Subgroups — C₂×C₇×#2852 #13213833

Order 1792.

3354 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2852	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#388	C₂	1	3	yes	maximal, complemented
5	C₂×C₇×#387	C₂	1	3	yes	maximal, complemented
6	C₂×C₇×#386	C₂	1	3	yes	maximal
7	C₂×C₇×#385	C₂	1	3	yes	maximal, complemented
8	C₂×C₇×#383	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
11	C₂×C₇×#126	—	2	8	no	—
12	C₂×C₇×#126	C₂²	1	8	yes	complemented
13	C₂²×C₇×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂²×C₇×C₄⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂²×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
17	C₂²×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
18	C₂²×C₄²×C₇	—	2	8	no	—
19	C₂²×C₄²×C₇	C₂²	1	3	yes	—
20	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₇	C₂²	1	8	yes	complemented
22	C₇×#388	C₂²	1	24	yes	complemented
23	C₇×#387	C₂²	1	24	yes	complemented
24	C₇×#386	C₂²	1	24	yes	—
25	C₇×#385	C₂²	1	24	yes	complemented
26	C₇×#383	C₂²	1	8	yes	complemented
27	C₂×C₇×#95	C₂²	1	3	yes	—
28	C₂×C₇×#95	C₂²	1	3	yes	—
29	C₂×C₇×#95	C₂²	1	3	yes	—
30	C₂×C₇×#95	C₂²	1	3	yes	—
31	C₂×#2852	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁵×C₇	C₂³	1	1	yes	characteristic
33	C₂⁵×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₇×D₄	—	2	12	no	—
35	C₂²×C₇×D₄	C₂³	1	4	yes	—
36	C₂³×C₄×C₇	—	2	12	no	—
37	C₂³×C₄×C₇	—	2	12	no	—
38	C₂³×C₄×C₇	—	2	12	no	—
39	C₂³×C₄×C₇	C₂³	1	12	yes	—
40	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₇	C₂³	1	3	yes	—
42	C₂³×C₄×C₇	C₂³	1	3	yes	—
43	C₂³×C₄×C₇	C₂³	1	3	yes	—
44	C₂³×C₄×C₇	C₂³	1	3	yes	—
45	C₇×#126	—	2	32	no	—
46	C₂×C₇×C₄⋊C₄	—	2	12	no	—
47	C₂×C₇×C₄⋊C₄	—	2	12	no	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
50	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
51	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
52	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
53	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	—
54	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	—
55	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	—
56	C₂×C₄²×C₇	—	2	12	no	—
57	C₂×C₄²×C₇	—	2	32	no	—
58	C₂×C₄²×C₇	D₄	1	8	yes	complemented
59	C₂×C₄²×C₇	C₂³	1	12	yes	—
60	C₂×C₄²×C₇	C₂³	1	4	yes	—
61	C₇×#95	C₂³	1	12	yes	—
62	C₇×#95	C₂³	1	12	yes	—
63	C₇×#95	C₂³	1	12	yes	—
64	C₇×#95	C₂³	1	12	yes	—
65	C₂²×#126	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
67	#2852	C₂×C₇	1	16	yes	direct product factor
68	C₂×#388	C₂×C₇	1	3	yes	complemented
69	C₂×#387	C₂×C₇	1	3	yes	complemented
70	C₂×#386	C₂×C₇	1	3	yes	—
71	C₂×#385	C₂×C₇	1	3	yes	complemented
72	C₂×#383	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂⁴×C₇	—	2	2	no	—
74	C₂⁴×C₇	—	2	6	no	—
75	C₂⁴×C₇	—	2	2	no	—
76	C₂⁴×C₇	—	2	6	no	—
77	C₂⁴×C₇	—	2	6	no	—
78	C₂⁴×C₇	#41	1	12	yes	—
79	C₂⁴×C₇	C₂⁴	1	2	yes	—
80	C₂⁴×C₇	C₂⁴	1	2	yes	—
81	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₇×D₄	—	2	24	no	—
