Subgroups — C₂×D₇×#347 #13506448

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×M₃₂	C₂	1	2	yes	maximal
2	C₂×D₇×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89307	C₂	1	2	yes	maximal, complemented
5	C₂×#89261	C₂	1	1	yes	maximal, characteristic, complemented
6	D₇×#347	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₈×D₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₇	C₄	1	1	yes	characteristic
10	C₂×C₇×M₃₂	C₂²	1	2	yes	—
11	C₂×C₇×#97	C₂²	1	1	yes	characteristic
12	C₂×#6389	C₄	1	1	yes	characteristic
13	C₂×#6389	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	2	yes	—
15	C₂×#6377	C₄	1	1	yes	characteristic
16	D₇×M₃₂	C₄	1	8	yes	complemented
17	D₇×M₃₂	C₂²	1	8	yes	—
18	C₂×#6372	C₄	1	4	yes	complemented
19	C₂×D₇×C₁₆	C₄	1	4	yes	complemented
20	D₇×#97	C₄	1	4	yes	—
21	D₇×#97	C₂²	1	4	yes	—
22	C₇×#347	C₂²	1	4	yes	complemented
23	#89307	C₂²	1	8	yes	complemented
24	#89261	C₂²	1	4	yes	complemented
25	C₂²×#347	—	7	1	no	maximal, Sylow
26	C₂²×C₄×D₇	—	2	1	no	—
27	C₂²×C₄×D₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇	—	2	1	no	—
32	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇	—	2	1	no	—
36	C₂×C₈×D₇	—	2	2	no	—
37	C₂×C₈×D₇	—	2	1	no	—
38	C₂×C₈×D₇	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₇	C₂³	1	2	yes	—
44	C₄²×D₇	—	2	2	no	—
45	C₄²×D₇	C₂×C₄	1	4	yes	—
46	C₇×M₃₂	C₂×C₄	1	4	yes	complemented
47	C₇×M₃₂	C₂³	1	4	yes	—
48	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	complemented
49	C₇×#97	C₂×C₄	1	2	yes	—
50	C₇×#97	C₂³	1	2	yes	—
51	#6389	C₂×C₄	1	4	yes	—
52	#6389	C₂×C₄	1	2	yes	—
53	#6389	C₂³	1	2	yes	—
54	#6386	C₂×C₄	1	4	yes	complemented
55	#6386	C₂³	1	4	yes	—
56	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
57	#6377	C₂×C₄	1	4	yes	—
58	#6372	C₂×C₄	1	16	yes	complemented
59	D₇×C₁₆	C₂×C₄	1	16	yes	complemented
60	C₂²×M₃₂	—	7	2	no	—
61	C₂²×#97	—	7	1	no	—
62	C₂×#347	—	7	2	no	—
63	C₂×#347	—	7	1	no	—
64	C₂×#347	—	7	8	no	—
65	C₂×#347	D₇	1	1	yes	characteristic, direct product factor
66	C₂³×D₇	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₇	—	2	1	no	—
68	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₇	—	2	1	no	—
70	C₂²×Dic₇	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₇	—	2	4	no	—
72	C₂×C₄×D₇	—	4	4	no	—
73	C₂×C₄×D₇	C₄²	1	2	yes	—
74	C₂×C₄×D₇	C₄²	1	1	yes	characteristic
75	C₂×C₄×D₇	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₇	M₁₆	1	2	yes	—
77	C₂×C₄×D₇	M₁₆	1	1	yes	characteristic
78	C₂×C₄×D₇	M₁₆	1	2	yes	—
