Subgroups — C₇×#23188 #13200404

Order 1792.

1226 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#442	C₂	1	2	yes	maximal, complemented
3	C₇×#2366	C₂	1	2	yes	maximal, complemented
4	C₇×#1982	C₂	1	2	yes	maximal, complemented
5	C₇×#1972	C₂	1	2	yes	maximal, complemented
6	C₇×#1879	C₂	1	4	yes	maximal, complemented
7	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	2	yes	—
10	C₂²×C₇×D₈	C₂²	1	2	yes	—
11	C₂×C₇×#127	C₂²	1	2	yes	—
12	C₂×C₄×C₇×D₄	—	2	2	no	—
13	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
14	C₇×#493	C₄	1	8	yes	complemented
15	C₇×#442	—	2	8	no	—
16	C₂×C₇×#106	C₂²	1	1	yes	characteristic
17	C₂×C₇×#106	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×#102	C₂²	1	4	yes	—
20	C₂×C₇×#102	C₂²	1	4	yes	—
21	C₂×C₇×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#390	—	2	2	no	—
24	C₇×#390	C₂²	1	4	yes	—
25	C₇×#386	—	2	4	no	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₇×#340	C₂²	1	2	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	#23188	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₇×D₄	—	2	2	no	—
32	C₂²×C₇×D₄	—	2	2	no	—
33	C₂²×C₇×D₄	C₂³	1	4	yes	—
34	C₂³×C₄×C₇	—	2	4	no	—
35	C₂×C₇×D₈	—	2	4	no	—
36	C₂×C₇×D₈	—	2	4	no	—
37	C₂×C₇×D₈	—	4	2	no	—
38	C₂×C₇×D₈	—	4	2	no	—
39	C₂×C₇×D₈	C₂×C₄	1	8	yes	complemented
40	C₂×C₇×D₈	C₂×C₄	1	8	yes	complemented
41	C₂²×C₇×C₈	C₂³	1	2	yes	—
42	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₇×#127	—	2	4	no	—
45	C₇×#127	C₂×C₄	1	8	yes	complemented
46	C₄×C₇×D₄	—	2	8	no	—
47	C₄×C₇×D₄	—	4	4	no	—
48	C₄×C₇×D₄	—	4	8	no	—
49	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
53	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
54	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
55	C₂×C₄²×C₇	—	2	2	no	—
56	C₂×C₄²×C₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₇×#106	—	2	2	no	—
59	C₇×#106	D₄	1	4	yes	complemented
60	C₇×C₄⋊C₈	—	2	2	no	—
61	C₇×#102	—	2	8	no	—
62	C₇×#102	—	4	4	no	—
63	C₇×#97	—	2	4	no	—
64	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
65	C₇×#95	—	2	2	no	—
66	C₇×#95	D₄	1	2	yes	—
67	C₂×#493	C₂×C₇	1	1	yes	characteristic
68	C₂×#442	C₂×C₇	1	2	yes	complemented
69	#2366	C₂×C₇	1	2	yes	complemented
70	#1982	C₂×C₇	1	2	yes	complemented
71	#1972	C₂×C₇	1	2	yes	complemented
72	#1879	C₂×C₇	1	4	yes	complemented
73	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
74	#1770	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂⁴×C₇	—	2	4	no	—
76	C₂×C₇×D₄	—	2	8	no	—
77	C₂×C₇×D₄	—	2	8	no	—
78	C₂×C₇×D₄	—	2	4	no	—
79	C₂×C₇×D₄	—	2	4	no	—
80	C₂×C₇×D₄	—	2	4	no	—
81	C₂×C₇×D₄	—	2	4	no	—
82	C₂×C₇×D₄	—	2	4	no	—
83	C₂×C₇×D₄	—	4	4	no	—
84	C₂×C₇×D₄	—	4	4	no	—
85	C₂×C₇×D₄	—	8	2	no	—
86	C₂×C₇×D₄	—	8	2	no	—
87	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
88	C₂²×C₄×C₇	—	2	2	no	—
89	C₂²×C₄×C₇	—	2	2	no	—
90	C₂²×C₄×C₇	—	4	8	no	—
91	C₂²×C₄×C₇	—	4	8	no	—
92	C₂²×C₄×C₇	—	8	4	no	—
93	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
97	C₇×D₈	—	2	16	no	—
98	C₇×D₈	—	4	8	no	—
99	C₂×C₇×C₈	—	2	1	no	—
100	C₂×C₇×C₈	—	2	1	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₇×C₈	#41	1	2	yes	—
108	C₇×C₄⋊C₄	—	2	4	no	—
109	C₇×C₄⋊C₄	—	2	2	no	—
110	C₇×C₄⋊C₄	—	2	2	no	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
113	C₇×C₂²⋊C₄	—	4	8	no	—
114	C₇×C₂²⋊C₄	—	4	8	no	—
115	C₇×C₂²⋊C₄	—	8	4	no	—
116	C₄²×C₇	—	2	2	no	—
117	C₄²×C₇	—	2	2	no	—
118	C₄²×C₇	—	4	4	no	—
119	C₄²×C₇	C₂²×C₄	1	2	yes	—
120	C₄²×C₇	#41	1	2	yes	—
121	C₂²×D₈	C₂²×C₇	1	2	yes	—
122	C₂²×D₈	C₂²×C₇	1	2	yes	—
123	C₂×#127	C₂²×C₇	1	2	yes	—
124	C₂×C₄×D₄	—	2	2	no	—
