	—
94	#1132	—	3	4	no	—
95	C₂×C₃×#41	—	7	1	no	—
96	C₂×#235	—	7	1	no	—
97	#951	—	7	2	no	—
98	#950	—	7	4	no	—
99	#949	—	7	2	no	—
100	#942	—	7	2	no	—
101	#932	—	7	2	no	—
102	C₂×#208	D₇	1	1	yes	characteristic, complemented
103	C₂²×C₃×C₇	—	4	1	no	—
104	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
107	C₂×D₂₁	—	2	2	no	—
108	C₂×D₂₁	—	4	2	no	—
109	C₂×C₃×D₇	—	2	2	no	—
110	C₂×C₃×D₇	—	4	2	no	—
111	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
113	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
114	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
115	Dic₂₁	—	2	2	no	—
116	C₃×Dic₇	—	2	2	no	—
117	#575	—	21	1	no	Sylow
118	C₂³×C₇	#67	1	1	yes	characteristic
119	C₂²×D₇	—	2	1	no	—
120	C₂²×D₇	—	6	1	no	—
121	C₂×C₄×C₇	#67	1	1	yes	characteristic
122	C₂×C₄×C₇	#67	1	1	yes	characteristic
123	C₂×C₄×C₇	#67	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
127	C₂²⋊D₇	—	2	2	no	—
128	C₂²⋊D₇	—	2	2	no	—
129	C₂²⋊D₇	—	6	2	no	—
130	C₂²⋊D₇	—	6	2	no	—
131	C₂×Dic₇	—	2	1	no	—
132	C₂×Dic₇	—	6	1	no	—
133	D₂₈	—	2	1	no	—
134	D₂₈	—	3	2	no	—
135	D₂₈	—	6	1	no	—
136	D₂₈	C₂²×D₃	1	2	yes	—
137	C₄×D₇	—	2	2	no	—
138	C₄×D₇	—	2	2	no	—
139	C₄×D₇	—	6	2	no	—
140	C₄×D₇	—	6	2	no	—
141	Dic₁₄	—	2	1	no	—
142	Dic₁₄	—	3	2	no	—
143	Dic₁₄	—	6	1	no	—
144	Dic₁₄	C₂²×D₃	1	2	yes	—
145	C₇×C₈	—	3	4	no	—
146	C₃×#41	—	7	2	no	—
147	C₃×#41	—	7	2	no	—
148	C₃×#41	—	14	1	no	—
149	C₃×#41	—	14	1	no	—
150	C₂×C₃×Q₈	—	7	1	no	—
151	C₂×C₃×D₄	—	7	1	no	—
152	C₂×C₃×D₄	—	14	1	no	—
153	C₂²×C₃×C₄	—	14	1	no	—
154	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
155	C₂×#67	—	14	1	no	—
156	#235	—	7	2	no	—
157	#235	—	7	2	no	—
158	#235	—	14	1	no	—
159	#235	—	14	1	no	—
160	C₂×D₁₂	—	7	1	no	—
161	C₂×C₄×D₃	—	14	1	no	—
162	C₂×Dic₆	—	7	1	no	—
163	#216	—	7	4	no	—
164	#215	—	7	4	no	—
165	#214	—	7	4	no	—
166	#213	—	7	4	no	—
167	#208	C₂×D₇	1	2	yes	complemented
168	#208	C₂×D₇	1	2	yes	complemented
169	C₂×C₃⋊C₈	C₂×D₇	1	2	yes	complemented
170	C₂×C₃×C₇	—	2	1	no	—
171	C₂×C₃×C₇	—	2	1	no	—
172	C₂×C₃×C₇	—	2	1	no	—
173	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
174	D₂₁	—	4	2	no	—
175	C₃×D₇	—	4	2	no	—
176	C₂×#41	—	7	1	no	—
177	C₂×#41	—	21	1	no	—
178	#137	—	21	2	no	—
179	#137	—	21	2	no	—
180	AGL(1,ℤ/8)	—	21	2	no	—
181	AGL(1,ℤ/8)	—	21	2	no	—
182	#135	—	21	4	no	—
183	C₂×M₁₆	—	3	1	no	—
184	C₂²×C₇	—	4	1	no	—
185	C₂²×C₇	C₂×#67	1	1	yes	characteristic
186	C₂²×C₇	C₂×#67	1	1	yes	characteristic
187	C₂²×C₇	C₂×#67	1	1	yes	characteristic
188	C₂×D₇	—	2	2	no	—
189	C₂×D₇	—	4	2	no	—
190	C₂×D₇	—	6	2	no	—
