Subgroups — C₇×#22670 #13199886

Order 1792.

1122 conjugacy classes of subgroups.

The subgroup lattice has 342 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×D₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₇×#1943	C₂	1	2	yes	maximal, complemented
5	C₇×#1864	C₂	1	4	yes	maximal, complemented
6	C₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1802	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	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
13	C₇×#442	—	2	8	no	—
14	C₄×C₇×D₈	C₄	1	8	yes	complemented
15	C₂×C₇×#107	C₂²	1	1	yes	characteristic
16	C₂×C₇×#106	C₂²	1	2	yes	—
17	C₂×C₇×#106	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×#102	C₂²	1	4	yes	—
20	C₂×C₇×#102	C₂²	1	2	yes	—
21	C₂×C₇×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#389	C₂²	1	4	yes	—
24	C₇×#385	—	2	4	no	—
25	C₇×#384	C₂²	1	2	yes	—
26	C₇×#380	—	2	2	no	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₇×#340	C₂²	1	4	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	#22670	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	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	4	no	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	—	4	8	no	—
48	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#107	C₂×C₄	1	4	yes	complemented
56	C₇×#106	—	2	4	no	—
57	C₇×#106	—	4	2	no	—
58	C₇×C₄⋊C₈	—	2	2	no	—
59	C₇×#102	—	2	8	no	—
60	C₇×#102	C₂×C₄	1	8	yes	complemented
61	C₇×#97	—	2	4	no	—
62	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
63	C₇×#95	—	2	4	no	—
64	C₇×#95	D₄	1	2	yes	—
65	C₂×#442	C₂×C₇	1	2	yes	—
66	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
67	D₄×C₄⋊C₄	C₂×C₇	1	2	yes	complemented
68	#1943	C₂×C₇	1	2	yes	complemented
69	#1864	C₂×C₇	1	4	yes	complemented
70	#1844	C₂×C₇	1	1	yes	characteristic, complemented
71	#1802	C₂×C₇	1	2	yes	complemented
72	#1770	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	—	4	4	no	—
86	C₂²×C₄×C₇	—	4	4	no	—
87	C₂²×C₄×C₇	—	8	4	no	—
88	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
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	—	4	8	no	—
107	C₇×C₄⋊C₄	—	4	4	no	—
108	C₇×C₄⋊C₄	—	8	2	no	—
109	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
110	C₇×C₂²⋊C₄	—	2	8	no	—
111	C₇×C₂²⋊C₄	—	4	8	no	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	2	2	no	—
114	C₄²×C₇	C₂²×C₄	1	2	yes	—
115	C₄²×C₇	#41	1	2	yes	—
116	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
118	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
119	C₂²×C₄⋊C₄	—	2	2	no	—
120	#442	—	2	8	no	—
121	C₄×D₈	C₄×C₇	1	8	yes	complemented
122	C₂×#107	C₂²×C₇	1	1	yes	characteristic
123	C₂×#106	C₂²×C₇	1	2	yes	—
124	C₂×#106	C₂²×C₇	1	2	yes	—
125	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
126	C₂×#102	C₂²×C₇	1	4	yes	—
127	C₂×#102	C₂²×C₇	1	2	yes	—
128	C₂×#97	C₂²×C₇	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
130	#389	C₂²×C₇	1	4	yes	—
131	#385	—	2	4	no	—
132	#384	C₂²×C₇	1	2	yes	—
133	#380	—	2	2	no	—
134	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
135	#340	C₂²×C₇	1	4	yes	—
136	#336	C₂²×C₇	1	2	yes	—
137	C₂³×C₇	—	2	4	no	—
138	C₂³×C₇	—	2	4	no	—
139	C₂³×C₇	—	2	4	no	—
140	C₂³×C₇	—	4	2	no	—
141	C₂³×C₇	—	4	2	no	—
142	C₂³×C₇	—	4	2	no	—
143	C₂³×C₇	—	4	2	no	—
144	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₇×D₄	—	2	4	no	—
146	C₇×D₄	—	2	4	no	—
147	C₇×D₄	—	2	4	no	—
148	C₇×D₄	C₂×C₄⋊C₄	1	8	yes	complemented
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	2	4	no	—
152	C₂×C₄×C₇	—	2	4	no	—
153	C₂×C₄×C₇	—	2	4	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	16	no	—
157	C₂×C₄×C₇	—	4	4	no	—
158	C₂×C₄×C₇	—	4	4	no	—
159	C₂×C₄×C₇	—	4	4	no	—
160	C₂×C₄×C₇	—	8	16	no	—
161	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
163	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
164	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
170	C₇×C₈	—	2	2	no	—
171	C₇×C₈	—	4	2	no	—
172	C₇×C₈	C₂×C₄⋊C₄	1	4	yes	complemented
173	C₂²×D₄	C₂³×C₇	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	—	2	4	no	—
176	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
177	C₂×D₈	C₇×D₄	1	4	yes	—
178	C₂×D₈	C₇×Q₈	1	4	yes	—
179	C₂²×C₈	C₂³×C₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
182	C₄×D₄	—	2	8	no	—
183	C₄×D₄	—	2	4	no	—
184	C₄×D₄	—	4	4	no	—
185	C₄×D₄	C₂×C₄×C₇	1	8	yes	complemented
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	4	8	no	—
190	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	4	no	—
195	C₂×C₄²	C₂³×C₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
197	#107	C₂×C₄×C₇	1	4	yes	complemented
198	#106	—	2	4	no	—
199	#106	—	4	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	#102	—	2	8	no	—
202	#102	C₂×C₄×C₇	1	8	yes	complemented
203	#97	—	2	4	no	—
