Subgroups — C₂×C₇×#1946 #13198378

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#424	C₂	1	2	yes	maximal
4	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
5	C₇×#1946	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₇×#339	C₂	1	2	yes	maximal
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₇×#108	—	2	4	no	—
13	C₂×C₇×#108	C₂²	1	4	yes	—
14	C₇×#427	C₄	1	4	yes	complemented
15	C₇×#427	C₂²	1	4	yes	—
16	C₂×C₇×#107	C₄	1	2	yes	complemented
17	C₂×C₇×#106	C₄	1	2	yes	complemented
18	C₇×#424	—	2	4	no	—
19	C₇×#424	C₂²	1	8	yes	—
20	C₂×C₇×C₄⋊C₈	—	2	4	no	—
21	C₂×C₇×#97	—	2	4	no	—
22	C₂×C₇×#97	C₂²	1	4	yes	—
23	C₇×#339	C₂²	1	8	yes	—
24	C₂×#1946	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₇	—	2	1	no	—
26	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	2	4	no	—
30	C₂×C₇×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₇×C₈	D₄	1	2	yes	—
38	C₂²×C₇×C₈	D₄	1	2	yes	—
39	C₂²×C₇×C₈	D₄	1	4	yes	—
40	C₂²×C₇×C₈	Q₈	1	4	yes	—
41	C₂²×C₇×C₈	C₂³	1	2	yes	—
42	C₂²×C₇×C₈	C₂³	1	2	yes	—
43	C₇×#117	—	2	2	no	—
44	C₇×#117	—	2	2	no	—
45	C₇×#117	C₂×C₄	1	4	yes	—
46	C₇×#117	C₂³	1	4	yes	—
47	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₇	—	2	2	no	—
50	C₂×C₄²×C₇	—	2	2	no	—
51	C₂×C₄²×C₇	—	2	1	no	—
52	C₂×C₄²×C₇	Q₈	1	4	yes	—
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₇×#108	—	2	8	no	—
55	C₇×#107	C₂×C₄	1	8	yes	complemented
56	C₇×#106	C₂×C₄	1	8	yes	complemented
57	C₇×C₄⋊C₈	—	2	16	no	—
58	C₇×#97	—	2	8	no	—
59	C₂²×#108	C₂×C₇	1	1	yes	characteristic
60	C₂×#427	C₂×C₇	1	1	yes	characteristic
61	C₂×#424	C₂×C₇	1	2	yes	—
62	C₂²×#97	C₂×C₇	1	1	yes	characteristic
63	#1946	C₂×C₇	1	8	yes	direct product factor
64	C₂×#339	C₂×C₇	1	2	yes	—
65	C₂⁴×C₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	—	2	4	no	—
68	C₂²×C₄×C₇	—	2	2	no	—
69	C₂²×C₄×C₇	—	4	4	no	—
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₇	#41	1	2	yes	—
79	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
80	C₇×M₁₆	—	2	4	no	—
81	C₇×M₁₆	—	2	4	no	—
82	C₇×M₁₆	—	4	4	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	—	4	4	no	—
89	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
90	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
91	C₂×C₇×C₈	C₄⋊C₄	1	8	yes	—
92	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
95	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
96	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
99	C₇×C₂²⋊C₄	—	2	4	no	—
100	C₇×C₂²⋊C₄	—	2	4	no	—
101	C₄²×C₇	—	2	4	no	—
102	C₄²×C₇	—	2	4	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₄²×C₇	—	2	4	no	—
105	C₄²×C₇	C₂×Q₈	1	4	yes	—
106	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
107	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
109	C₂×#117	C₂²×C₇	1	2	yes	—
110	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
111	C₂×#108	—	2	4	no	—
112	C₂×#108	C₂²×C₇	1	4	yes	—
113	#427	C₄×C₇	1	4	yes	complemented
114	#427	C₂²×C₇	1	4	yes	—
115	C₂×#107	C₄×C₇	1	2	yes	complemented
116	C₂×#106	C₄×C₇	1	2	yes	complemented
117	#424	—	2	4	no	—
118	#424	C₂²×C₇	1	8	yes	—
119	C₂×C₄⋊C₈	—	2	4	no	—
120	C₂×#97	—	2	4	no	—
121	C₂×#97	C₂²×C₇	1	4	yes	—
122	#339	C₂²×C₇	1	8	yes	—
123	C₂³×C₇	—	2	4	no	—
124	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
125	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₇	C₂²⋊Q₈	1	2	yes	—
127	C₂³×C₇	#125	1	2	yes	—
128	C₂³×C₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₇	—	2	8	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	4	8	no	—
133	C₂×C₄×C₇	—	4	8	no	—
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₇	#121	1	2	yes	—
141	C₂×C₄×C₇	#121	1	2	yes	—
142	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₇	#125	1	2	yes	—
144	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
145	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
152	C₇×C₈	—	2	4	no	—
153	C₇×C₈	—	4	4	no	—
154	C₇×C₈	C₂×C₄⋊C₄	1	4	yes	—
155	C₇×C₈	C₂×C₄⋊C₄	1	4	yes	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	C₂³×C₇	1	4	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
168	C₂²×C₈	C₇×D₄	1	2	yes	—
169	C₂²×C₈	C₇×D₄	1	2	yes	—
170	C₂²×C₈	C₇×D₄	1	4	yes	—
171	C₂²×C₈	C₇×Q₈	1	4	yes	—
172	C₂²×C₈	C₂³×C₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₇	1	2	yes	—
174	#117	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	C₂×C₄×C₇	1	4	yes	—