191	C₂×D₇	—	12	2	no	—
192	C₄×C₇	C₂×#67	1	1	yes	characteristic
193	C₄×C₇	C₂×#67	1	1	yes	characteristic
194	C₄×C₇	C₂×#67	1	1	yes	characteristic
195	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
196	Dic₇	—	2	2	no	—
197	Dic₇	—	6	2	no	—
198	C₂³×C₃	—	14	1	no	—
199	C₂³×C₃	D₂₈	1	1	yes	characteristic
200	C₂²×D₃	—	14	1	no	—
201	C₃×Q₈	—	7	2	no	—
202	C₃×Q₈	—	14	1	no	—
203	C₃×D₄	—	7	2	no	—
204	C₃×D₄	—	14	2	no	—
205	C₃×D₄	—	14	2	no	—
206	C₃×D₄	—	14	1	no	—
207	C₂×C₃×C₄	—	14	1	no	—
208	C₂×C₃×C₄	—	14	2	no	—
209	C₂×C₃×C₄	—	14	2	no	—
210	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
211	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
212	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
216	#67	—	14	2	no	—
217	#67	—	14	2	no	—
218	C₂×Dic₃	—	14	1	no	—
219	D₁₂	—	7	2	no	—
220	D₁₂	—	14	1	no	—
221	C₄×D₃	—	14	2	no	—
222	C₄×D₃	—	14	2	no	—
223	Dic₆	—	7	2	no	—
224	Dic₆	—	14	1	no	—
225	C₃⋊C₈	C₂²×D₇	1	4	yes	—
226	C₃×C₇	#575	1	1	yes	characteristic, complemented, Hall
227	#41	—	7	2	no	—
228	#41	—	7	2	no	—
229	#41	—	14	1	no	—
230	#41	—	14	1	no	—
231	#41	—	21	2	no	—
232	#41	—	21	2	no	—
233	#41	—	42	1	no	—
234	#41	—	42	1	no	—
235	C₂×Q₈	—	7	1	no	—
236	C₂×Q₈	—	21	1	no	—
237	C₂×D₄	—	7	1	no	—
238	C₂×D₄	—	14	1	no	—
239	C₂×D₄	—	21	1	no	—
240	C₂×D₄	—	42	1	no	—
241	C₂²×C₄	—	14	1	no	—
242	C₂²×C₄	—	42	1	no	—
243	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
244	Q₁₆	—	21	4	no	—
245	SD₁₆	—	21	4	no	—
246	SD₁₆	—	21	4	no	—
247	D₈	—	21	4	no	—
248	M₁₆	—	3	2	no	—
249	M₁₆	—	3	2	no	—
250	C₂×C₈	—	3	2	no	—
251	C₂×C₇	—	2	1	no	—
252	C₂×C₇	—	2	1	no	—
253	C₂×C₇	—	2	1	no	—
254	C₂×C₇	C₂²×#67	1	1	yes	characteristic
255	D₇	—	4	2	no	—
256	D₇	—	12	2	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	—	14	2	no	—
259	C₂²×C₃	—	28	2	no	—
260	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
261	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
262	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
263	C₂×D₃	—	14	2	no	—
264	C₂×D₃	—	28	2	no	—
265	C₃×C₄	—	14	2	no	—
266	C₃×C₄	C₂×D₂₈	1	1	yes	characteristic
267	C₃×C₄	C₂×D₂₈	1	1	yes	characteristic
268	C₃×C₄	C₂×D₂₈	1	1	yes	characteristic
269	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
270	Dic₃	—	14	2	no	—
271	C₂³	—	14	1	no	—
272	C₂³	—	42	1	no	—
273	C₂³	#4246	1	1	yes	characteristic
274	Q₈	—	7	2	no	—
275	Q₈	—	14	1	no	—
276	Q₈	—	21	2	no	—
277	Q₈	—	42	1	no	—
278	D₄	—	7	2	no	—
279	D₄	—	14	2	no	—
280	D₄	—	14	2	no	—
281	D₄	—	14	1	no	—
282	D₄	—	21	2	no	—
283	D₄	—	42	2	no	—
284	D₄	—	42	2	no	—
