Subgroups — C₇×#23833 #13201049

Order 1792.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2955	C₂	1	1	yes	maximal, characteristic
3	C₇×#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2080	C₂	1	2	yes	maximal, complemented
6	C₇×#2053	C₂	1	1	yes	maximal, characteristic
7	C₇×#2050	C₂	1	1	yes	maximal, characteristic
8	C₇×#2031	C₂	1	2	yes	maximal, complemented
9	C₇×#1764	C₂	1	2	yes	maximal, complemented
10	C₇×#1757	C₂	1	2	yes	maximal, complemented
11	C₇×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#575	C₂²	1	1	yes	characteristic
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₇×#533	C₂²	1	1	yes	characteristic
15	C₇×#446	—	2	2	no	—
16	C₇×C₈×Q₈	—	2	2	no	—
17	C₇×#444	—	2	2	no	—
18	C₇×#443	—	2	2	no	—
19	C₇×#430	C₂²	1	1	yes	characteristic
20	C₇×#424	C₂²	1	1	yes	characteristic
21	C₇×#424	C₂²	1	1	yes	characteristic
22	C₇×#423	C₂²	1	1	yes	characteristic
23	C₇×#421	C₂²	1	2	yes	—
24	C₂×C₇×#104	C₂²	1	2	yes	—
25	C₇×#416	C₂²	1	2	yes	—
26	C₇×#413	C₂²	1	2	yes	—
27	C₇×#410	C₂²	1	2	yes	—
28	C₇×#405	C₂²	1	2	yes	—
29	C₇×#405	C₂²	1	1	yes	characteristic
30	C₇×#405	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#344	C₂²	1	2	yes	—
33	C₇×#344	C₂²	1	2	yes	—
34	C₇×#341	C₂²	1	2	yes	—
35	C₇×#339	C₂²	1	2	yes	—
36	C₇×#338	C₂²	1	1	yes	characteristic
37	C₇×H(3,ℤ/4)	C₂²	1	2	yes	—
38	C₇×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	#23833	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇×#41	C₂³	1	2	yes	—
41	C₇×#137	—	2	2	no	—
42	C₇×AGL(1,ℤ/8)	—	2	2	no	—
43	C₇×#135	—	2	2	no	—
44	C₇×#131	—	2	2	no	—
45	C₇×#131	—	2	2	no	—
46	C₇×#131	—	2	2	no	—
47	C₇×#131	—	2	2	no	—
48	C₂×C₇×M₁₆	—	2	2	no	—
49	C₂×C₇×M₁₆	C₂³	1	2	yes	—
50	C₂×C₇×M₁₆	C₂³	1	2	yes	—
51	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
53	C₂²×C₇×C₈	—	2	2	no	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₇×C₄⋊Q₈	—	2	1	no	—
57	C₇×#125	—	2	1	no	—
58	C₇×C₂²⋊Q₈	—	2	2	no	—
59	C₄×C₇×Q₈	—	2	2	no	—
60	C₇×#117	C₂³	1	2	yes	—
61	C₇×#117	C₂³	1	1	yes	characteristic
62	C₇×#117	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
64	C₇×#108	—	2	2	no	—
65	C₇×C₄⋊C₈	—	2	2	no	—
66	C₇×C₄⋊C₈	—	2	2	no	—
67	C₇×C₄⋊C₈	—	2	2	no	—
68	C₇×#104	—	2	4	no	—
69	C₇×#104	—	2	2	no	—
70	C₇×#104	—	2	2	no	—
71	C₇×#103	—	2	2	no	—
72	C₇×#102	—	2	2	no	—
73	C₇×#101	—	2	2	no	—
74	C₇×#100	—	2	2	no	—
75	C₇×#99	—	2	4	no	—
76	C₇×#97	—	2	1	no	—
77	C₇×#97	—	2	2	no	—
78	C₇×#97	D₄	1	2	yes	—
79	C₄×C₇×C₈	—	2	1	no	—
80	C₄×C₇×C₈	—	2	1	no	—
81	C₄×C₇×C₈	—	2	1	no	—
82	C₄×C₇×C₈	—	2	2	no	—
83	C₄×C₇×C₈	D₄	1	2	yes	—
84	#2960	C₂×C₇	1	1	yes	characteristic, complemented
85	#2955	C₂×C₇	1	1	yes	characteristic
86	#2103	C₂×C₇	1	1	yes	characteristic, complemented
87	#2102	C₂×C₇	1	1	yes	characteristic, complemented
88	#2080	C₂×C₇	1	2	yes	complemented
89	#2053	C₂×C₇	1	1	yes	characteristic
90	#2050	C₂×C₇	1	1	yes	characteristic
91	#2031	C₂×C₇	1	2	yes	complemented
92	#1764	C₂×C₇	1	2	yes	complemented
93	#1757	C₂×C₇	1	2	yes	complemented
94	#1747	C₂×C₇	1	1	yes	characteristic, complemented
95	C₇×#41	—	2	2	no	—
96	C₇×#41	—	2	2	no	—
97	C₇×#41	—	2	2	no	—
98	C₇×#41	—	2	2	no	—
99	C₂×C₇×Q₈	—	2	2	no	—
100	C₂×C₇×Q₈	#41	1	2	yes	—
101	C₂×C₇×D₄	—	2	2	no	—
102	C₂×C₇×D₄	#41	1	2	yes	—
103	C₂²×C₄×C₇	—	2	1	no	—
104	C₂²×C₄×C₇	—	2	2	no	—
105	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