177	#117	C₂³×C₇	1	4	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	C₇×Q₈	1	4	yes	—
184	C₂×C₄²	C₂³×C₇	1	2	yes	—
185	#108	—	2	8	no	—
186	#107	C₂×C₄×C₇	1	8	yes	complemented
187	#106	C₂×C₄×C₇	1	8	yes	complemented
188	C₄⋊C₈	—	2	16	no	—
189	#97	—	2	8	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	—	2	2	no	—
192	C₂²×C₇	—	2	4	no	—
193	C₂²×C₇	—	2	4	no	—
194	C₂²×C₇	#389	1	2	yes	—
195	C₂²×C₇	#389	1	2	yes	—
196	C₂²×C₇	#444	1	4	yes	—
197	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂²×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
199	C₂²×C₇	C₂×#125	1	1	yes	characteristic
200	C₄×C₇	—	4	4	no	—
201	C₄×C₇	—	4	8	no	—
202	C₄×C₇	#389	1	2	yes	—
203	C₄×C₇	#389	1	2	yes	—
204	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
205	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
206	C₄×C₇	C₂×#121	1	1	yes	characteristic
207	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
208	C₂⁴	C₇×#41	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
216	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
219	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
220	C₂²×C₄	C₇×#41	1	1	yes	characteristic
221	C₂²×C₄	C₇×#41	1	2	yes	—
222	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
233	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
234	C₂×C₈	C₇×C₄⋊C₄	1	8	yes	—
235	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
237	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
238	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
239	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	4	no	—
248	C₄²	C₂×C₇×Q₈	1	4	yes	—
249	C₂×C₇	—	2	1	no	—
250	C₂×C₇	—	2	1	no	—
251	C₂×C₇	—	2	2	no	—
252	C₂×C₇	#1946	1	4	yes	direct product factor
253	C₂×C₇	C₂×#389	1	1	yes	characteristic
254	C₂×C₇	C₂×#444	1	2	yes	—
255	C₂³	—	2	4	no	—
256	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
257	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
258	C₂³	C₇×C₂²⋊Q₈	1	2	yes	—
259	C₂³	C₇×#125	1	2	yes	—
260	C₂³	C₂×C₇×#41	1	1	yes	characteristic
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
267	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
268	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
269	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
270	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
271	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
272	C₂×C₄	C₇×#121	1	2	yes	—
273	C₂×C₄	C₇×#121	1	2	yes	—
274	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
275	C₂×C₄	C₇×#125	1	2	yes	—
276	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
277	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
278	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	4	4	no	—
286	C₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
287	C₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
288	C₇	C₂×#1946	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	C₇×#389	1	2	yes	—
294	C₂²	C₇×#389	1	2	yes	—
295	C₂²	C₇×#444	1	4	yes	—
296	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
297	C₂²	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂²	C₂×C₇×#125	1	1	yes	characteristic
299	C₄	—	4	4	no	—
300	C₄	—	4	8	no	—
301	C₄	C₇×#389	1	2	yes	—
302	C₄	C₇×#389	1	2	yes	—
303	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
304	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
305	C₄	C₂×C₇×#121	1	1	yes	characteristic
306	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	C₇×#1946	1	4	yes	direct product factor
311	C₂	C₂×C₇×#389	1	1	yes	characteristic
312	C₂	C₂×C₇×#444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₄×D₄
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇×#1946	C₂	direct (N × H)
14. C₇×#427	C₄	semidirect
16. C₂×C₇×#107	C₄	semidirect
17. C₂×C₇×#106	C₄	semidirect
24. C₂×#1946	C₇	direct (N × H)
55. C₇×#107	C₂×C₄	semidirect
56. C₇×#106	C₂×C₄	semidirect
63. #1946	C₂×C₇	direct (N × H)
113. #427	C₄×C₇	semidirect
115. C₂×#107	C₄×C₇	semidirect
116. C₂×#106	C₄×C₇	semidirect
186. #107	C₂×C₄×C₇	semidirect
187. #106	C₂×C₄×C₇	semidirect
252. C₂×C₇	#1946	direct (N × H)
288. C₇	C₂×#1946	direct (N × H)
310. C₂	C₇×#1946	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#108	2
C₂×C₇×#427	2
C₂×C₇×#424	2
C₂²×C₇×#97	2
C₇×#1946	2
C₂×C₇×#339	2
C₂×#1946	7

Subgroups — C2×C7×#1946 #13198378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1946 #13198378