Subgroups — C₂²×C₇×#485 #13237719

Order 1792.

2358 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#485	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₂²⋊Q₈	—	2	6	no	—
14	C₂×C₇×C₂²⋊Q₈	C₂²	1	12	yes	—
15	C₂×C₇×#117	—	2	6	no	—
16	C₂×C₇×#117	C₂²	1	12	yes	—
17	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
18	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
22	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
23	C₂²×C₄²×C₇	—	2	1	no	—
24	C₇×#485	C₂²	1	64	yes	direct product factor
25	C₂×C₇×#107	C₂²	1	12	yes	complemented
26	C₂×C₇×#106	C₂²	1	12	yes	complemented
27	C₂×C₇×#103	C₂²	1	12	yes	complemented
28	C₂×C₇×#103	C₂²	1	12	yes	complemented
29	C₂×C₇×C₂²⋊C₈	C₂²	1	12	yes	—
30	C₂²×#485	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂⁵×C₇	D₄	1	1	yes	characteristic
32	C₂²×C₇×Q₈	—	2	3	no	—
33	C₂²×C₇×Q₈	—	2	4	no	—
34	C₂²×C₇×Q₈	—	2	4	no	—
35	C₂²×C₇×Q₈	C₂³	1	6	yes	—
36	C₂³×C₄×C₇	—	2	4	no	—
37	C₂³×C₄×C₇	—	2	4	no	—
38	C₂³×C₄×C₇	—	2	1	no	—
39	C₂³×C₄×C₇	—	2	1	no	—
40	C₂³×C₄×C₇	—	2	1	no	—
41	C₂³×C₄×C₇	—	2	1	no	—
42	C₂³×C₄×C₇	D₄	1	6	yes	—
43	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₇	C₂³	1	6	yes	—
45	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂²×C₇×C₈	—	2	4	no	—
47	C₂²×C₇×C₈	—	2	4	no	—
48	C₂²×C₇×C₈	C₂³	1	6	yes	—
49	C₂²×C₇×C₈	C₂³	1	6	yes	—
50	C₇×C₂²⋊Q₈	—	2	24	no	—
51	C₇×C₂²⋊Q₈	C₂³	1	16	yes	—
52	C₇×#117	—	2	24	no	—
53	C₇×#117	C₂³	1	16	yes	—
54	C₂×C₇×C₄⋊C₄	—	2	12	no	—
55	C₂×C₇×C₄⋊C₄	—	2	3	no	—
56	C₂×C₇×C₄⋊C₄	—	2	3	no	—
57	C₂×C₇×C₄⋊C₄	—	2	3	no	—
58	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
59	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
60	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
61	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
62	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
63	C₂×C₄²×C₇	—	2	12	no	—
64	C₇×#107	C₂³	1	16	yes	complemented
65	C₇×#106	C₂³	1	16	yes	complemented
66	C₇×#103	C₂³	1	16	yes	complemented
67	C₇×#103	C₂³	1	16	yes	complemented
68	C₇×C₂²⋊C₈	C₂³	1	16	yes	—
69	C₂²×C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic
70	C₂²×#117	C₂×C₇	1	1	yes	characteristic
71	C₂×#485	C₂×C₇	1	24	yes	direct product factor
72	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂²×#106	C₂×C₇	1	1	yes	characteristic, complemented
74	C₂²×#103	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂²×#103	C₂×C₇	1	1	yes	characteristic, complemented
76	C₂²×C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic
77	C₂⁴×C₇	—	2	4	no	—
78	C₂⁴×C₇	—	2	4	no	—
79	C₂⁴×C₇	—	2	4	no	—
80	C₂⁴×C₇	C₂×D₄	1	6	yes	—
81	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₇×Q₈	—	2	6	no	—
83	C₂×C₇×Q₈	—	2	12	no	—
84	C₂×C₇×Q₈	—	2	12	no	—
85	C₂×C₇×Q₈	—	4	12	no	—
86	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
87	C₂²×C₄×C₇	—	2	4	no	—
88	C₂²×C₄×C₇	—	2	24	no	—
89	C₂²×C₄×C₇	—	2	12	no	—
