Subgroups — C₇×#22682 #13199898

Order 1792.

1062 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#2349	C₂	1	2	yes	maximal, complemented
4	C₇×#1942	C₂	1	2	yes	maximal, complemented
5	C₇×#1864	C₂	1	4	yes	maximal, complemented
6	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1804	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
11	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
12	C₄×C₇×D₈	—	2	8	no	—
13	C₄×C₇×D₈	C₄	1	8	yes	complemented
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₂×C₇×#107	C₂²	1	1	yes	characteristic
16	C₂×C₇×#106	C₂²	1	2	yes	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×#102	C₂²	1	4	yes	—
19	C₂×C₇×#102	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₇×#389	C₂²	1	4	yes	—
23	C₇×#385	—	2	4	no	—
24	C₇×#382	C₂²	1	2	yes	—
25	C₇×#380	—	2	2	no	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₇×#95	—	2	2	no	—
28	C₇×#340	C₂²	1	4	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	#22682	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₇×D₄	C₂³	1	2	yes	—
32	C₂³×C₄×C₇	—	2	2	no	—
33	C₂³×C₄×C₇	—	2	4	no	—
34	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
35	C₂×C₇×D₈	D₄	1	4	yes	—
36	C₂×C₇×D₈	Q₈	1	4	yes	—
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₇×D₄	—	2	8	no	—
41	C₄×C₇×D₄	—	2	4	no	—
42	C₄×C₇×D₄	—	4	4	no	—
43	C₄×C₇×D₄	C₂×C₄	1	8	yes	complemented
44	C₂×C₇×C₄⋊C₄	—	2	4	no	—
45	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
50	C₂×C₄²×C₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₇×#107	—	2	4	no	—
53	C₇×#107	C₂×C₄	1	4	yes	complemented
54	C₇×#106	—	4	2	no	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×#102	—	2	8	no	—
57	C₇×#102	C₂×C₄	1	8	yes	complemented
58	C₄×C₇×C₈	—	2	4	no	—
59	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
60	C₇×#95	—	2	2	no	—
61	C₇×#95	—	2	4	no	—
62	C₇×#95	—	2	2	no	—
63	C₇×#95	D₄	1	1	yes	characteristic
64	C₇×#95	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈	C₂×C₇	1	2	yes	—
66	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
67	#2349	C₂×C₇	1	2	yes	complemented
68	#1942	C₂×C₇	1	2	yes	complemented
69	#1864	C₂×C₇	1	4	yes	complemented
70	#1846	C₂×C₇	1	1	yes	characteristic, complemented
71	#1804	C₂×C₇	1	2	yes	complemented
72	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂⁴×C₇	—	2	2	no	—
74	C₂×C₇×D₄	—	2	2	no	—
75	C₂×C₇×D₄	—	2	2	no	—
76	C₂×C₇×D₄	—	2	2	no	—
77	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
78	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
79	C₂×C₇×D₄	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	8	no	—
82	C₂²×C₄×C₇	—	2	4	no	—
83	C₂²×C₄×C₇	—	4	8	no	—
84	C₂²×C₄×C₇	—	4	8	no	—
85	C₂²×C₄×C₇	—	8	4	no	—
86	C₂²×C₄×C₇	—	8	4	no	—
87	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
92	C₇×D₈	C₄⋊C₄	1	16	yes	complemented
93	C₂×C₇×C₈	—	2	1	no	—
94	C₂×C₇×C₈	—	2	1	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	2	4	no	—
105	C₇×C₄⋊C₄	—	2	4	no	—
106	C₇×C₄⋊C₄	—	8	4	no	—
107	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
108	C₇×C₂²⋊C₄	—	2	8	no	—
109	C₇×C₂²⋊C₄	—	4	8	no	—
110	C₄²×C₇	—	2	2	no	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₇	#41	1	2	yes	—
114	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
115	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
116	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
117	C₄×D₈	—	2	8	no	—
118	C₄×D₈	C₄×C₇	1	8	yes	complemented
119	C₂×#107	C₂²×C₇	1	2	yes	—
120	C₂×#107	C₂²×C₇	1	1	yes	characteristic
121	C₂×#106	C₂²×C₇	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
123	C₂×#102	C₂²×C₇	1	4	yes	—
124	C₂×#102	C₂²×C₇	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
127	#389	C₂²×C₇	1	4	yes	—
128	#385	—	2	4	no	—
129	#382	C₂²×C₇	1	2	yes	—
130	#380	—	2	2	no	—
131	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
132	C₂×#95	—	2	2	no	—
133	#340	C₂²×C₇	1	4	yes	—
134	#336	C₂²×C₇	1	2	yes	—
135	C₂³×C₇	—	2	4	no	—
136	C₂³×C₇	—	2	4	no	—
137	C₂³×C₇	—	2	4	no	—
138	C₂³×C₇	—	4	4	no	—
139	C₂³×C₇	—	4	4	no	—
140	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₇×D₄	—	2	4	no	—
142	C₇×D₄	—	2	4	no	—
143	C₇×D₄	—	2	4	no	—
144	C₇×D₄	C₂×C₄⋊C₄	1	8	yes	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	4	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	4	4	no	—
154	C₂×C₄×C₇	—	4	4	no	—
155	C₂×C₄×C₇	—	4	4	no	—
156	C₂×C₄×C₇	—	4	4	no	—
157	C₂×C₄×C₇	—	8	8	no	—
158	C₂×C₄×C₇	—	8	16	no	—
159	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
160	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₇	#135	1	4	yes	—
162	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
168	C₇×C₈	—	2	2	no	—
169	C₇×C₈	—	4	2	no	—
170	C₇×C₈	C₂×C₄⋊C₄	1	4	yes	—
171	C₂²×D₄	C₂³×C₇	1	2	yes	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	2	4	no	—
174	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
