Subgroups — C₂×C₇×#3185 #13237908

Order 1792.

1954 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	2	yes	maximal
2	C₇×#3185	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#487	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#470	C₂	1	4	yes	maximal
7	C₂×C₇×#441	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	C₂²	1	4	yes	—
10	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
11	C₇×#533	—	2	8	no	—
12	C₇×#533	C₂²	1	16	yes	—
13	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₇×#125	—	2	2	no	—
16	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
17	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₂×C₄×C₇×Q₈	—	2	2	no	—
19	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
20	C₂×C₇×#117	—	2	2	no	—
21	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#498	C₂²	1	8	yes	complemented
23	C₇×#494	C₂²	1	8	yes	complemented
24	C₇×#487	C₂²	1	16	yes	complemented
25	C₇×#470	C₂²	1	32	yes	—
26	C₇×#441	C₂²	1	32	yes	complemented
27	C₂×C₇×#106	C₂²	1	4	yes	—
28	C₂×C₇×#103	C₂²	1	8	yes	—
29	C₄×C₇×M₁₆	C₂²	1	8	yes	—
30	C₂×C₇×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
33	C₂×#3185	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₇×Q₈	—	2	2	no	—
35	C₂²×C₇×Q₈	C₂³	1	4	yes	—
36	C₂³×C₄×C₇	D₄	1	2	yes	—
37	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₇×Q₁₆	—	2	16	no	—
39	C₂×C₇×Q₁₆	C₂³	1	16	yes	—
40	C₂×C₇×M₁₆	—	2	8	no	—
41	C₂×C₇×M₁₆	C₂³	1	8	yes	—
42	C₂²×C₇×C₈	C₂³	1	2	yes	—
43	C₂²×C₇×C₈	C₂³	1	2	yes	—
44	C₇×C₄⋊Q₈	—	2	4	no	—
45	C₇×C₄⋊Q₈	—	2	4	no	—
46	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
47	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
48	C₇×#125	—	2	16	no	—
49	C₇×C₂²⋊Q₈	—	2	16	no	—
50	C₇×C₂²⋊Q₈	—	2	8	no	—
51	C₇×C₂²⋊Q₈	C₂³	1	16	yes	—
52	C₄×C₇×Q₈	—	2	16	no	—
53	C₄×C₇×Q₈	—	2	8	no	—
54	C₄×C₇×Q₈	C₂³	1	16	yes	—
55	C₇×#117	—	2	16	no	—
56	C₂×C₇×C₄⋊C₄	—	2	2	no	—
57	C₂×C₇×C₄⋊C₄	—	2	2	no	—
58	C₂×C₇×C₄⋊C₄	—	2	2	no	—
59	C₂×C₇×C₄⋊C₄	—	2	4	no	—
60	C₂×C₇×C₄⋊C₄	—	2	4	no	—
61	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
62	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
63	C₂×C₄²×C₇	—	2	4	no	—
64	C₂×C₄²×C₇	D₄	1	2	yes	—
65	C₂×C₄²×C₇	D₄	1	4	yes	—
66	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇	C₂³	1	4	yes	—
69	C₇×#106	C₂³	1	16	yes	—
70	C₇×#103	C₂³	1	32	yes	—
71	C₇×#97	C₂³	1	8	yes	—
72	C₄×C₇×C₈	C₂³	1	4	yes	—
73	C₄×C₇×C₈	C₂³	1	4	yes	—
74	C₂×#533	C₂×C₇	1	2	yes	—
75	#3185	C₂×C₇	1	16	yes	direct product factor
76	C₂×#498	C₂×C₇	1	1	yes	characteristic, complemented
77	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂×#487	C₂×C₇	1	2	yes	complemented
79	C₂×#470	C₂×C₇	1	4	yes	—
80	C₂×#441	C₂×C₇	1	4	yes	complemented
81	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
82	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₇×Q₈	—	2	4	no	—
84	C₂×C₇×Q₈	—	2	8	no	—
85	C₂×C₇×Q₈	—	2	16	no	—
86	C₂×C₇×Q₈	—	4	8	no	—
87	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
88	C₂²×C₄×C₇	—	2	8	no	—
89	C₂²×C₄×C₇	—	2	4	no	—
90	C₂²×C₄×C₇	—	2	4	no	—
91	C₂²×C₄×C₇	—	2	4	no	—
92	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