285	D₄	—	42	1	no	—
286	C₂×C₄	—	14	1	no	—
287	C₂×C₄	—	14	2	no	—
288	C₂×C₄	—	14	2	no	—
289	C₂×C₄	—	42	1	no	—
290	C₂×C₄	—	42	2	no	—
291	C₂×C₄	—	42	2	no	—
292	C₂×C₄	#4246	1	1	yes	characteristic
293	C₂×C₄	#4246	1	1	yes	characteristic
294	C₂×C₄	#4246	1	1	yes	characteristic
295	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
296	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
297	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
298	C₈	—	3	4	no	—
299	C₇	#5822	1	1	yes	characteristic, complemented, Sylow
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	28	2	no	—
304	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
305	D₃	—	28	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	14	2	no	—
308	C₂²	—	28	2	no	—
309	C₂²	—	42	2	no	—
310	C₂²	—	84	2	no	—
311	C₂²	C₂×#4246	1	1	yes	characteristic
312	C₂²	C₂×#4246	1	1	yes	characteristic
313	C₂²	C₂×#4246	1	1	yes	characteristic
314	C₄	—	14	2	no	—
315	C₄	—	42	2	no	—
316	C₄	C₂×#4246	1	1	yes	characteristic
317	C₄	C₂×#4246	1	1	yes	characteristic
318	C₄	C₂×#4246	1	1	yes	characteristic
319	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
320	C₃	#90439	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	28	2	no	—
325	C₂	—	84	2	no	—
326	C₂	C₂²×#4246	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4246
commutator, cosocle	C₃×C₄×C₇	C₂⁴
Frattini	C₄	C₂²×D₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#575
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#66728	C₂	semidirect
2. C₂×C₃×#1157	C₂	semidirect
3. C₂×C₇×#208	C₂	semidirect
4. #10620877	C₂	semidirect
5. #10620875	C₂	semidirect
6. #10620874	C₂	semidirect
7. #10620873	C₂	semidirect
8. #10620863	C₂	semidirect
15. #66728	C₂²	semidirect
16. C₂×D₈₄	C₂²	semidirect
18. C₂×Dic₄₂	C₂²	semidirect
23. C₃×#1157	C₂²	semidirect
24. C₂×C₃×D₂₈	C₂²	semidirect
26. C₂×C₃×Dic₁₄	C₂²	semidirect
27. C₇×#208	C₂²	semidirect
28. C₇×#208	C₂²	semidirect
29. C₂×C₇×C₃⋊C₈	C₂²	semidirect
30. #66570	C₂²	semidirect
31. #66566	C₂²	semidirect
32. #66565	C₂²	semidirect
33. #66561	C₂²	semidirect
35. C₂×#1157	D₃	semidirect
81. #1157	C₂×D₃	semidirect
83. C₂×D₂₈	C₂×D₃	semidirect
87. C₂×Dic₁₄	C₂×D₃	semidirect
102. C₂×#208	D₇	semidirect
167. #208	C₂×D₇	semidirect
168. #208	C₂×D₇	semidirect
169. C₂×C₃⋊C₈	C₂×D₇	semidirect
226. C₃×C₇	#575	semidirect
299. C₇	#5822	semidirect
320. C₃	#90439	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66728	2
C₂×C₃×#1157	2
C₂×C₇×#208	2
#10620877	2
#10620875	2
#10620874	2
#10620873	2
#10620863	2
#90439	3
#5822	7