125	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
126	#493	C₄×C₇	1	8	yes	complemented
127	#442	—	2	8	no	—
128	C₂×#106	C₂²×C₇	1	1	yes	characteristic
129	C₂×#106	C₂²×C₇	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
131	C₂×#102	C₂²×C₇	1	4	yes	—
132	C₂×#102	C₂²×C₇	1	4	yes	—
133	C₂×#97	C₂²×C₇	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
135	#390	—	2	2	no	—
136	#390	C₂²×C₇	1	4	yes	—
137	#386	—	2	4	no	—
138	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
139	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
140	#340	C₂²×C₇	1	2	yes	—
141	#336	C₂²×C₇	1	2	yes	—
142	C₂³×C₇	—	2	8	no	—
143	C₂³×C₇	—	4	4	no	—
144	C₂³×C₇	—	4	4	no	—
145	C₂³×C₇	—	4	4	no	—
146	C₂³×C₇	—	4	4	no	—
147	C₂³×C₇	—	4	4	no	—
148	C₂³×C₇	—	4	4	no	—
149	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₇×D₄	—	2	8	no	—
151	C₇×D₄	—	2	8	no	—
152	C₇×D₄	—	4	4	no	—
153	C₇×D₄	—	4	4	no	—
154	C₇×D₄	—	4	16	no	—
155	C₇×D₄	—	8	8	no	—
156	C₂×C₄×C₇	—	2	2	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	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	4	2	no	—
164	C₂×C₄×C₇	—	4	2	no	—
165	C₂×C₄×C₇	—	4	2	no	—
166	C₂×C₄×C₇	—	8	16	no	—
167	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
169	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
170	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
176	C₇×C₈	—	2	2	no	—
177	C₇×C₈	—	4	2	no	—
178	C₇×C₈	C₄×D₄	1	4	yes	complemented
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	C₂³×C₇	1	4	yes	—
182	C₂³×C₄	—	2	4	no	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂×D₈	—	4	2	no	—
187	C₂×D₈	C₂×C₄×C₇	1	8	yes	complemented
188	C₂×D₈	C₂×C₄×C₇	1	8	yes	complemented
189	C₂²×C₈	C₂³×C₇	1	2	yes	—
190	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
192	#127	—	2	4	no	—
193	#127	C₂×C₄×C₇	1	8	yes	complemented
194	C₄×D₄	—	2	8	no	—
195	C₄×D₄	—	4	4	no	—
196	C₄×D₄	—	4	8	no	—
197	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
206	#106	—	2	2	no	—
207	#106	C₇×D₄	1	4	yes	complemented
208	C₄⋊C₈	—	2	2	no	—
209	#102	—	2	8	no	—
210	#102	—	4	4	no	—
211	#97	—	2	4	no	—
212	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
213	#95	—	2	2	no	—
214	#95	C₇×D₄	1	2	yes	—
215	C₂²×C₇	—	4	8	no	—
216	C₂²×C₇	—	4	8	no	—
217	C₂²×C₇	—	4	8	no	—
218	C₂²×C₇	—	8	4	no	—
219	C₂²×C₇	—	8	4	no	—
220	C₂²×C₇	—	8	4	no	—
221	C₂²×C₇	—	8	4	no	—
222	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂²×C₇	D₄²	1	1	yes	characteristic
224	C₂²×C₇	#550	1	1	yes	characteristic
225	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
226	C₂²×C₇	#576	1	2	yes	—
227	C₄×C₇	—	2	2	no	—
228	C₄×C₇	—	4	4	no	—
229	C₄×C₇	—	4	4	no	—
230	C₄×C₇	—	8	2	no	—
231	C₄×C₇	#442	1	4	yes	complemented
232	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
233	C₄×C₇	#518	1	1	yes	characteristic
234	C₄×C₇	#550	1	1	yes	characteristic
235	C₄×C₇	#559	1	1	yes	characteristic
236	C₂⁴	—	2	4	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	8	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	4	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	8	2	no	—
247	C₂×D₄	—	8	2	no	—
248	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	—	4	8	no	—
253	C₂²×C₄	—	8	4	no	—
254	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
255	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
257	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
258	D₈	—	2	16	no	—
259	D₈	—	4	8	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
266	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
267	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