90	C₂²×C₄×C₇	—	2	12	no	—
91	C₂²×C₄×C₇	—	2	6	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	—	4	4	no	—
96	C₂²×C₄×C₇	—	4	4	no	—
97	C₂²×C₄×C₇	—	4	4	no	—
98	C₂²×C₄×C₇	—	4	4	no	—
99	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
100	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
101	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
102	C₂²×C₄×C₇	#41	1	4	yes	—
103	C₂²×C₄×C₇	#41	1	4	yes	—
104	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
105	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
106	C₂×C₇×C₈	—	2	12	no	—
107	C₂×C₇×C₈	—	2	12	no	—
108	C₂×C₇×C₈	C₂⁴	1	4	yes	—
109	C₂×C₇×C₈	C₂⁴	1	4	yes	—
110	C₇×C₄⋊C₄	—	2	16	no	—
111	C₇×C₄⋊C₄	—	2	6	no	—
112	C₇×C₄⋊C₄	—	2	6	no	—
113	C₇×C₄⋊C₄	—	2	6	no	—
114	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
115	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
116	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
117	C₇×C₂²⋊C₄	—	2	16	no	—
118	C₇×C₂²⋊C₄	—	2	16	no	—
119	C₄²×C₇	—	2	16	no	—
120	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
121	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
122	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
123	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
124	C₂×C₂²⋊Q₈	—	2	6	no	—
125	C₂×C₂²⋊Q₈	C₂²×C₇	1	12	yes	—
126	C₂×#117	—	2	6	no	—
127	C₂×#117	C₂²×C₇	1	12	yes	—
128	C₂²×C₄⋊C₄	—	2	1	no	—
129	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
130	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
131	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	—	2	1	no	—
133	C₂²×C₂²⋊C₄	—	2	1	no	—
134	C₂²×C₄²	—	2	1	no	—
135	#485	C₂²×C₇	1	64	yes	direct product factor
136	C₂×#107	C₂²×C₇	1	12	yes	complemented
137	C₂×#106	C₂²×C₇	1	12	yes	complemented
138	C₂×#103	C₂²×C₇	1	12	yes	complemented
139	C₂×#103	C₂²×C₇	1	12	yes	complemented
140	C₂×C₂²⋊C₈	C₂²×C₇	1	12	yes	—
141	C₂³×C₇	—	2	12	no	—
142	C₂³×C₇	—	2	12	no	—
143	C₂³×C₇	—	2	12	no	—
144	C₂³×C₇	—	4	16	no	—
145	C₂³×C₇	#123	1	4	yes	—
146	C₂³×C₇	#135	1	4	yes	—
147	C₂³×C₇	#137	1	4	yes	—
148	C₂³×C₇	C₂²×D₄	1	4	yes	—
149	C₂³×C₇	C₂²×D₄	1	3	yes	—
150	C₇×Q₈	—	2	4	no	—
151	C₇×Q₈	—	2	4	no	—
152	C₇×Q₈	—	4	12	no	—
153	C₂×C₄×C₇	—	2	12	no	—
154	C₂×C₄×C₇	—	2	24	no	—
155	C₂×C₄×C₇	—	2	4	no	—
156	C₂×C₄×C₇	—	2	4	no	—
157	C₂×C₄×C₇	—	2	4	no	—
158	C₂×C₄×C₇	—	2	4	no	—
159	C₂×C₄×C₇	—	2	4	no	—
160	C₂×C₄×C₇	—	2	4	no	—
161	C₂×C₄×C₇	—	4	12	no	—
162	C₂×C₄×C₇	—	4	12	no	—
163	C₂×C₄×C₇	—	4	12	no	—
164	C₂×C₄×C₇	—	4	12	no	—
165	C₂×C₄×C₇	#123	1	12	yes	—
166	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
167	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
168	C₂×C₄×C₇	C₂×#41	1	6	yes	—
169	C₂×C₄×C₇	C₂×#41	1	6	yes	—
170	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
171	C₇×C₈	—	2	4	no	—
172	C₇×C₈	—	2	4	no	—
173	C₂⁵	C₇×D₄	1	1	yes	characteristic
174	C₂²×Q₈	—	2	3	no	—
175	C₂²×Q₈	—	2	4	no	—