204	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
205	#95	—	2	4	no	—
206	#95	C₇×D₄	1	2	yes	—
207	C₂²×C₇	—	2	8	no	—
208	C₂²×C₇	—	4	4	no	—
209	C₂²×C₇	—	4	4	no	—
210	C₂²×C₇	—	4	4	no	—
211	C₂²×C₇	—	4	4	no	—
212	C₂²×C₇	—	4	4	no	—
213	C₂²×C₇	—	4	4	no	—
214	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₇	D₄²	1	1	yes	characteristic
216	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
217	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
218	C₂²×C₇	#577	1	2	yes	—
219	C₄×C₇	—	2	2	no	—
220	C₄×C₇	—	4	4	no	—
221	C₄×C₇	—	4	4	no	—
222	C₄×C₇	—	8	4	no	—
223	C₄×C₇	#442	1	4	yes	complemented
224	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
225	C₄×C₇	#520	1	1	yes	characteristic
226	C₄×C₇	#547	1	1	yes	characteristic
227	C₄×C₇	#554	1	1	yes	characteristic
228	C₂⁴	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
233	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
234	C₂×D₄	C₂×C₇×Q₈	1	4	yes	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	8	no	—
239	C₂²×C₄	—	4	8	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	8	4	no	—
243	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
244	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
246	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
247	D₈	C₇×C₄⋊C₄	1	16	yes	complemented
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
254	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
255	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
256	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	4	8	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	C₂²×C₄×C₇	1	2	yes	—
270	C₄²	C₇×#41	1	2	yes	—
271	C₂×C₇	—	4	8	no	—
272	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₇	C₂×#442	1	1	yes	characteristic
274	C₂×C₇	#2952	1	1	yes	characteristic
275	C₂×C₇	#3272	1	2	yes	—
276	C₂×C₇	#3384	1	2	yes	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	4	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	4	2	no	—
284	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
285	D₄	—	2	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	2	4	no	—
288	D₄	C₂×C₇×C₄⋊C₄	1	8	yes	complemented
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	16	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	16	no	—
301	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
302	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
303	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
304	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
307	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
308	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₂×C₇×C₄⋊C₄	1	4	yes	complemented
313	C₇	#22670	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	8	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	4	4	no	—
321	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
322	C₂²	C₇×D₄²	1	1	yes	characteristic
323	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
324	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
325	C₂²	C₇×#577	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	4	no	—
330	C₄	C₇×#442	1	4	yes	complemented
331	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
332	C₄	C₇×#520	1	1	yes	characteristic
333	C₄	C₇×#547	1	1	yes	characteristic
334	C₄	C₇×#554	1	1	yes	characteristic
335	C₂	—	4	8	no	—
336	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
337	C₂	C₂×C₇×#442	1	1	yes	characteristic
338	C₂	C₇×#2952	1	1	yes	characteristic
339	C₂	C₇×#3272	1	2	yes	—
340	C₂	C₇×#3384	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×D₄×C₄⋊C₄	C₂	semidirect
4. C₇×#1943	C₂	semidirect
5. C₇×#1864	C₂	semidirect
6. C₇×#1844	C₂	semidirect
7. C₇×#1802	C₂	semidirect
8. C₇×#1770	C₂	semidirect
14. C₄×C₇×D₈	C₄	semidirect
30. #22670	C₇	direct (N × H)
43. C₄×C₇×D₄	C₂×C₄	semidirect
55. C₇×#107	C₂×C₄	semidirect
60. C₇×#102	C₂×C₄	semidirect
62. C₄×C₇×C₈	C₂×C₄	semidirect
67. D₄×C₄⋊C₄	C₂×C₇	semidirect
68. #1943	C₂×C₇	semidirect
69. #1864	C₂×C₇	semidirect
70. #1844	C₂×C₇	semidirect
71. #1802	C₂×C₇	semidirect
72. #1770	C₂×C₇	semidirect
92. C₇×D₈	C₄⋊C₄	semidirect
121. C₄×D₈	C₄×C₇	semidirect
148. C₇×D₄	C₂×C₄⋊C₄	semidirect
172. C₇×C₈	C₂×C₄⋊C₄	semidirect
185. C₄×D₄	C₂×C₄×C₇	semidirect
197. #107	C₂×C₄×C₇	semidirect
202. #102	C₂×C₄×C₇	semidirect
204. C₄×C₈	C₂×C₄×C₇	semidirect
223. C₄×C₇	#442	semidirect
247. D₈	C₇×C₄⋊C₄	semidirect
288. D₄	C₂×C₇×C₄⋊C₄	semidirect
312. C₈	C₂×C₇×C₄⋊C₄	semidirect
313. C₇	#22670	direct (N × H)
330. C₄	C₇×#442	semidirect

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×D₄×C₄⋊C₄	2
C₇×#1943	2
C₇×#1864	2
C₇×#1844	2
C₇×#1802	2
C₇×#1770	2
#22670	7

Subgroups — C7×#22670 #13199886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22670 #13199886