79	C₂×C₄×D₇	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
81	C₂×C₇×C₈	—	2	1	no	—
82	C₂×C₇×C₈	—	2	1	no	—
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
88	C₄²×C₇	—	2	1	no	—
89	C₄²×C₇	C₂²×C₄	1	2	yes	—
90	C₄×Dic₇	—	2	1	no	—
91	C₄×Dic₇	—	2	4	no	—
92	C₄×Dic₇	C₂²×C₄	1	2	yes	—
93	C₂×C₇⋊C₈	—	2	1	no	—
94	C₂×C₇⋊C₈	—	2	1	no	—
95	C₂×C₇⋊C₈	C₄²	1	2	yes	—
96	C₂×C₇⋊C₈	C₄²	1	2	yes	—
97	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
99	C₈×D₇	—	2	4	no	—
100	C₈×D₇	—	2	4	no	—
101	C₈×D₇	C₄²	1	4	yes	—
102	C₈×D₇	C₄²	1	4	yes	—
103	C₈×D₇	C₂²×C₄	1	4	yes	—
104	C₈×D₇	C₂²×C₄	1	4	yes	—
105	C₇×C₁₆	C₂²×C₄	1	8	yes	complemented
106	C₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
107	C₂³×C₈	—	7	1	no	—
108	C₂³×C₈	—	7	1	no	—
109	C₂²×C₄²	—	7	1	no	—
110	C₂×M₃₂	—	7	2	no	—
111	C₂×M₃₂	—	7	8	no	—
112	C₂×M₃₂	—	7	8	no	—
113	C₂×M₃₂	—	7	4	no	—
114	C₂×M₃₂	C₂×D₇	1	2	yes	—
115	C₂²×C₁₆	—	7	4	no	—
116	C₂×#97	—	7	1	no	—
117	C₂×#97	—	7	1	no	—
118	C₂×#97	—	7	1	no	—
119	C₂×#97	—	7	4	no	—
120	C₂×#97	—	7	4	no	—
121	C₂×#97	C₂×D₇	1	1	yes	characteristic
122	#347	—	7	8	no	—
123	#347	—	7	4	no	—
124	#347	C₂×D₇	1	4	yes	direct product factor
125	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₇	—	2	4	no	—
127	C₂²×D₇	#97	1	1	yes	characteristic
128	C₂²×D₇	#97	1	2	yes	—
129	C₂²×D₇	#97	1	1	yes	characteristic
130	C₂²×D₇	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₇	—	2	2	no	—
140	C₂×Dic₇	—	4	2	no	—
141	C₂×Dic₇	#97	1	2	yes	—
142	C₂×Dic₇	#97	1	1	yes	characteristic
143	C₂×Dic₇	#97	1	1	yes	characteristic
144	C₂×Dic₇	C₂×M₁₆	1	2	yes	—
145	C₄×D₇	—	4	8	no	—
146	C₄×D₇	#97	1	2	yes	—
147	C₄×D₇	#97	1	2	yes	—
148	C₄×D₇	#97	1	2	yes	—
149	C₄×D₇	#97	1	2	yes	—
150	C₄×D₇	C₂×C₄²	1	2	yes	—
151	C₄×D₇	C₂×C₄²	1	2	yes	—
152	C₄×D₇	C₂×M₁₆	1	2	yes	—
153	C₄×D₇	C₂×M₁₆	1	2	yes	—
154	C₇×C₈	—	2	2	no	—
155	C₇×C₈	C₂×C₄²	1	2	yes	—
156	C₇×C₈	C₂³×C₄	1	2	yes	—
157	C₇⋊C₈	—	2	2	no	—
158	C₇⋊C₈	C₂×C₄²	1	4	yes	—
159	C₂³×C₄	—	7	1	no	—
160	C₂³×C₄	—	14	1	no	—
161	C₂²×C₈	—	7	1	no	—
162	C₂²×C₈	—	7	2	no	—
163	C₂²×C₈	—	7	2	no	—
164	C₂²×C₈	—	7	4	no	—
165	C₂²×C₈	—	7	2	no	—
166	C₂²×C₈	—	7	2	no	—
167	C₂²×C₈	—	7	1	no	—
168	C₂²×C₈	—	7	4	no	—
169	C₂²×C₈	—	14	1	no	—
170	C₂²×C₈	—	14	1	no	—
171	C₂²×C₈	—	14	2	no	—
172	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
174	C₂×C₄²	—	7	1	no	—
175	C₂×C₄²	—	7	4	no	—
176	C₂×C₄²	—	14	2	no	—
177	C₂×C₄²	—	14	1	no	—
178	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
179	M₃₂	—	7	4	no	—
