Subgroups — C₇×#22672 #13199888

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#442	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×#1943	C₂	1	2	yes	maximal, complemented
5	C₇×#1862	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×#1770	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	C₇×#442	—	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×C₇×#97	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	#22672	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	—	4	2	no	—
53	C₇×#107	C₂×C₄	1	4	yes	complemented
54	C₇×#106	—	2	4	no	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×#102	—	2	8	no	—
57	C₇×#102	C₂×C₄	1	8	yes	complemented
58	C₇×#97	—	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	2	yes	—
64	C₂×#442	C₂×C₇	1	2	yes	—
65	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
66	#2349	C₂×C₇	1	2	yes	complemented
67	#1943	C₂×C₇	1	2	yes	complemented
68	#1862	C₂×C₇	1	4	yes	complemented
69	#1846	C₂×C₇	1	1	yes	characteristic, complemented
70	#1804	C₂×C₇	1	2	yes	complemented
71	#1770	C₂×C₇	1	1	yes	characteristic, complemented
72	C₂⁴×C₇	—	2	2	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	—	2	2	no	—
75	C₂×C₇×D₄	—	2	2	no	—
76	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
77	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
78	C₂×C₇×D₄	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₇	—	2	2	no	—
80	C₂²×C₄×C₇	—	2	8	no	—
81	C₂²×C₄×C₇	—	2	4	no	—
82	C₂²×C₄×C₇	—	4	8	no	—
83	C₂²×C₄×C₇	—	4	8	no	—
84	C₂²×C₄×C₇	—	8	4	no	—
85	C₂²×C₄×C₇	—	8	4	no	—
86	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₇×D₈	C₄⋊C₄	1	16	yes	complemented
91	C₂×C₇×C₈	—	2	1	no	—
92	C₂×C₇×C₈	—	2	1	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	—	4	2	no	—
104	C₇×C₄⋊C₄	—	8	4	no	—
105	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
106	C₇×C₂²⋊C₄	—	2	8	no	—
107	C₇×C₂²⋊C₄	—	4	8	no	—
108	C₄²×C₇	—	2	2	no	—
109	C₄²×C₇	—	2	2	no	—
110	C₄²×C₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₇	#41	1	2	yes	—
112	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
114	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
115	#442	—	2	8	no	—
116	C₄×D₈	C₄×C₇	1	8	yes	complemented
117	C₂×#107	C₂²×C₇	1	2	yes	—
118	C₂×#107	C₂²×C₇	1	1	yes	characteristic
119	C₂×#106	C₂²×C₇	1	2	yes	—
120	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
121	C₂×#102	C₂²×C₇	1	4	yes	—
122	C₂×#102	C₂²×C₇	1	2	yes	—
123	C₂×#97	C₂²×C₇	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
125	#389	C₂²×C₇	1	4	yes	—
126	#385	—	2	4	no	—
127	#382	C₂²×C₇	1	2	yes	—
128	#380	—	2	2	no	—
129	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
130	C₂×#95	—	2	2	no	—
131	#340	C₂²×C₇	1	4	yes	—
132	#336	C₂²×C₇	1	2	yes	—
133	C₂³×C₇	—	2	4	no	—
134	C₂³×C₇	—	2	4	no	—
135	C₂³×C₇	—	2	4	no	—
136	C₂³×C₇	—	4	4	no	—
137	C₂³×C₇	—	4	4	no	—
138	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₇×D₄	—	2	4	no	—
140	C₇×D₄	—	2	4	no	—
141	C₇×D₄	—	2	4	no	—
142	C₇×D₄	C₂×C₄⋊C₄	1	8	yes	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	4	no	—
146	C₂×C₄×C₇	—	2	4	no	—
147	C₂×C₄×C₇	—	2	4	no	—
148	C₂×C₄×C₇	—	2	4	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	—	4	4	no	—
151	C₂×C₄×C₇	—	4	4	no	—
152	C₂×C₄×C₇	—	4	4	no	—
153	C₂×C₄×C₇	—	8	8	no	—
154	C₂×C₄×C₇	—	8	16	no	—
155	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
157	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
163	C₇×C₈	—	2	2	no	—
164	C₇×C₈	—	4	2	no	—
165	C₇×C₈	C₂×C₄⋊C₄	1	4	yes	—
166	C₂²×D₄	C₂³×C₇	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	—	2	4	no	—
169	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
170	C₂×D₈	C₇×D₄	1	4	yes	—
171	C₂×D₈	C₇×Q₈	1	4	yes	—
172	C₂²×C₈	C₂³×C₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
175	C₄×D₄	—	2	8	no	—
176	C₄×D₄	—	2	4	no	—
177	C₄×D₄	—	4	4	no	—
178	C₄×D₄	C₂×C₄×C₇	1	8	yes	complemented
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₄²	C₂³×C₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
187	#107	—	4	2	no	—
188	#107	C₂×C₄×C₇	1	4	yes	complemented