106	C₇×Q₁₆	—	2	2	no	—
107	C₇×SD₁₆	—	2	4	no	—
108	C₇×D₈	—	2	2	no	—
109	C₇×M₁₆	—	2	2	no	—
110	C₇×M₁₆	—	2	2	no	—
111	C₇×M₁₆	—	2	2	no	—
112	C₇×M₁₆	—	2	2	no	—
113	C₇×M₁₆	—	2	2	no	—
114	C₇×M₁₆	—	2	2	no	—
115	C₇×M₁₆	—	4	2	no	—
116	C₇×M₁₆	—	4	2	no	—
117	C₂×C₇×C₈	—	2	1	no	—
118	C₂×C₇×C₈	—	2	1	no	—
119	C₂×C₇×C₈	—	2	2	no	—
120	C₂×C₇×C₈	—	2	2	no	—
121	C₂×C₇×C₈	—	2	2	no	—
122	C₂×C₇×C₈	—	4	2	no	—
123	C₂×C₇×C₈	—	4	2	no	—
124	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₇×C₈	#41	1	2	yes	—
128	C₂×C₇×C₈	#41	1	2	yes	—
129	C₇×C₄⋊C₄	—	2	4	no	—
130	C₇×C₄⋊C₄	—	2	2	no	—
131	C₇×C₄⋊C₄	—	2	1	no	—
132	C₇×C₄⋊C₄	—	2	1	no	—
133	C₇×C₄⋊C₄	—	2	2	no	—
134	C₇×C₂²⋊C₄	—	2	1	no	—
135	C₇×C₂²⋊C₄	—	2	2	no	—
136	C₇×C₂²⋊C₄	#41	1	2	yes	—
137	C₄²×C₇	—	2	1	no	—
138	C₄²×C₇	—	2	1	no	—
139	C₄²×C₇	—	2	2	no	—
140	C₄²×C₇	C₂×D₄	1	2	yes	—
141	C₄²×C₇	#41	1	1	yes	characteristic
142	C₄²×C₇	#41	1	1	yes	characteristic
143	#575	C₂²×C₇	1	1	yes	characteristic
144	C₂×#131	C₂²×C₇	1	2	yes	—
145	#533	C₂²×C₇	1	1	yes	characteristic
146	#446	—	2	2	no	—
147	C₈×Q₈	—	2	2	no	—
148	#444	—	2	2	no	—
149	#443	—	2	2	no	—
150	#430	C₂²×C₇	1	1	yes	characteristic
151	#424	C₂²×C₇	1	1	yes	characteristic
152	#424	C₂²×C₇	1	1	yes	characteristic
153	#423	C₂²×C₇	1	1	yes	characteristic
154	#421	C₂²×C₇	1	2	yes	—
155	C₂×#104	C₂²×C₇	1	2	yes	—
156	#416	C₂²×C₇	1	2	yes	—
157	#413	C₂²×C₇	1	2	yes	—
158	#410	C₂²×C₇	1	2	yes	—
159	#405	C₂²×C₇	1	1	yes	characteristic
160	#405	C₂²×C₇	1	2	yes	—
161	#405	C₂²×C₇	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
163	#344	C₂²×C₇	1	2	yes	—
164	#344	C₂²×C₇	1	2	yes	—
165	#341	C₂²×C₇	1	2	yes	—
166	#339	C₂²×C₇	1	2	yes	—
167	#338	C₂²×C₇	1	1	yes	characteristic
168	H(3,ℤ/4)	C₂²×C₇	1	1	yes	characteristic
169	H(3,ℤ/4)	C₂²×C₇	1	2	yes	—
170	C₂³×C₇	—	2	2	no	—
171	C₂³×C₇	C₂×#41	1	1	yes	characteristic
172	C₇×Q₈	—	2	2	no	—
173	C₇×Q₈	—	2	2	no	—
174	C₇×Q₈	—	4	4	no	—
175	C₇×D₄	—	2	2	no	—
176	C₇×D₄	—	2	2	no	—
177	C₇×D₄	—	4	2	no	—
178	C₇×D₄	—	4	2	no	—
179	C₂×C₄×C₇	—	2	1	no	—
180	C₂×C₄×C₇	—	2	1	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	2	2	no	—
184	C₂×C₄×C₇	—	4	2	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	4	2	no	—
187	C₂×C₄×C₇	—	4	2	no	—
188	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂×#41	1	2	yes	—
190	C₂×C₄×C₇	C₂×#41	1	2	yes	—
191	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
192	C₇×C₈	—	2	2	no	—
193	C₇×C₈	—	4	1	no	—
194	C₇×C₈	—	4	1	no	—
195	C₇×C₈	—	4	2	no	—
196	C₇×C₈	—	4	2	no	—
197	C₇×C₈	#124	1	2	yes	—
198	C₇×C₈	#124	1	1	yes	characteristic
199	C₇×C₈	#124	1	1	yes	characteristic
200	C₂×#41	C₂³×C₇	1	2	yes	—
201	#137	—	2	2	no	—
202	AGL(1,ℤ/8)	—	2	2	no	—
203	#135	—	2	2	no	—
204	#131	—	2	2	no	—
205	#131	—	2	2	no	—
206	#131	—	2	2	no	—
207	#131	—	2	2	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	C₂³×C₇	1	2	yes	—
210	C₂×M₁₆	C₂³×C₇	1	2	yes	—
211	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
212	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	#125	—	2	1	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	#117	C₂³×C₇	1	2	yes	—
221	#117	C₂³×C₇	1	1	yes	characteristic
222	#117	C₂³×C₇	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
224	#108	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	#104	—	2	4	no	—
229	#104	—	2	2	no	—
