Subgroups — C₂×C₇×#1810 #13197881

Order 1792.

1566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×#108	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#407	C₂	1	2	yes	maximal
4	C₇×#1810	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#341	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
8	C₂³×C₇×C₈	—	2	8	no	—
9	C₂³×C₇×C₈	C₄	1	4	yes	—
10	C₂³×C₇×C₈	C₄	1	4	yes	—
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₇×C₈	C₂²	1	4	yes	—
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×#108	—	2	8	no	—
15	C₂×C₇×#108	C₂²	1	8	yes	complemented
16	C₇×#407	—	2	4	no	—
17	C₇×#407	C₂²	1	8	yes	—
18	C₂×C₇×C₂²⋊C₈	—	2	4	no	—
19	C₇×#341	C₂²	1	8	yes	complemented
20	C₂×#1810	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂⁵×C₇	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	4	no	—
23	C₂³×C₄×C₇	—	2	4	no	—
24	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₇	D₄	1	2	yes	—
27	C₂³×C₄×C₇	D₄	1	2	yes	—
28	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₇	Q₈	1	2	yes	—
30	C₂³×C₄×C₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₇×M₁₆	—	2	4	no	—
33	C₂×C₇×M₁₆	—	2	8	no	—
34	C₂×C₇×M₁₆	—	2	8	no	—
35	C₂×C₇×M₁₆	C₂³	1	8	yes	—
36	C₂²×C₇×C₈	—	2	4	no	—
37	C₂²×C₇×C₈	—	2	16	no	—
38	C₂²×C₇×C₈	—	2	16	no	—
39	C₂²×C₇×C₈	—	2	16	no	—
40	C₂²×C₇×C₈	—	2	4	no	—
41	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₇×C₈	C₂×C₄	1	16	yes	—
44	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₇×C₈	D₄	1	8	yes	—
47	C₂²×C₇×C₈	C₂³	1	2	yes	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₇×#108	—	2	16	no	—
50	C₇×C₂²⋊C₈	—	2	16	no	—
51	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic
52	C₂²×#108	C₂×C₇	1	2	yes	complemented
53	C₂×#407	C₂×C₇	1	2	yes	—
54	#1810	C₂×C₇	1	8	yes	direct product factor
55	C₂×#341	C₂×C₇	1	2	yes	complemented
56	C₂⁴×C₇	—	2	8	no	—
57	C₂⁴×C₇	—	2	4	no	—
58	C₂⁴×C₇	C₄⋊C₄	1	2	yes	—
59	C₂⁴×C₇	C₄⋊C₄	1	2	yes	—
60	C₂⁴×C₇	C₂×Q₈	1	2	yes	—
61	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	—	2	8	no	—
68	C₂²×C₄×C₇	—	2	8	no	—
69	C₂²×C₄×C₇	—	2	8	no	—
70	C₂²×C₄×C₇	—	2	4	no	—
71	C₂²×C₄×C₇	—	2	4	no	—
72	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₇	#41	1	4	yes	—
86	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
87	C₇×M₁₆	—	2	8	no	—
88	C₇×M₁₆	—	2	8	no	—
89	C₇×M₁₆	—	4	8	no	—
90	C₂×C₇×C₈	—	2	4	no	—
91	C₂×C₇×C₈	—	2	4	no	—
92	C₂×C₇×C₈	—	2	4	no	—
93	C₂×C₇×C₈	—	2	32	no	—
94	C₂×C₇×C₈	—	2	8	no	—
95	C₂×C₇×C₈	—	4	8	no	—
96	C₂×C₇×C₈	C₂²⋊C₄	1	8	yes	—
97	C₂×C₇×C₈	C₂²⋊C₄	1	8	yes	—
98	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
101	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
102	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
103	C₂³×C₈	—	2	8	no	—
104	C₂³×C₈	C₄×C₇	1	4	yes	—
105	C₂³×C₈	C₄×C₇	1	4	yes	—
106	C₂³×C₈	C₂²×C₇	1	4	yes	—
107	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
109	C₂×#108	—	2	8	no	—
110	C₂×#108	C₂²×C₇	1	8	yes	complemented
111	#407	—	2	4	no	—
112	#407	C₂²×C₇	1	8	yes	—
113	C₂×C₂²⋊C₈	—	2	4	no	—
114	#341	C₂²×C₇	1	8	yes	complemented
115	C₂³×C₇	—	2	32	no	—
116	C₂³×C₇	—	2	4	no	—
117	C₂³×C₇	—	2	4	no	—
118	C₂³×C₇	—	2	4	no	—
119	C₂³×C₇	—	2	4	no	—
120	C₂³×C₇	—	2	4	no	—
121	C₂³×C₇	—	2	8	no	—
122	C₂³×C₇	—	2	8	no	—
123	C₂³×C₇	#108	1	8	yes	complemented
124	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
126	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₇	C₂²⋊Q₈	1	4	yes	—
128	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	4	no	—
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	8	no	—
135	C₂×C₄×C₇	—	2	8	no	—
136	C₂×C₄×C₇	—	2	8	no	—
137	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₇	#117	1	2	yes	—
143	C₂×C₄×C₇	#117	1	2	yes	—
144	C₂×C₄×C₇	#121	1	4	yes	—
145	C₂×C₄×C₇	#121	1	4	yes	—
146	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
147	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
149	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×#41	1	2	yes	—
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	—	4	8	no	—
153	C₇×C₈	C₂×C₂²⋊C₄	1	8	yes	—
154	C₂⁵	C₇×Q₈	1	1	yes	characteristic
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	—	2	4	no	—
157	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
158	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
159	C₂³×C₄	C₇×D₄	1	2	yes	—
160	C₂³×C₄	C₇×D₄	1	2	yes	—
161	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
162	C₂³×C₄	C₇×Q₈	1	2	yes	—
163	C₂³×C₄	C₂³×C₇	1	2	yes	—
164	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	8	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂×M₁₆	C₂³×C₇	1	8	yes	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	2	16	no	—
171	C₂²×C₈	—	2	16	no	—
172	C₂²×C₈	—	2	16	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