176	C₂²×Q₈	—	2	4	no	—
177	C₂²×Q₈	C₂³×C₇	1	6	yes	—
178	C₂³×C₄	—	2	4	no	—
179	C₂³×C₄	—	2	4	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	—	2	1	no	—
182	C₂³×C₄	—	2	1	no	—
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
185	C₂³×C₄	C₇×D₄	1	6	yes	—
186	C₂³×C₄	C₂³×C₇	1	6	yes	—
187	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	C₂³×C₇	1	6	yes	—
191	C₂²×C₈	C₂³×C₇	1	6	yes	—
192	C₂²⋊Q₈	—	2	24	no	—
193	C₂²⋊Q₈	C₂³×C₇	1	16	yes	—
194	#117	—	2	24	no	—
195	#117	C₂³×C₇	1	16	yes	—
196	C₂×C₄⋊C₄	—	2	12	no	—
197	C₂×C₄⋊C₄	—	2	3	no	—
198	C₂×C₄⋊C₄	—	2	3	no	—
199	C₂×C₄⋊C₄	—	2	3	no	—
200	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
201	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
202	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
203	C₂×C₂²⋊C₄	—	2	12	no	—
204	C₂×C₂²⋊C₄	—	2	12	no	—
205	C₂×C₄²	—	2	12	no	—
206	#107	C₂³×C₇	1	16	yes	complemented
207	#106	C₂³×C₇	1	16	yes	complemented
208	#103	C₂³×C₇	1	16	yes	complemented
209	#103	C₂³×C₇	1	16	yes	complemented
210	C₂²⋊C₈	C₂³×C₇	1	16	yes	—
211	C₂²×C₇	—	2	4	no	—
212	C₂²×C₇	—	2	4	no	—
213	C₂²×C₇	—	2	4	no	—
214	C₂²×C₇	—	4	24	no	—
215	C₂²×C₇	#485	1	16	yes	direct product factor
216	C₂²×C₇	C₂×#123	1	6	yes	—
217	C₂²×C₇	C₂×#135	1	6	yes	—
218	C₂²×C₇	C₂×#137	1	6	yes	—
219	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
220	C₄×C₇	—	2	4	no	—
221	C₄×C₇	—	4	4	no	—
222	C₄×C₇	—	4	4	no	—
223	C₄×C₇	—	4	4	no	—
224	C₄×C₇	—	4	4	no	—
225	C₄×C₇	C₂×#123	1	6	yes	—
226	C₄×C₇	C₂²×#41	1	1	yes	characteristic
227	C₄×C₇	C₂²×#41	1	1	yes	characteristic
228	C₂⁴	—	2	4	no	—
229	C₂⁴	—	2	4	no	—
230	C₂⁴	—	2	4	no	—
231	C₂⁴	C₂×C₇×D₄	1	6	yes	—
232	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
233	C₂×Q₈	—	2	6	no	—
234	C₂×Q₈	—	2	12	no	—
235	C₂×Q₈	—	2	12	no	—
236	C₂×Q₈	—	4	12	no	—
237	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	24	no	—
240	C₂²×C₄	—	2	12	no	—
241	C₂²×C₄	—	2	12	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	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
251	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
252	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
253	C₂²×C₄	C₇×#41	1	4	yes	—
254	C₂²×C₄	C₇×#41	1	4	yes	—
255	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
256	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
257	C₂×C₈	—	2	12	no	—
258	C₂×C₈	—	2	12	no	—
259	C₂×C₈	C₂⁴×C₇	1	4	yes	—
260	C₂×C₈	C₂⁴×C₇	1	4	yes	—
261	C₄⋊C₄	—	2	16	no	—
262	C₄⋊C₄	—	2	6	no	—
263	C₄⋊C₄	—	2	6	no	—
264	C₄⋊C₄	—	2	6	no	—
265	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
266	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
267	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
268	C₂²⋊C₄	—	2	16	no	—
269	C₂²⋊C₄	—	2	16	no	—
270	C₄²	—	2	16	no	—
271	C₂×C₇	—	4	4	no	—
272	C₂×C₇	C₂×#485	1	12	yes	direct product factor
273	C₂×C₇	C₂²×#123	1	1	yes	characteristic