83	C₂×C₇×D₄	—	4	16	no	—
84	C₂²×C₄×C₇	—	2	96	no	—
85	C₂²×C₄×C₇	—	2	6	no	—
86	C₂²×C₄×C₇	—	2	24	no	—
87	C₂²×C₄×C₇	—	2	6	no	—
88	C₂²×C₄×C₇	—	2	6	no	—
89	C₂²×C₄×C₇	—	2	6	no	—
90	C₂²×C₄×C₇	—	2	24	no	—
91	C₂²×C₄×C₇	—	2	24	no	—
92	C₂²×C₄×C₇	—	2	6	no	—
93	C₂²×C₄×C₇	—	2	6	no	—
94	C₂²×C₄×C₇	—	2	6	no	—
95	C₂²×C₄×C₇	—	2	6	no	—
96	C₂²×C₄×C₇	—	2	6	no	—
97	C₂²×C₄×C₇	—	2	6	no	—
98	C₂²×C₄×C₇	—	2	6	no	—
99	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
100	C₂²×C₄×C₇	#41	1	12	yes	—
101	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
103	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
104	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
105	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
106	C₇×C₄⋊C₄	—	2	24	no	—
107	C₇×C₄⋊C₄	—	2	24	no	—
108	C₇×C₂²⋊C₄	—	2	24	no	—
109	C₇×C₂²⋊C₄	—	2	24	no	—
110	C₇×C₂²⋊C₄	—	2	24	no	—
111	C₄²×C₇	—	2	24	no	—
112	C₄²×C₇	C₂×D₄	1	16	yes	complemented
113	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
114	C₂⁴×C₄	C₂²×C₇	1	3	yes	—
115	C₂×#126	—	2	8	no	—
116	C₂×#126	C₂²×C₇	1	8	yes	complemented
117	C₂²×C₄⋊C₄	C₂²×C₇	1	3	yes	—
118	C₂²×C₄⋊C₄	C₂²×C₇	1	3	yes	—
119	C₂²×C₂²⋊C₄	C₂²×C₇	1	3	yes	—
120	C₂²×C₂²⋊C₄	C₂²×C₇	1	3	yes	—
121	C₂²×C₂²⋊C₄	C₂²×C₇	1	3	yes	—
122	C₂²×C₄²	—	2	8	no	—
123	C₂²×C₄²	C₂²×C₇	1	3	yes	—
124	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
125	C₂²×C₄²	C₂²×C₇	1	8	yes	complemented
126	#388	C₂²×C₇	1	24	yes	complemented
127	#387	C₂²×C₇	1	24	yes	complemented
128	#386	C₂²×C₇	1	24	yes	—
129	#385	C₂²×C₇	1	24	yes	complemented
130	#383	C₂²×C₇	1	8	yes	complemented
131	C₂×#95	C₂²×C₇	1	3	yes	—
132	C₂×#95	C₂²×C₇	1	3	yes	—
133	C₂×#95	C₂²×C₇	1	3	yes	—
134	C₂×#95	C₂²×C₇	1	3	yes	—
135	C₂³×C₇	—	2	24	no	—
136	C₂³×C₇	—	2	24	no	—
137	C₂³×C₇	—	2	2	no	—
138	C₂³×C₇	—	2	6	no	—
139	C₂³×C₇	—	2	2	no	—
140	C₂³×C₇	—	2	6	no	—
141	C₂³×C₇	—	2	6	no	—
142	C₂³×C₇	—	4	12	no	—
143	C₂³×C₇	—	4	4	no	—
144	C₂³×C₇	—	4	12	no	—
145	C₂³×C₇	#126	1	16	yes	complemented
146	C₂³×C₇	C₂²×D₄	1	2	yes	—
147	C₂³×C₇	C₂×#41	1	12	yes	—
148	C₂³×C₇	C₂×#41	1	6	yes	—
149	C₂³×C₇	C₂×#41	1	6	yes	—
150	C₂³×C₇	C₂⁵	1	1	yes	characteristic
151	C₇×D₄	—	4	16	no	—
152	C₂×C₄×C₇	—	2	96	no	—
153	C₂×C₄×C₇	—	2	24	no	—
154	C₂×C₄×C₇	—	2	24	no	—
155	C₂×C₄×C₇	—	2	6	no	—
156	C₂×C₄×C₇	—	2	6	no	—
157	C₂×C₄×C₇	—	2	6	no	—
158	C₂×C₄×C₇	—	2	6	no	—
159	C₂×C₄×C₇	—	2	6	no	—
160	C₂×C₄×C₇	—	2	6	no	—
161	C₂×C₄×C₇	—	2	6	no	—
162	C₂×C₄×C₇	—	2	6	no	—
163	C₂×C₄×C₇	—	2	6	no	—
164	C₂×C₄×C₇	—	2	6	no	—
165	C₂×C₄×C₇	—	2	6	no	—
166	C₂×C₄×C₇	—	4	12	no	—
167	C₂×C₄×C₇	—	4	4	no	—
168	C₂×C₄×C₇	—	4	12	no	—
169	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
170	C₂×C₄×C₇	C₂×#41	1	12	yes	—
171	C₂⁵	C₂³×C₇	1	1	yes	characteristic
172	C₂⁵	C₂³×C₇	1	1	yes	characteristic
173	C₂²×D₄	—	2	12	no	—
174	C₂²×D₄	C₂³×C₇	1	4	yes	—
175	C₂³×C₄	—	2	12	no	—
176	C₂³×C₄	—	2	12	no	—
177	C₂³×C₄	—	2	12	no	—
178	C₂³×C₄	C₂³×C₇	1	3	yes	—
179	C₂³×C₄	C₂³×C₇	1	3	yes	—