180	M₃₂	—	7	4	no	—
181	M₃₂	—	7	16	no	—
182	M₃₂	C₄×D₇	1	4	yes	complemented
183	M₃₂	C₂²×D₇	1	4	yes	—
184	C₂×C₁₆	—	7	4	no	—
185	C₂×C₁₆	—	7	16	no	—
186	C₂×C₁₆	C₄×D₇	1	4	yes	complemented
187	#97	—	7	4	no	—
188	#97	—	7	2	no	—
189	#97	—	7	2	no	—
190	#97	—	7	4	no	—
191	#97	C₄×D₇	1	2	yes	—
192	#97	C₂²×D₇	1	2	yes	—
193	C₂²×C₇	—	2	2	no	—
194	C₂²×C₇	C₂×#97	1	1	yes	characteristic
195	C₂²×C₇	C₂×#97	1	1	yes	characteristic
196	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
197	C₂×D₇	—	2	2	no	—
198	C₂×D₇	—	2	4	no	—
199	C₂×D₇	—	2	4	no	—
200	C₂×D₇	#347	1	4	yes	direct product factor
201	C₂×D₇	C₂×#97	1	2	yes	—
202	C₂×D₇	C₂×#97	1	2	yes	—
203	C₄×C₇	—	4	2	no	—
204	C₄×C₇	C₂×#97	1	1	yes	characteristic
205	C₄×C₇	C₂×#97	1	1	yes	characteristic
206	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
207	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
208	Dic₇	—	4	2	no	—
209	Dic₇	C₂×#97	1	2	yes	—
210	Dic₇	C₂×#97	1	2	yes	—
211	C₂⁴	—	7	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	7	1	no	—
214	C₂²×C₄	—	7	2	no	—
215	C₂²×C₄	—	7	1	no	—
216	C₂²×C₄	—	7	1	no	—
217	C₂²×C₄	—	7	2	no	—
218	C₂²×C₄	—	7	1	no	—
219	C₂²×C₄	—	7	2	no	—
220	C₂²×C₄	—	7	1	no	—
221	C₂²×C₄	—	7	2	no	—
222	C₂²×C₄	—	14	4	no	—
223	C₂²×C₄	—	14	1	no	—
224	C₂²×C₄	—	28	4	no	—
225	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	7	2	no	—
229	C₂×C₈	—	7	2	no	—
230	C₂×C₈	—	7	2	no	—
231	C₂×C₈	—	7	4	no	—
232	C₂×C₈	—	7	4	no	—
233	C₂×C₈	—	7	4	no	—
234	C₂×C₈	—	7	4	no	—
235	C₂×C₈	—	7	2	no	—
236	C₂×C₈	—	14	4	no	—
237	C₂×C₈	—	14	4	no	—
238	C₂×C₈	—	14	1	no	—
239	C₂×C₈	—	14	1	no	—
240	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
244	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
245	C₄²	—	2	1	no	—
246	C₄²	—	7	2	no	—
247	C₄²	—	14	4	no	—
248	C₄²	—	14	1	no	—
249	C₄²	C₂×C₄×D₇	1	2	yes	—
250	C₁₆	—	7	8	no	—
251	C₁₆	C₂×C₄×D₇	1	8	yes	complemented
252	C₂×C₇	—	2	1	no	—
253	C₂×C₇	—	2	1	no	—
254	C₂×C₇	C₂×#347	1	2	yes	complemented
255	C₂×C₇	C₂²×#97	1	1	yes	characteristic
256	D₇	—	2	2	no	—
257	D₇	C₂×#347	1	4	yes	direct product factor
258	C₂³	—	7	1	no	—
259	C₂³	—	7	2	no	—
260	C₂³	—	7	2	no	—
261	C₂³	—	7	1	no	—
262	C₂³	—	14	4	no	—
263	C₂³	D₇×M₁₆	1	1	yes	characteristic
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	7	2	no	—
267	C₂×C₄	—	7	2	no	—
268	C₂×C₄	—	7	1	no	—
269	C₂×C₄	—	7	1	no	—
270	C₂×C₄	—	7	2	no	—
271	C₂×C₄	—	7	2	no	—
272	C₂×C₄	—	7	2	no	—
273	C₂×C₄	—	7	2	no	—
274	C₂×C₄	—	7	2	no	—
275	C₂×C₄	—	7	2	no	—
276	C₂×C₄	—	7	2	no	—