268	C₂×C₈	C₇×#41	1	2	yes	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
274	C₂²⋊C₄	—	4	8	no	—
275	C₂²⋊C₄	—	4	8	no	—
276	C₂²⋊C₄	—	8	4	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	4	4	no	—
280	C₄²	C₂²×C₄×C₇	1	2	yes	—
281	C₄²	C₇×#41	1	2	yes	—
282	C₂×C₇	—	8	8	no	—
283	C₂×C₇	#2366	1	1	yes	characteristic
284	C₂×C₇	C₂×#442	1	1	yes	characteristic
285	C₂×C₇	#2948	1	1	yes	characteristic
286	C₂×C₇	#3274	1	2	yes	—
287	C₂×C₇	#3399	1	2	yes	—
288	C₂³	—	2	8	no	—
289	C₂³	—	4	4	no	—
290	C₂³	—	4	4	no	—
291	C₂³	—	4	4	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	4	4	no	—
295	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
296	D₄	—	2	8	no	—
297	D₄	—	2	8	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	16	no	—
301	D₄	—	8	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	16	no	—
313	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
314	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
315	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
316	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
318	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
319	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
320	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
321	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	4	2	no	—
324	C₈	C₄×C₇×D₄	1	4	yes	complemented
325	C₇	#23188	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	4	8	no	—
327	C₂²	—	4	8	no	—
328	C₂²	—	4	8	no	—
329	C₂²	—	8	4	no	—
330	C₂²	—	8	4	no	—
331	C₂²	—	8	4	no	—
332	C₂²	—	8	4	no	—
333	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
334	C₂²	C₇×D₄²	1	1	yes	characteristic
335	C₂²	C₇×#550	1	1	yes	characteristic
336	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
337	C₂²	C₇×#576	1	2	yes	—
338	C₄	—	2	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	4	4	no	—
341	C₄	—	8	2	no	—
342	C₄	C₇×#442	1	4	yes	complemented
343	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
344	C₄	C₇×#518	1	1	yes	characteristic
345	C₄	C₇×#550	1	1	yes	characteristic
346	C₄	C₇×#559	1	1	yes	characteristic
347	C₂	—	8	8	no	—
348	C₂	C₇×#2366	1	1	yes	characteristic
349	C₂	C₂×C₇×#442	1	1	yes	characteristic
350	C₂	C₇×#2948	1	1	yes	characteristic
351	C₂	C₇×#3274	1	2	yes	—
352	C₂	C₇×#3399	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	C₂×C₇×#493	C₂
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
2. C₂×C₇×#442	C₂	semidirect
3. C₇×#2366	C₂	semidirect
4. C₇×#1982	C₂	semidirect
5. C₇×#1972	C₂	semidirect
6. C₇×#1879	C₂	semidirect
7. C₄×C₇×#106	C₂	semidirect
8. C₇×#1770	C₂	semidirect
14. C₇×#493	C₄	semidirect
30. #23188	C₇	direct (N × H)
39. C₂×C₇×D₈	C₂×C₄	semidirect
40. C₂×C₇×D₈	C₂×C₄	semidirect
45. C₇×#127	C₂×C₄	semidirect
59. C₇×#106	D₄	semidirect
64. C₄×C₇×C₈	C₂×C₄	semidirect
68. C₂×#442	C₂×C₇	semidirect
69. #2366	C₂×C₇	semidirect
70. #1982	C₂×C₇	semidirect
71. #1972	C₂×C₇	semidirect
72. #1879	C₂×C₇	semidirect
73. C₄×#106	C₂×C₇	semidirect
74. #1770	C₂×C₇	semidirect
126. #493	C₄×C₇	semidirect
178. C₇×C₈	C₄×D₄	semidirect
187. C₂×D₈	C₂×C₄×C₇	semidirect
188. C₂×D₈	C₂×C₄×C₇	semidirect
193. #127	C₂×C₄×C₇	semidirect
207. #106	C₇×D₄	semidirect
212. C₄×C₈	C₂×C₄×C₇	semidirect
231. C₄×C₇	#442	semidirect
324. C₈	C₄×C₇×D₄	semidirect
325. C₇	#23188	direct (N × H)
342. 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×#493	2
C₂×C₇×#442	2
C₇×#2366	2
C₇×#1982	2
C₇×#1972	2
C₇×#1879	2
C₄×C₇×#106	2
C₇×#1770	2
#23188	7

Subgroups — C7×#23188 #13200404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23188 #13200404