175	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
176	C₂²×C₈	C₂×C₄×C₇	1	16	yes	—
177	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
178	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
179	C₂²×C₈	C₇×D₄	1	8	yes	—
180	C₂²×C₈	C₂³×C₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₇	1	2	yes	—
182	#108	—	2	16	no	—
183	C₂²⋊C₈	—	2	16	no	—
184	C₂²×C₇	—	2	16	no	—
185	C₂²×C₇	—	2	16	no	—
186	C₂²×C₇	—	2	16	no	—
187	C₂²×C₇	—	2	8	no	—
188	C₂²×C₇	—	2	2	no	—
189	C₂²×C₇	—	2	4	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	—	2	4	no	—
192	C₂²×C₇	#380	1	2	yes	—
193	C₂²×C₇	#380	1	2	yes	—
194	C₂²×C₇	C₂×#108	1	4	yes	—
195	C₂²×C₇	C₂×#108	1	8	yes	complemented
196	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
198	C₄×C₇	—	2	4	no	—
199	C₄×C₇	#380	1	2	yes	—
200	C₄×C₇	#380	1	2	yes	—
201	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
202	C₄×C₇	C₂×#117	1	1	yes	characteristic
203	C₄×C₇	C₂×#121	1	2	yes	—
204	C₂⁴	—	2	8	no	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	C₇×C₄⋊C₄	1	2	yes	—
207	C₂⁴	C₇×C₄⋊C₄	1	2	yes	—
208	C₂⁴	C₂×C₇×Q₈	1	2	yes	—
209	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
221	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
222	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
223	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
228	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
229	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
232	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
233	C₂²×C₄	C₇×#41	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
235	M₁₆	—	2	8	no	—
236	M₁₆	—	2	8	no	—
237	M₁₆	—	4	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	32	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	C₇×C₂²⋊C₄	1	8	yes	—
245	C₂×C₈	C₇×C₂²⋊C₄	1	8	yes	—
246	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
247	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
248	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
249	C₂×C₇	—	2	2	no	—
250	C₂×C₇	—	2	2	no	—
251	C₂×C₇	—	2	8	no	—
252	C₂×C₇	#1810	1	4	yes	direct product factor
253	C₂×C₇	C₂×#380	1	1	yes	characteristic
254	C₂×C₇	C₂²×#108	1	2	yes	—
255	C₂³	—	2	32	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	8	no	—
262	C₂³	—	2	8	no	—
263	C₂³	C₇×#108	1	8	yes	complemented
264	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
265	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
266	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
267	C₂³	C₇×C₂²⋊Q₈	1	4	yes	—
268	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
279	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
280	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₇×#117	1	2	yes	—
283	C₂×C₄	C₇×#117	1	2	yes	—
284	C₂×C₄	C₇×#121	1	4	yes	—
285	C₂×C₄	C₇×#121	1	4	yes	—
286	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
287	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
289	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₇×#41	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	4	8	no	—
293	C₈	C₂×C₇×C₂²⋊C₄	1	8	yes	—
294	C₇	C₂×#1810	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	16	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	16	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	2	4	no	—
303	C₂²	C₇×#380	1	2	yes	—
304	C₂²	C₇×#380	1	2	yes	—
305	C₂²	C₂×C₇×#108	1	4	yes	—
306	C₂²	C₂×C₇×#108	1	8	yes	complemented
307	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
308	C₂²	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	C₇×#380	1	2	yes	—
311	C₄	C₇×#380	1	2	yes	—
312	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
313	C₄	C₂×C₇×#117	1	1	yes	characteristic
314	C₄	C₂×C₇×#121	1	2	yes	—
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	8	no	—
318	C₂	C₇×#1810	1	4	yes	direct product factor
319	C₂	C₂×C₇×#380	1	1	yes	characteristic
320	C₂	C₂²×C₇×#108	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	C₂×C₄⋊C₄
prime radical	C₂⁵×C₇	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×#108	C₂	semidirect
4. C₇×#1810	C₂	direct (N × H)
5. C₂×C₇×#341	C₂	semidirect
15. C₂×C₇×#108	C₂²	semidirect
19. C₇×#341	C₂²	semidirect
20. C₂×#1810	C₇	direct (N × H)
52. C₂²×#108	C₂×C₇	semidirect
54. #1810	C₂×C₇	direct (N × H)
55. C₂×#341	C₂×C₇	semidirect
110. C₂×#108	C₂²×C₇	semidirect
114. #341	C₂²×C₇	semidirect
123. C₂³×C₇	#108	semidirect
195. C₂²×C₇	C₂×#108	semidirect
252. C₂×C₇	#1810	direct (N × H)
263. C₂³	C₇×#108	semidirect
294. C₇	C₂×#1810	direct (N × H)
306. C₂²	C₂×C₇×#108	semidirect
318. C₂	C₇×#1810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×C₈	2
C₂²×C₇×#108	2
C₂×C₇×#407	2
C₇×#1810	2
C₂×C₇×#341	2
C₂×#1810	7

Subgroups — C2×C7×#1810 #13197881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1810 #13197881