277	C₂×C₄	—	7	2	no	—
278	C₂×C₄	—	14	2	no	—
279	C₂×C₄	—	28	8	no	—
280	C₂×C₄	—	28	2	no	—
281	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
282	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
283	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
284	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
285	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	7	4	no	—
289	C₈	—	14	2	no	—
290	C₈	C₄²×D₇	1	2	yes	—
291	C₈	C₂²×C₄×D₇	1	2	yes	—
292	C₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	2	2	no	—
294	C₂²	—	7	2	no	—
295	C₂²	—	7	4	no	—
296	C₂²	—	7	2	no	—
297	C₂²	—	14	4	no	—
298	C₂²	—	14	4	no	—
299	C₂²	—	14	2	no	—
300	C₂²	D₇×#97	1	1	yes	characteristic
301	C₂²	D₇×#97	1	1	yes	characteristic
302	C₂²	C₂×D₇×M₁₆	1	1	yes	characteristic
303	C₄	—	4	2	no	—
304	C₄	—	7	2	no	—
305	C₄	—	7	2	no	—
306	C₄	—	28	2	no	—
307	C₄	D₇×#97	1	1	yes	characteristic
308	C₄	D₇×#97	1	1	yes	characteristic
309	C₄	C₂×C₄²×D₇	1	1	yes	characteristic
310	C₄	C₂×D₇×M₁₆	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	7	4	no	—
314	C₂	—	14	2	no	—
315	C₂	D₇×#347	1	2	yes	direct product factor
316	C₂	C₂×D₇×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#347	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×D₇	M₁₆
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#347	D₇
hypocenter	C₇	C₂²×#347
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#347	C₂	semidirect
4. C₂×#89307	C₂	semidirect
5. C₂×#89261	C₂	semidirect
6. D₇×#347	C₂	direct (N × H)
16. D₇×M₃₂	C₄	semidirect
18. C₂×#6372	C₄	semidirect
19. C₂×D₇×C₁₆	C₄	semidirect
22. C₇×#347	C₂²	semidirect
23. #89307	C₂²	semidirect
24. #89261	C₂²	semidirect
46. C₇×M₃₂	C₂×C₄	semidirect
48. C₂×C₇×C₁₆	C₂×C₄	semidirect
54. #6386	C₂×C₄	semidirect
56. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
58. #6372	C₂×C₄	semidirect
59. D₇×C₁₆	C₂×C₄	semidirect
65. C₂×#347	D₇	direct (N × H)
105. C₇×C₁₆	C₂²×C₄	semidirect
106. C₇⋊C₁₆	C₂²×C₄	semidirect
124. #347	C₂×D₇	direct (N × H)
182. M₃₂	C₄×D₇	semidirect
186. C₂×C₁₆	C₄×D₇	semidirect
200. C₂×D₇	#347	direct (N × H)
251. C₁₆	C₂×C₄×D₇	semidirect
254. C₂×C₇	C₂×#347	semidirect
257. D₇	C₂×#347	direct (N × H)
292. C₇	C₂²×#347	semidirect
315. C₂	D₇×#347	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₇×M₃₂	2
C₂×D₇×#97	2
C₂×C₇×#347	2
C₂×#89307	2
C₂×#89261	2
D₇×#347	2
C₂²×#347	7

Subgroups — C2×D7×#347 #13506448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇×#347 #13506448