175	C₂×D₈	C₇×D₄	1	4	yes	—
176	C₂×D₈	C₇×Q₈	1	4	yes	—
177	C₂²×C₈	C₂³×C₇	1	2	yes	—
178	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
180	C₄×D₄	—	2	8	no	—
181	C₄×D₄	—	2	4	no	—
182	C₄×D₄	—	4	4	no	—
183	C₄×D₄	C₂×C₄×C₇	1	8	yes	complemented
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
187	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	4	no	—
190	C₂×C₄²	C₂³×C₇	1	2	yes	—
191	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	C₂×C₄×C₇	1	4	yes	complemented
194	#106	—	4	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	#102	—	2	8	no	—
197	#102	C₂×C₄×C₇	1	8	yes	complemented
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
200	#95	—	2	2	no	—
201	#95	—	2	4	no	—
202	#95	—	2	2	no	—
203	#95	C₇×D₄	1	1	yes	characteristic
204	#95	C₇×D₄	1	1	yes	characteristic
205	C₂²×C₇	—	2	8	no	—
206	C₂²×C₇	—	4	8	no	—
207	C₂²×C₇	—	4	4	no	—
208	C₂²×C₇	—	4	4	no	—
209	C₂²×C₇	—	4	8	no	—
210	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₇	D₄²	1	1	yes	characteristic
212	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
213	C₂²×C₇	C₂×#135	1	2	yes	—
214	C₂²×C₇	#576	1	2	yes	—
215	C₄×C₇	—	2	2	no	—
216	C₄×C₇	—	2	2	no	—
217	C₄×C₇	—	2	4	no	—
218	C₄×C₇	—	4	4	no	—
219	C₄×C₇	—	4	2	no	—
220	C₄×C₇	—	8	4	no	—
221	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₄×C₇	#520	1	1	yes	characteristic
223	C₄×C₇	#547	1	1	yes	characteristic
224	C₄×C₇	#554	1	1	yes	characteristic
225	C₂⁴	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
230	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
231	C₂×D₄	C₂×C₇×Q₈	1	4	yes	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	8	no	—
236	C₂²×C₄	—	4	8	no	—
237	C₂²×C₄	—	8	4	no	—
238	C₂²×C₄	—	8	4	no	—
239	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
240	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
244	D₈	C₇×C₄⋊C₄	1	16	yes	complemented
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
251	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
252	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
253	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	8	4	no	—
259	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
260	C₂²⋊C₄	—	2	8	no	—
261	C₂²⋊C₄	—	4	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	C₂²×C₄×C₇	1	2	yes	—
265	C₄²	C₇×#41	1	2	yes	—
266	C₂×C₇	—	4	8	no	—
267	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
268	C₂×C₇	C₄×#135	1	1	yes	characteristic
269	C₂×C₇	#2948	1	1	yes	characteristic
270	C₂×C₇	#3280	1	2	yes	—
271	C₂×C₇	#3380	1	2	yes	—
272	C₂³	—	2	4	no	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
278	D₄	—	2	4	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	8	8	no	—
295	C₂×C₄	—	8	16	no	—
296	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
297	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
298	C₂×C₄	C₇×#135	1	4	yes	—
299	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
303	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
308	C₇	#22682	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	8	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	8	no	—
313	C₂²	—	4	8	no	—
314	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
315	C₂²	C₇×D₄²	1	1	yes	characteristic
316	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
317	C₂²	C₂×C₇×#135	1	2	yes	—
318	C₂²	C₇×#576	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	4	no	—
325	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
326	C₄	C₇×#520	1	1	yes	characteristic
327	C₄	C₇×#547	1	1	yes	characteristic
328	C₄	C₇×#554	1	1	yes	characteristic
329	C₂	—	4	8	no	—
330	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
331	C₂	C₄×C₇×#135	1	1	yes	characteristic
332	C₂	C₇×#2948	1	1	yes	characteristic
333	C₂	C₇×#3280	1	2	yes	—
334	C₂	C₇×#3380	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2349	C₂	semidirect
4. C₇×#1942	C₂	semidirect
5. C₇×#1864	C₂	semidirect
6. C₇×#1846	C₂	semidirect
7. C₇×#1804	C₂	semidirect
8. C₇×C₈×C₄⋊C₄	C₂	semidirect
13. C₄×C₇×D₈	C₄	semidirect
30. #22682	C₇	direct (N × H)
43. C₄×C₇×D₄	C₂×C₄	semidirect
53. C₇×#107	C₂×C₄	semidirect
57. C₇×#102	C₂×C₄	semidirect
59. C₄×C₇×C₈	C₂×C₄	semidirect
67. #2349	C₂×C₇	semidirect
68. #1942	C₂×C₇	semidirect
69. #1864	C₂×C₇	semidirect
70. #1846	C₂×C₇	semidirect
71. #1804	C₂×C₇	semidirect
72. C₈×C₄⋊C₄	C₂×C₇	semidirect
92. C₇×D₈	C₄⋊C₄	semidirect
118. C₄×D₈	C₄×C₇	semidirect
183. C₄×D₄	C₂×C₄×C₇	semidirect
193. #107	C₂×C₄×C₇	semidirect
197. #102	C₂×C₄×C₇	semidirect
199. C₄×C₈	C₂×C₄×C₇	semidirect
244. D₈	C₇×C₄⋊C₄	semidirect
308. C₇	#22682	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×C₄×C₇×D₈	2
C₇×#2349	2
C₇×#1942	2
C₇×#1864	2
C₇×#1846	2
C₇×#1804	2
C₇×C₈×C₄⋊C₄	2
#22682	7

Subgroups — C7×#22682 #13199898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22682 #13199898