274	C₂×C₇	C₂²×#135	1	1	yes	characteristic
275	C₂×C₇	C₂²×#137	1	1	yes	characteristic
276	C₂³	—	2	12	no	—
277	C₂³	—	2	12	no	—
278	C₂³	—	2	12	no	—
279	C₂³	—	4	16	no	—
280	C₂³	C₇×#123	1	4	yes	—
281	C₂³	C₇×#135	1	4	yes	—
282	C₂³	C₇×#137	1	4	yes	—
283	C₂³	C₂²×C₇×D₄	1	4	yes	—
284	C₂³	C₂²×C₇×D₄	1	3	yes	—
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	12	no	—
288	C₂×C₄	—	2	12	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	24	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	12	no	—
297	C₂×C₄	—	4	12	no	—
298	C₂×C₄	—	4	12	no	—
299	C₂×C₄	—	4	12	no	—
300	C₂×C₄	C₇×#123	1	12	yes	—
301	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
302	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
303	C₂×C₄	C₂×C₇×#41	1	6	yes	—
304	C₂×C₄	C₂×C₇×#41	1	6	yes	—
305	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
306	C₈	—	2	4	no	—
307	C₈	—	2	4	no	—
308	C₇	C₂²×#485	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	24	no	—
313	C₂²	C₇×#485	1	16	yes	direct product factor
314	C₂²	C₂×C₇×#123	1	6	yes	—
315	C₂²	C₂×C₇×#135	1	6	yes	—
316	C₂²	C₂×C₇×#137	1	6	yes	—
317	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
318	C₄	—	2	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	4	no	—
323	C₄	C₂×C₇×#123	1	6	yes	—
324	C₄	C₂²×C₇×#41	1	1	yes	characteristic
325	C₄	C₂²×C₇×#41	1	1	yes	characteristic
326	C₂	—	4	4	no	—
327	C₂	C₂×C₇×#485	1	12	yes	direct product factor
328	C₂	C₂²×C₇×#123	1	1	yes	characteristic
329	C₂	C₂²×C₇×#135	1	1	yes	characteristic
330	C₂	C₂²×C₇×#137	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#485	C₂	direct (N × H)
4. C₂²×C₇×#107	C₂	semidirect
5. C₂²×C₇×#106	C₂	semidirect
6. C₂²×C₇×#103	C₂	semidirect
7. C₂²×C₇×#103	C₂	semidirect
24. C₇×#485	C₂²	direct (N × H)
25. C₂×C₇×#107	C₂²	semidirect
26. C₂×C₇×#106	C₂²	semidirect
27. C₂×C₇×#103	C₂²	semidirect
28. C₂×C₇×#103	C₂²	semidirect
30. C₂²×#485	C₇	direct (N × H)
64. C₇×#107	C₂³	semidirect
65. C₇×#106	C₂³	semidirect
66. C₇×#103	C₂³	semidirect
67. C₇×#103	C₂³	semidirect
71. C₂×#485	C₂×C₇	direct (N × H)
72. C₂²×#107	C₂×C₇	semidirect
73. C₂²×#106	C₂×C₇	semidirect
74. C₂²×#103	C₂×C₇	semidirect
75. C₂²×#103	C₂×C₇	semidirect
135. #485	C₂²×C₇	direct (N × H)
136. C₂×#107	C₂²×C₇	semidirect
137. C₂×#106	C₂²×C₇	semidirect
138. C₂×#103	C₂²×C₇	semidirect
139. C₂×#103	C₂²×C₇	semidirect
206. #107	C₂³×C₇	semidirect
207. #106	C₂³×C₇	semidirect
208. #103	C₂³×C₇	semidirect
209. #103	C₂³×C₇	semidirect
215. C₂²×C₇	#485	direct (N × H)
272. C₂×C₇	C₂×#485	direct (N × H)
308. C₇	C₂²×#485	direct (N × H)
313. C₂²	C₇×#485	direct (N × H)
327. C₂	C₂×C₇×#485	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×C₂²⋊Q₈	2
C₂²×C₇×#117	2
C₂×C₇×#485	2
C₂²×C₇×#107	2
C₂²×C₇×#106	2
C₂²×C₇×#103	2
C₂²×C₇×C₂²⋊C₈	2
C₂²×#485	7

Subgroups — C22×C7×#485 #13237719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#485 #13237719