230	#104	—	2	2	no	—
231	#103	—	2	2	no	—
232	#102	—	2	2	no	—
233	#101	—	2	2	no	—
234	#100	—	2	2	no	—
235	#99	—	2	4	no	—
236	#97	—	2	1	no	—
237	#97	—	2	2	no	—
238	#97	C₇×D₄	1	2	yes	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	2	2	no	—
243	C₄×C₈	C₇×D₄	1	2	yes	—
244	C₂²×C₇	—	4	1	no	—
245	C₂²×C₇	—	4	2	no	—
246	C₂²×C₇	—	8	2	no	—
247	C₂²×C₇	#525	1	1	yes	characteristic
248	C₂²×C₇	#529	1	2	yes	—
249	C₄×C₇	—	2	2	no	—
250	C₄×C₇	—	4	1	no	—
251	C₄×C₇	—	4	2	no	—
252	C₄×C₇	—	4	2	no	—
253	C₄×C₇	—	4	2	no	—
254	C₄×C₇	—	8	2	no	—
255	C₄×C₇	#525	1	1	yes	characteristic
256	C₄×C₇	C₂×#124	1	1	yes	characteristic
257	C₄×C₇	#529	1	2	yes	—
258	#41	—	2	2	no	—
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	C₇×#41	1	2	yes	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	C₇×#41	1	2	yes	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
269	Q₁₆	—	2	2	no	—
270	SD₁₆	—	2	4	no	—
271	D₈	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	4	2	no	—
279	M₁₆	—	4	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
288	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
289	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
290	C₂×C₈	C₇×#41	1	2	yes	—
291	C₂×C₈	C₇×#41	1	2	yes	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	C₇×#41	1	2	yes	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	2	no	—
303	C₄²	C₂×C₇×D₄	1	2	yes	—
304	C₄²	C₇×#41	1	1	yes	characteristic
305	C₄²	C₇×#41	1	1	yes	characteristic
306	C₂×C₇	—	2	1	no	—
307	C₂×C₇	—	2	2	no	—
308	C₂×C₇	—	8	2	no	—
309	C₂×C₇	#2398	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	C₂×C₇×#41	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	4	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
329	C₂×C₄	C₂×C₇×#41	1	2	yes	—
330	C₂×C₄	C₂×C₇×#41	1	2	yes	—
331	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	4	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	C₇×#124	1	2	yes	—
338	C₈	C₇×#124	1	1	yes	characteristic
339	C₈	C₇×#124	1	1	yes	characteristic
340	C₇	#23833	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	—	4	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₇×#525	1	1	yes	characteristic
345	C₂²	C₇×#529	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₇×#525	1	1	yes	characteristic
353	C₄	C₂×C₇×#124	1	1	yes	characteristic
354	C₄	C₇×#529	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	8	2	no	—
358	C₂	C₇×#2398	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	#2398
prime radical	C₇×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#2960	C₂	semidirect
3. C₇×#2103	C₂	semidirect
4. C₇×#2102	C₂	semidirect
5. C₇×#2080	C₂	semidirect
8. C₇×#2031	C₂	semidirect
9. C₇×#1764	C₂	semidirect
10. C₇×#1757	C₂	semidirect
11. C₇×#1747	C₂	semidirect
39. #23833	C₇	direct (N × H)
84. #2960	C₂×C₇	semidirect
86. #2103	C₂×C₇	semidirect
87. #2102	C₂×C₇	semidirect
88. #2080	C₂×C₇	semidirect
91. #2031	C₂×C₇	semidirect
92. #1764	C₂×C₇	semidirect
93. #1757	C₂×C₇	semidirect
94. #1747	C₂×C₇	semidirect
340. C₇	#23833	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#2960	2
C₇×#2955	2
C₇×#2103	2
C₇×#2102	2
C₇×#2080	2
C₇×#2053	2
C₇×#2050	2
C₇×#2031	2
C₇×#1764	2
C₇×#1757	2
C₇×#1747	2
#23833	7

Subgroups — C7×#23833 #13201049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23833 #13201049