Subgroups — #13059442

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×C₇×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620884	C₂	1	2	yes	maximal, complemented
5	#10620883	C₂	1	2	yes	maximal, complemented
6	#10620867	C₂	1	2	yes	maximal, complemented
7	#10620866	C₂	1	2	yes	maximal, complemented
8	#10620865	C₂	1	4	yes	maximal, complemented
9	#90513	—	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×C₇×#67	—	2	1	no	—
20	C₇×#235	—	2	1	no	—
21	C₇×#235	—	2	1	no	—
22	C₇×#235	C₂²	1	2	yes	—
23	C₇×#235	C₂²	1	2	yes	complemented
24	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₄×D₃×C₇	—	2	1	no	—
26	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic, complemented
27	C₃×#1136	C₂²	1	2	yes	complemented
28	C₃×#1136	C₂²	1	2	yes	complemented
29	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	complemented
30	#66571	C₂²	1	4	yes	complemented
31	#66568	C₂²	1	4	yes	complemented
32	#66567	C₂²	1	4	yes	complemented
33	#66562	C₂²	1	4	yes	complemented
34	C₂×C₇×#41	—	3	1	no	—
35	C₂×#1157	—	3	1	no	—
36	#6514	—	3	2	no	—
37	#6513	—	3	4	no	—
38	#6512	—	3	2	no	—
39	#6505	—	3	2	no	—
40	#6495	—	3	2	no	—
41	C₂×#1136	D₃	1	1	yes	characteristic, complemented
42	#5748	—	7	1	no	maximal, Hall
43	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
44	C₂²×D₂₁	—	2	1	no	—
45	C₂²×D₃×C₇	—	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	C₇×#67	—	2	2	no	—
62	C₇×#67	—	2	2	no	—
63	C₂×C₇×Dic₃	—	2	1	no	—
64	C₇×D₁₂	—	2	1	no	—
65	C₇×D₁₂	C₂³	1	2	yes	—
66	C₄×D₃×C₇	—	2	2	no	—
67	C₄×D₃×C₇	—	2	2	no	—
68	C₇×Dic₆	—	2	1	no	—
69	C₇×Dic₆	C₂³	1	2	yes	—
70	C₃×C₇⋊C₈	C₂³	1	4	yes	—
71	C₇×#41	—	3	2	no	—
72	C₇×#41	—	3	2	no	—
73	C₇×#41	—	6	1	no	—
74	C₇×#41	—	6	1	no	—
75	C₂×C₇×Q₈	—	3	1	no	—
76	C₂×C₇×D₄	—	3	1	no	—
77	C₂×C₇×D₄	—	6	1	no	—
78	C₂²×C₄×C₇	—	6	1	no	—
79	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
80	C₂×C₂²⋊D₇	—	6	1	no	—
81	#1157	—	3	2	no	—
82	#1157	—	3	2	no	—
83	#1157	—	6	1	no	—
84	#1157	—	6	1	no	—
85	C₂×D₂₈	—	3	1	no	—
86	C₂×C₄×D₇	—	6	1	no	—
87	C₂×Dic₁₄	—	3	1	no	—
88	#1144	—	3	4	no	—
89	#1143	—	3	4	no	—
90	#1142	—	3	4	no	—
91	#1141	—	3	4	no	—
92	#1136	C₂×D₃	1	2	yes	complemented
93	#1136	C₂×D₃	1	2	yes	complemented
94	C₂×C₇⋊C₈	C₂×D₃	1	2	yes	complemented
95	C₂×#235	—	7	1	no	—
96	C₂×#235	D₇	1	1	yes	characteristic, complemented
97	C₂×C₃×M₁₆	—	7	1	no	—
98	#909	—	7	2	no	—
99	#909	—	7	2	no	—
100	#908	—	7	2	no	—
101	#908	—	7	2	no	—
102	#904	—	7	4	no	—
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×D₃×C₇	—	2	2	no	—
110	C₂×D₃×C₇	—	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	C₇×Dic₃	—	2	2	no	—
117	#575	—	21	1	no	Sylow
118	C₂³×C₇	—	6	1	no	—
119	C₂³×C₇	D₁₂	1	1	yes	characteristic
120	C₂²×D₇	—	6	1	no	—
121	C₇×Q₈	—	3	2	no	—
122	C₇×Q₈	—	6	1	no	—
123	C₇×D₄	—	3	2	no	—
124	C₇×D₄	—	6	2	no	—
125	C₇×D₄	—	6	2	no	—
126	C₇×D₄	—	6	1	no	—
127	C₂×C₄×C₇	—	6	1	no	—
128	C₂×C₄×C₇	—	6	2	no	—
129	C₂×C₄×C₇	—	6	2	no	—