94	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
99	C₇×Q₁₆	—	2	32	no	—
100	C₇×M₁₆	—	2	16	no	—
101	C₂×C₇×C₈	—	2	4	no	—
102	C₂×C₇×C₈	#41	1	8	yes	—
103	C₂×C₇×C₈	C₂⁴	1	4	yes	—
104	C₂×C₇×C₈	C₂⁴	1	4	yes	—
105	C₇×C₄⋊C₄	—	2	8	no	—
106	C₇×C₄⋊C₄	—	2	8	no	—
107	C₇×C₄⋊C₄	—	2	8	no	—
108	C₇×C₄⋊C₄	—	2	16	no	—
109	C₇×C₄⋊C₄	—	2	16	no	—
110	C₇×C₄⋊C₄	—	2	4	no	—
111	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
112	C₇×C₂²⋊C₄	—	2	16	no	—
113	C₄²×C₇	—	2	16	no	—
114	C₄²×C₇	C₂×D₄	1	2	yes	—
115	C₄²×C₇	C₂×D₄	1	8	yes	—
116	C₄²×C₇	C₂×D₄	1	2	yes	—
117	C₄²×C₇	C₂⁴	1	2	yes	—
118	C₄²×C₇	C₂⁴	1	2	yes	—
119	C₂²×Q₁₆	C₂²×C₇	1	4	yes	—
120	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
121	#533	—	2	8	no	—
122	#533	C₂²×C₇	1	16	yes	—
123	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
124	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
125	C₂×#125	—	2	2	no	—
126	C₂×C₂²⋊Q₈	—	2	2	no	—
127	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
128	C₂×C₄×Q₈	—	2	2	no	—
129	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
130	C₂×#117	—	2	2	no	—
131	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
132	#498	C₂²×C₇	1	8	yes	complemented
133	#494	C₂²×C₇	1	8	yes	complemented
134	#487	C₂²×C₇	1	16	yes	complemented
135	#470	C₂²×C₇	1	32	yes	—
136	#441	C₂²×C₇	1	32	yes	complemented
137	C₂×#106	C₂²×C₇	1	4	yes	—
138	C₂×#103	C₂²×C₇	1	8	yes	—
139	C₄×M₁₆	C₂²×C₇	1	8	yes	—
140	C₂×#97	C₂²×C₇	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
143	C₂³×C₇	—	2	2	no	—
144	C₂³×C₇	—	2	4	no	—
145	C₂³×C₇	C₂²×D₄	1	2	yes	—
146	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
147	C₇×Q₈	—	2	16	no	—
148	C₇×Q₈	—	4	8	no	—
149	C₇×Q₈	—	4	16	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	4	no	—
152	C₂×C₄×C₇	—	2	4	no	—
153	C₂×C₄×C₇	—	2	16	no	—
154	C₂×C₄×C₇	—	2	4	no	—
155	C₂×C₄×C₇	—	2	8	no	—
156	C₂×C₄×C₇	—	2	8	no	—
157	C₂×C₄×C₇	—	4	8	no	—
158	C₂×C₄×C₇	—	4	8	no	—
159	C₂×C₄×C₇	#137	1	8	yes	—
160	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₇	C₂×#41	1	4	yes	—
165	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
166	C₇×C₈	—	2	4	no	—
167	C₇×C₈	C₂×#41	1	8	yes	—
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×Q₈	C₂³×C₇	1	4	yes	—
170	C₂³×C₄	C₇×D₄	1	2	yes	—
171	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
172	C₂×Q₁₆	—	2	16	no	—
173	C₂×Q₁₆	C₂³×C₇	1	16	yes	—
174	C₂×M₁₆	—	2	8	no	—
175	C₂×M₁₆	C₂³×C₇	1	8	yes	—
176	C₂²×C₈	C₂³×C₇	1	2	yes	—
177	C₂²×C₈	C₂³×C₇	1	2	yes	—
178	C₄⋊Q₈	—	2	4	no	—
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
181	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
182	#125	—	2	16	no	—
183	C₂²⋊Q₈	—	2	16	no	—
184	C₂²⋊Q₈	—	2	8	no	—
185	C₂²⋊Q₈	C₂³×C₇	1	16	yes	—
186	C₄×Q₈	—	2	16	no	—
187	C₄×Q₈	—	2	8	no	—
188	C₄×Q₈	C₂³×C₇	1	16	yes	—
189	#117	—	2	16	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	C₇×D₄	1	2	yes	—
199	C₂×C₄²	C₇×D₄	1	4	yes	—
200	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₇	1	4	yes	—
203	#106	C₂³×C₇	1	16	yes	—
204	#103	C₂³×C₇	1	32	yes	—
205	#97	C₂³×C₇	1	8	yes	—
206	C₄×C₈	C₂³×C₇	1	4	yes	—
207	C₄×C₈	C₂³×C₇	1	4	yes	—
208	C₂²×C₇	—	2	2	no	—
209	C₂²×C₇	—	2	4	no	—