189	#106	—	2	4	no	—
190	C₄⋊C₈	—	2	2	no	—
191	#102	—	2	8	no	—
192	#102	C₂×C₄×C₇	1	8	yes	complemented
193	#97	—	2	4	no	—
194	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
195	#95	—	2	2	no	—
196	#95	—	2	4	no	—
197	#95	—	2	2	no	—
198	#95	C₇×D₄	1	2	yes	—
199	C₂²×C₇	—	2	8	no	—
200	C₂²×C₇	—	4	8	no	—
201	C₂²×C₇	—	4	4	no	—
202	C₂²×C₇	—	4	4	no	—
203	C₂²×C₇	—	4	8	no	—
204	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₇	D₄²	1	1	yes	characteristic
206	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
207	C₂²×C₇	#575	1	2	yes	—
208	C₂²×C₇	#576	1	2	yes	—
209	C₄×C₇	—	2	2	no	—
210	C₄×C₇	—	2	2	no	—
211	C₄×C₇	—	4	4	no	—
212	C₄×C₇	—	4	4	no	—
213	C₄×C₇	—	8	4	no	—
214	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
215	C₄×C₇	#520	1	1	yes	characteristic
216	C₄×C₇	#547	1	1	yes	characteristic
217	C₄×C₇	#554	1	1	yes	characteristic
218	C₂⁴	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
223	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
224	C₂×D₄	C₂×C₇×Q₈	1	4	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	—	4	8	no	—
230	C₂²×C₄	—	8	4	no	—
231	C₂²×C₄	—	8	4	no	—
232	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
233	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
236	D₈	C₇×C₄⋊C₄	1	16	yes	complemented
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
244	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
245	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	8	4	no	—
251	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	4	8	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂²×C₄×C₇	1	2	yes	—
257	C₄²	C₇×#41	1	2	yes	—
258	C₂×C₇	—	4	8	no	—
259	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
260	C₂×C₇	#2943	1	1	yes	characteristic
261	C₂×C₇	#2948	1	1	yes	characteristic
262	C₂×C₇	#3273	1	2	yes	—
263	C₂×C₇	#3389	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	8	no	—
285	C₂×C₄	—	8	16	no	—
286	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
287	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
288	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
292	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
293	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	4	2	no	—
296	C₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
297	C₇	#22672	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	2	8	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	4	8	no	—
303	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
304	C₂²	C₇×D₄²	1	1	yes	characteristic
305	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
306	C₂²	C₇×#575	1	2	yes	—
307	C₂²	C₇×#576	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	4	no	—
313	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
314	C₄	C₇×#520	1	1	yes	characteristic
315	C₄	C₇×#547	1	1	yes	characteristic
316	C₄	C₇×#554	1	1	yes	characteristic
317	C₂	—	4	8	no	—
318	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
319	C₂	C₇×#2943	1	1	yes	characteristic
320	C₂	C₇×#2948	1	1	yes	characteristic
321	C₂	C₇×#3273	1	2	yes	—
322	C₂	C₇×#3389	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×#1943	C₂	semidirect
5. C₇×#1862	C₂	semidirect
6. C₇×#1846	C₂	semidirect
7. C₇×#1804	C₂	semidirect
8. C₇×#1770	C₂	semidirect
13. C₄×C₇×D₈	C₄	semidirect
30. #22672	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
66. #2349	C₂×C₇	semidirect
67. #1943	C₂×C₇	semidirect
68. #1862	C₂×C₇	semidirect
69. #1846	C₂×C₇	semidirect
70. #1804	C₂×C₇	semidirect
71. #1770	C₂×C₇	semidirect
90. C₇×D₈	C₄⋊C₄	semidirect
116. C₄×D₈	C₄×C₇	semidirect
178. C₄×D₄	C₂×C₄×C₇	semidirect
188. #107	C₂×C₄×C₇	semidirect
192. #102	C₂×C₄×C₇	semidirect
194. C₄×C₈	C₂×C₄×C₇	semidirect
236. D₈	C₇×C₄⋊C₄	semidirect
297. C₇	#22672	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×C₇×#442	2
C₂×C₄×C₇×D₈	2
C₇×#2349	2
C₇×#1943	2
C₇×#1862	2
C₇×#1846	2
C₇×#1804	2
C₇×#1770	2
#22672	7

Subgroups — C7×#22672 #13199888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22672 #13199888