130	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
131	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
132	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
136	C₂²⋊D₇	—	6	2	no	—
137	C₂²⋊D₇	—	6	2	no	—
138	C₂×Dic₇	—	6	1	no	—
139	D₂₈	—	3	2	no	—
140	D₂₈	—	6	1	no	—
141	C₄×D₇	—	6	2	no	—
142	C₄×D₇	—	6	2	no	—
143	Dic₁₄	—	3	2	no	—
144	Dic₁₄	—	6	1	no	—
145	C₇⋊C₈	C₂²×D₃	1	4	yes	—
146	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
147	C₂×#67	—	2	1	no	—
148	C₂×#67	—	14	1	no	—
149	#235	—	2	1	no	—
150	#235	—	2	1	no	—
151	#235	—	7	2	no	—
152	#235	—	7	2	no	—
153	#235	—	14	1	no	—
154	#235	—	14	1	no	—
155	#235	C₂×D₇	1	2	yes	—
156	#235	C₂×D₇	1	2	yes	complemented
157	C₂×D₁₂	—	7	1	no	—
158	C₂×D₁₂	C₂×D₇	1	1	yes	characteristic, complemented
159	C₂×C₄×D₃	—	2	1	no	—
160	C₂×C₄×D₃	—	14	1	no	—
161	C₂×Dic₆	—	7	1	no	—
162	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic, complemented
163	C₃×M₁₆	—	7	2	no	—
164	C₃×M₁₆	—	7	2	no	—
165	C₂×C₃×C₈	—	7	2	no	—
166	Dic₁₂	—	7	4	no	—
167	D₂₄	—	7	4	no	—
168	#204	—	7	4	no	—
169	#204	—	7	4	no	—
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	D₃×C₇	—	4	2	no	—
176	C₂×#41	—	3	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	—	7	1	no	—
184	C₂²×C₇	—	4	1	no	—
185	C₂²×C₇	—	6	2	no	—
186	C₂²×C₇	—	12	2	no	—
187	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
188	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
189	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
190	C₂×D₇	—	6	2	no	—
191	C₂×D₇	—	12	2	no	—
192	C₄×C₇	—	6	2	no	—
193	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
194	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
195	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
196	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
197	Dic₇	—	6	2	no	—
198	C₂³×C₃	C₂²⋊D₇	1	1	yes	characteristic
199	C₂²×D₃	—	2	1	no	—
200	C₂²×D₃	—	14	1	no	—
201	C₂×C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
207	#67	—	2	2	no	—
208	#67	—	2	2	no	—
209	#67	—	14	2	no	—
210	#67	—	14	2	no	—
211	C₂×Dic₃	—	2	1	no	—
212	C₂×Dic₃	—	14	1	no	—
213	D₁₂	—	2	1	no	—
214	D₁₂	—	7	2	no	—
215	D₁₂	—	14	1	no	—
216	D₁₂	C₂²×D₇	1	2	yes	—
217	C₄×D₃	—	2	2	no	—
218	C₄×D₃	—	2	2	no	—
219	C₄×D₃	—	14	2	no	—
220	C₄×D₃	—	14	2	no	—
221	Dic₆	—	2	1	no	—
222	Dic₆	—	7	2	no	—
223	Dic₆	—	14	1	no	—
224	Dic₆	C₂²×D₇	1	2	yes	—
225	C₃×C₈	—	7	4	no	—
226	C₃×C₇	#575	1	1	yes	characteristic, complemented, Hall
227	#41	—	3	2	no	—
228	#41	—	3	2	no	—
229	#41	—	6	1	no	—
230	#41	—	6	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	—	3	1	no	—
236	C₂×Q₈	—	21	1	no	—
237	C₂×D₄	—	3	1	no	—
238	C₂×D₄	—	6	1	no	—
239	C₂×D₄	—	21	1	no	—
240	C₂×D₄	—	42	1	no	—
241	C₂²×C₄	—	6	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	—	7	2	no	—
249	M₁₆	—	7	2	no	—
250	C₂×C₈	—	7	2	no	—
251	C₂×C₇	—	2	1	no	—
252	C₂×C₇	—	2	1	no	—
253	C₂×C₇	—	2	1	no	—
254	C₂×C₇	—	12	2	no	—
255	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