210	C₂²×C₇	—	4	4	no	—
211	C₂²×C₇	#532	1	2	yes	—
212	C₂²×C₇	C₂×#137	1	4	yes	—
213	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
214	C₄×C₇	—	2	2	no	—
215	C₄×C₇	—	2	4	no	—
216	C₄×C₇	—	4	8	no	—
217	C₄×C₇	—	4	8	no	—
218	C₄×C₇	#532	1	2	yes	—
219	C₄×C₇	C₂×#137	1	8	yes	—
220	C₄×C₇	C₂²×#41	1	2	yes	—
221	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	2	16	no	—
225	C₂×Q₈	—	4	8	no	—
226	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
234	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
235	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
236	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
238	Q₁₆	—	2	32	no	—
239	M₁₆	—	2	16	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	C₇×#41	1	8	yes	—
242	C₂×C₈	C₂⁴×C₇	1	4	yes	—
243	C₂×C₈	C₂⁴×C₇	1	4	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	16	no	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
251	C₂²⋊C₄	—	2	16	no	—
252	C₄²	—	2	16	no	—
253	C₄²	C₂×C₇×D₄	1	2	yes	—
254	C₄²	C₂×C₇×D₄	1	8	yes	—
255	C₄²	C₂×C₇×D₄	1	2	yes	—
256	C₄²	C₂⁴×C₇	1	2	yes	—
257	C₄²	C₂⁴×C₇	1	2	yes	—
258	C₂×C₇	—	4	2	no	—
259	C₂×C₇	#3185	1	4	yes	direct product factor
260	C₂×C₇	C₂×#532	1	1	yes	characteristic
261	C₂×C₇	C₂²×#137	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₂²×C₇×D₄	1	2	yes	—
265	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
266	Q₈	—	2	16	no	—
267	Q₈	—	4	8	no	—
268	Q₈	—	4	16	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₇×#137	1	8	yes	—
279	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
280	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
281	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
283	C₂×C₄	C₂×C₇×#41	1	4	yes	—
284	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	C₂×C₇×#41	1	8	yes	—
287	C₇	C₂×#3185	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₇×#532	1	2	yes	—
292	C₂²	C₂×C₇×#137	1	4	yes	—
293	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	4	8	no	—
298	C₄	C₇×#532	1	2	yes	—
299	C₄	C₂×C₇×#137	1	8	yes	—
300	C₄	C₂²×C₇×#41	1	2	yes	—
301	C₂	—	4	2	no	—
302	C₂	C₇×#3185	1	4	yes	direct product factor
303	C₂	C₂×C₇×#532	1	1	yes	characteristic
304	C₂	C₂²×C₇×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3185	C₂	direct (N × H)
3. C₂×C₇×#498	C₂	semidirect
4. C₂×C₇×#494	C₂	semidirect
5. C₂×C₇×#487	C₂	semidirect
7. C₂×C₇×#441	C₂	semidirect
22. C₇×#498	C₂²	semidirect
23. C₇×#494	C₂²	semidirect
24. C₇×#487	C₂²	semidirect
26. C₇×#441	C₂²	semidirect
33. C₂×#3185	C₇	direct (N × H)
75. #3185	C₂×C₇	direct (N × H)
76. C₂×#498	C₂×C₇	semidirect
77. C₂×#494	C₂×C₇	semidirect
78. C₂×#487	C₂×C₇	semidirect
80. C₂×#441	C₂×C₇	semidirect
132. #498	C₂²×C₇	semidirect
133. #494	C₂²×C₇	semidirect
134. #487	C₂²×C₇	semidirect
136. #441	C₂²×C₇	semidirect
259. C₂×C₇	#3185	direct (N × H)
287. C₇	C₂×#3185	direct (N × H)
302. C₂	C₇×#3185	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×C₇×#533	2
C₇×#3185	2
C₂×C₇×#498	2
C₂×C₇×#494	2
C₂×C₇×#487	2
C₂×C₇×#470	2
C₂×C₇×#441	2
C₂×C₄×C₇×M₁₆	2
C₂×#3185	7

Subgroups — C2×C7×#3185 #13237908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3185 #13237908