180	C₂³×C₄	C₂³×C₇	1	12	yes	—
181	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₇	1	3	yes	—
183	C₂³×C₄	C₂³×C₇	1	3	yes	—
184	#126	—	2	32	no	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	2	12	no	—
187	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
188	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
189	C₂×C₂²⋊C₄	—	2	12	no	—
190	C₂×C₂²⋊C₄	—	2	12	no	—
191	C₂×C₂²⋊C₄	—	2	12	no	—
192	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	—
193	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	—
194	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	—
195	C₂×C₄²	—	2	12	no	—
196	C₂×C₄²	—	2	32	no	—
197	C₂×C₄²	C₇×D₄	1	8	yes	complemented
198	C₂×C₄²	C₂³×C₇	1	12	yes	—
199	C₂×C₄²	C₂³×C₇	1	4	yes	—
200	#95	C₂³×C₇	1	12	yes	—
201	#95	C₂³×C₇	1	12	yes	—
202	#95	C₂³×C₇	1	12	yes	—
203	#95	C₂³×C₇	1	12	yes	—
204	C₂²×C₇	—	2	32	no	—
205	C₂²×C₇	—	2	12	no	—
206	C₂²×C₇	—	2	12	no	—
207	C₂²×C₇	—	4	6	no	—
208	C₂²×C₇	—	4	2	no	—
209	C₂²×C₇	—	4	6	no	—
210	C₂²×C₇	—	8	8	no	—
211	C₂²×C₇	#525	1	12	yes	—
212	C₂²×C₇	C₂×#126	1	4	yes	—
213	C₂²×C₇	C₂×#126	1	8	yes	complemented
214	C₂²×C₇	#529	1	12	yes	—
215	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
216	C₂²×C₇	C₂²×#41	1	3	yes	—
217	C₂²×C₇	C₂²×#41	1	3	yes	—
218	C₄×C₇	—	2	12	no	—
219	C₄×C₇	—	2	12	no	—
220	C₄×C₇	—	4	6	no	—
221	C₄×C₇	—	4	2	no	—
222	C₄×C₇	—	4	6	no	—
223	C₂⁴	—	2	2	no	—
224	C₂⁴	—	2	6	no	—
225	C₂⁴	—	2	2	no	—
226	C₂⁴	—	2	6	no	—
227	C₂⁴	—	2	6	no	—
228	C₂⁴	C₇×#41	1	12	yes	—
229	C₂⁴	C₂⁴×C₇	1	2	yes	—
230	C₂⁴	C₂⁴×C₇	1	2	yes	—
231	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
232	C₂×D₄	—	2	24	no	—
233	C₂×D₄	—	4	16	no	—
234	C₂²×C₄	—	2	96	no	—
235	C₂²×C₄	—	2	6	no	—
236	C₂²×C₄	—	2	24	no	—
237	C₂²×C₄	—	2	6	no	—
238	C₂²×C₄	—	2	6	no	—
239	C₂²×C₄	—	2	6	no	—
240	C₂²×C₄	—	2	24	no	—
241	C₂²×C₄	—	2	24	no	—
242	C₂²×C₄	—	2	6	no	—
243	C₂²×C₄	—	2	6	no	—
244	C₂²×C₄	—	2	6	no	—
245	C₂²×C₄	—	2	6	no	—
246	C₂²×C₄	—	2	6	no	—
247	C₂²×C₄	—	2	6	no	—
248	C₂²×C₄	—	2	6	no	—
249	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
250	C₂²×C₄	C₇×#41	1	12	yes	—
251	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
253	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
254	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
255	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
256	C₄⋊C₄	—	2	24	no	—
257	C₄⋊C₄	—	2	24	no	—
258	C₂²⋊C₄	—	2	24	no	—
259	C₂²⋊C₄	—	2	24	no	—
260	C₂²⋊C₄	—	2	24	no	—
261	C₄²	—	2	24	no	—
262	C₄²	C₂×C₇×D₄	1	16	yes	complemented
263	C₂×C₇	—	2	8	no	—
264	C₂×C₇	—	8	2	no	—
265	C₂×C₇	#2852	1	8	yes	direct product factor
266	C₂×C₇	C₂×#525	1	3	yes	—
267	C₂×C₇	C₂²×#126	1	1	yes	characteristic
268	C₂×C₇	C₂×#529	1	3	yes	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	6	no	—
271	C₂³	—	2	24	no	—
272	C₂³	—	2	24	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	6	no	—
275	C₂³	—	2	6	no	—
276	C₂³	—	4	12	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	4	12	no	—