256	D₇	—	12	2	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
259	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
260	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
261	C₂×D₃	—	2	2	no	—
262	C₂×D₃	—	4	2	no	—
263	C₂×D₃	—	14	2	no	—
264	C₂×D₃	—	28	2	no	—
265	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
266	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
267	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
268	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
269	Dic₃	—	2	2	no	—
270	Dic₃	—	14	2	no	—
271	C₂³	—	6	1	no	—
272	C₂³	—	42	1	no	—
273	C₂³	#4247	1	1	yes	characteristic
274	Q₈	—	3	2	no	—
275	Q₈	—	6	1	no	—
276	Q₈	—	21	2	no	—
277	Q₈	—	42	1	no	—
278	D₄	—	3	2	no	—
279	D₄	—	6	2	no	—
280	D₄	—	6	2	no	—
281	D₄	—	6	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	—	6	1	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	42	1	no	—
290	C₂×C₄	—	42	2	no	—
291	C₂×C₄	—	42	2	no	—
292	C₂×C₄	#4247	1	1	yes	characteristic
293	C₂×C₄	#4247	1	1	yes	characteristic
294	C₂×C₄	#4247	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	—	7	4	no	—
299	C₇	#5748	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	C₂²×C₂²⋊D₇	1	1	yes	characteristic
304	D₃	—	4	2	no	—
305	D₃	—	28	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	6	2	no	—
308	C₂²	—	12	2	no	—
309	C₂²	—	42	2	no	—
310	C₂²	—	84	2	no	—
311	C₂²	C₂×#4247	1	1	yes	characteristic
312	C₂²	C₂×#4247	1	1	yes	characteristic
313	C₂²	C₂×#4247	1	1	yes	characteristic
314	C₄	—	6	2	no	—
315	C₄	—	42	2	no	—
316	C₄	C₂×#4247	1	1	yes	characteristic
317	C₄	C₂×#4247	1	1	yes	characteristic
318	C₄	C₂×#4247	1	1	yes	characteristic
319	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
320	C₃	#90513	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	12	2	no	—
325	C₂	—	84	2	no	—
326	C₂	C₂²×#4247	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4247
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×C₇×#235	C₂	semidirect
3. C₂×C₃×#1136	C₂	semidirect
4. #10620884	C₂	semidirect
5. #10620883	C₂	semidirect
6. #10620867	C₂	semidirect
7. #10620866	C₂	semidirect
8. #10620865	C₂	semidirect
15. #66728	C₂²	semidirect
16. C₂×D₈₄	C₂²	semidirect
18. C₂×Dic₄₂	C₂²	semidirect
23. C₇×#235	C₂²	semidirect
24. C₂×C₇×D₁₂	C₂²	semidirect
26. C₂×C₇×Dic₆	C₂²	semidirect
27. C₃×#1136	C₂²	semidirect
28. C₃×#1136	C₂²	semidirect
29. C₂×C₃×C₇⋊C₈	C₂²	semidirect
30. #66571	C₂²	semidirect
31. #66568	C₂²	semidirect
32. #66567	C₂²	semidirect
33. #66562	C₂²	semidirect
41. C₂×#1136	D₃	semidirect
92. #1136	C₂×D₃	semidirect
93. #1136	C₂×D₃	semidirect
94. C₂×C₇⋊C₈	C₂×D₃	semidirect
96. C₂×#235	D₇	semidirect
156. #235	C₂×D₇	semidirect
158. C₂×D₁₂	C₂×D₇	semidirect
162. C₂×Dic₆	C₂×D₇	semidirect
226. C₃×C₇	#575	semidirect
299. C₇	#5748	semidirect
320. C₃	#90513	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×C₇×#235	2
C₂×C₃×#1136	2
#10620884	2
#10620883	2
#10620867	2
#10620866	2
#10620865	2
#90513	3
#5748	7