279	C₂³	C₇×#126	1	16	yes	complemented
280	C₂³	C₂²×C₇×D₄	1	2	yes	—
281	C₂³	C₂×C₇×#41	1	12	yes	—
282	C₂³	C₂×C₇×#41	1	6	yes	—
283	C₂³	C₂×C₇×#41	1	6	yes	—
284	C₂³	C₂⁵×C₇	1	1	yes	characteristic
285	D₄	—	4	16	no	—
286	C₂×C₄	—	2	96	no	—
287	C₂×C₄	—	2	6	no	—
288	C₂×C₄	—	2	6	no	—
289	C₂×C₄	—	2	6	no	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	2	24	no	—
292	C₂×C₄	—	2	24	no	—
293	C₂×C₄	—	2	6	no	—
294	C₂×C₄	—	2	6	no	—
295	C₂×C₄	—	2	6	no	—
296	C₂×C₄	—	2	6	no	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	2	6	no	—
300	C₂×C₄	—	4	12	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	12	no	—
303	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
304	C₂×C₄	C₂×C₇×#41	1	12	yes	—
305	C₇	C₂×#2852	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	12	no	—
307	C₂²	—	2	12	no	—
308	C₂²	—	2	32	no	—
309	C₂²	—	4	6	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	6	no	—
312	C₂²	—	8	8	no	—
313	C₂²	C₇×#525	1	12	yes	—
314	C₂²	C₂×C₇×#126	1	4	yes	—
315	C₂²	C₂×C₇×#126	1	8	yes	complemented
316	C₂²	C₇×#529	1	12	yes	—
317	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
318	C₂²	C₂²×C₇×#41	1	3	yes	—
319	C₂²	C₂²×C₇×#41	1	3	yes	—
320	C₄	—	2	12	no	—
321	C₄	—	2	12	no	—
322	C₄	—	4	6	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	6	no	—
325	C₂	—	2	8	no	—
326	C₂	—	8	2	no	—
327	C₂	C₇×#2852	1	8	yes	direct product factor
328	C₂	C₂×C₇×#525	1	3	yes	—
329	C₂	C₂²×C₇×#126	1	1	yes	characteristic
330	C₂	C₂×C₇×#529	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×#126	C₂	semidirect
2. C₂³×C₄²×C₇	C₂	semidirect
3. C₇×#2852	C₂	direct (N × H)
4. C₂×C₇×#388	C₂	semidirect
5. C₂×C₇×#387	C₂	semidirect
7. C₂×C₇×#385	C₂	semidirect
8. C₂×C₇×#383	C₂	semidirect
12. C₂×C₇×#126	C₂²	semidirect
21. C₂²×C₄²×C₇	C₂²	semidirect
22. C₇×#388	C₂²	semidirect
23. C₇×#387	C₂²	semidirect
25. C₇×#385	C₂²	semidirect
26. C₇×#383	C₂²	semidirect
31. C₂×#2852	C₇	direct (N × H)
58. C₂×C₄²×C₇	D₄	semidirect
65. C₂²×#126	C₂×C₇	semidirect
66. C₂³×C₄²	C₂×C₇	semidirect
67. #2852	C₂×C₇	direct (N × H)
68. C₂×#388	C₂×C₇	semidirect
69. C₂×#387	C₂×C₇	semidirect
71. C₂×#385	C₂×C₇	semidirect
72. C₂×#383	C₂×C₇	semidirect
112. C₄²×C₇	C₂×D₄	semidirect
116. C₂×#126	C₂²×C₇	semidirect
125. C₂²×C₄²	C₂²×C₇	semidirect
126. #388	C₂²×C₇	semidirect
127. #387	C₂²×C₇	semidirect
129. #385	C₂²×C₇	semidirect
130. #383	C₂²×C₇	semidirect
145. C₂³×C₇	#126	semidirect
197. C₂×C₄²	C₇×D₄	semidirect
213. C₂²×C₇	C₂×#126	semidirect
262. C₄²	C₂×C₇×D₄	semidirect
265. C₂×C₇	#2852	direct (N × H)
279. C₂³	C₇×#126	semidirect
305. C₇	C₂×#2852	direct (N × H)
315. C₂²	C₂×C₇×#126	semidirect
327. C₂	C₇×#2852	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×#126	2
C₂³×C₄²×C₇	2
C₇×#2852	2
C₂×C₇×#388	2
C₂×C₇×#387	2
C₂×C₇×#386	2
C₂×C₇×#385	2
C₂×C₇×#383	2
C₂×#2852	7

Subgroups — C2×C7×#2852 #13213833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2852 #13213833