Subgroups — C₇×#19193 #13196409

Order 1792.

650 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3227	C₂	1	2	yes	maximal
2	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₇×#1695	C₂	1	4	yes	maximal, complemented
5	C₇×#1675	C₂	1	4	yes	maximal, complemented
6	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1548	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#131	—	2	1	no	—
9	C₇×#532	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₇×#500	—	2	1	no	—
12	C₇×#500	—	2	1	no	—
13	C₇×#500	C₂²	1	2	yes	—
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#498	—	2	2	no	—
16	C₇×#486	—	2	2	no	—
17	C₇×#481	—	2	4	no	—
18	C₇×#476	C₂²	1	8	yes	—
19	C₇×#436	—	2	1	no	—
20	C₇×#436	C₂²	1	2	yes	—
21	C₇×C₈×D₄	—	2	1	no	—
22	C₇×C₈×D₄	C₂²	1	2	yes	—
23	C₇×#431	C₂²	1	2	yes	—
24	C₂×C₇×#107	—	2	1	no	—
25	C₂×C₇×#107	—	2	1	no	—
26	C₇×#423	C₂²	1	1	yes	characteristic
27	C₇×#408	—	2	1	no	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#335	C₂²	1	2	yes	—
30	C₇×#334	C₂²	1	2	yes	—
31	C₇×#333	C₂²	1	8	yes	—
32	#19193	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₇×#41	—	2	1	no	—
34	C₂²×C₇×Q₈	—	2	2	no	—
35	C₇×#131	—	4	4	no	—
36	C₂×C₇×M₁₆	—	2	1	no	—
37	C₂×C₇×M₁₆	—	2	2	no	—
38	C₂²×C₇×C₈	—	2	2	no	—
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₇×C₄⋊Q₈	—	2	2	no	—
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₇×#127	D₄	1	1	yes	characteristic
47	C₇×#124	—	2	1	no	—
48	C₇×#121	—	2	1	no	—
49	C₇×#121	D₄	1	2	yes	—
50	C₄×C₇×D₄	—	2	1	no	—
51	C₄×C₇×D₄	C₂³	1	2	yes	—
52	C₂×C₇×C₄⋊C₄	—	2	2	no	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#107	—	2	2	no	—
56	C₇×#107	—	2	2	no	—
57	C₇×#107	—	2	2	no	—
58	C₇×#107	—	2	2	no	—
59	C₇×#106	—	2	8	no	—
60	C₇×C₄⋊C₈	—	2	1	no	—
61	C₇×C₄⋊C₈	C₂³	1	4	yes	—
62	C₇×C₄⋊C₈	C₂³	1	2	yes	—
63	C₇×#102	—	2	8	no	—
64	C₇×C₂²⋊C₈	—	2	2	no	—
65	C₇×C₂²⋊C₈	—	2	2	no	—
66	C₇×C₂²⋊C₈	—	2	2	no	—
67	C₄×C₇×C₈	—	2	2	no	—
68	C₄×C₇×C₈	D₄	1	1	yes	characteristic
69	C₄×C₇×C₈	D₄	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	#3227	C₂×C₇	1	2	yes	—
73	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
74	#2931	C₂×C₇	1	1	yes	characteristic
75	#1695	C₂×C₇	1	4	yes	complemented
76	#1675	C₂×C₇	1	4	yes	complemented
77	#1564	C₂×C₇	1	1	yes	characteristic, complemented
78	#1548	C₂×C₇	1	2	yes	complemented
79	C₇×#41	—	4	2	no	—
80	C₇×#41	—	4	2	no	—
81	C₂×C₇×Q₈	—	2	1	no	—
82	C₂×C₇×Q₈	—	2	4	no	—
83	C₂×C₇×Q₈	—	4	2	no	—
84	C₂×C₇×Q₈	—	8	4	no	—
85	C₂×C₇×D₄	—	2	2	no	—
86	C₂×C₇×D₄	—	2	1	no	—
87	C₂×C₇×D₄	—	2	1	no	—
88	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₇	—	2	2	no	—
90	C₂²×C₄×C₇	—	4	2	no	—
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	#41	1	1	yes	characteristic
94	C₇×M₁₆	—	4	2	no	—
95	C₇×M₁₆	—	4	4	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	4	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	4	4	no	—
100	C₂×C₇×C₈	SD₁₆	1	4	yes	—
101	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
105	C₇×C₄⋊C₄	—	2	1	no	—
106	C₇×C₄⋊C₄	—	2	4	no	—
107	C₇×C₄⋊C₄	—	2	4	no	—
108	C₇×C₄⋊C₄	—	4	2	no	—
109	C₇×C₄⋊C₄	—	4	2	no	—
110	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
111	C₇×C₂²⋊C₄	—	2	2	no	—
112	C₇×C₂²⋊C₄	—	2	2	no	—
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇	#41	1	1	yes	characteristic
116	C₄²×C₇	C₂⁴	1	1	yes	characteristic
117	C₂×#131	—	2	1	no	—
118	#532	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
120	#500	—	2	1	no	—
121	#500	—	2	1	no	—
122	#500	C₂²×C₇	1	2	yes	—
123	#500	C₂²×C₇	1	2	yes	—
124	#498	—	2	2	no	—
125	#486	—	2	2	no	—
126	#481	—	2	4	no	—
127	#476	C₂²×C₇	1	8	yes	—
128	#436	—	2	1	no	—
129	#436	C₂²×C₇	1	2	yes	—
130	C₈×D₄	—	2	1	no	—
131	C₈×D₄	C₂²×C₇	1	2	yes	—
132	#431	C₂²×C₇	1	2	yes	—
133	C₂×#107	—	2	1	no	—
134	C₂×#107	—	2	1	no	—
135	#423	C₂²×C₇	1	1	yes	characteristic
136	#408	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
138	#335	C₂²×C₇	1	2	yes	—
139	#334	C₂²×C₇	1	2	yes	—
140	#333	C₂²×C₇	1	8	yes	—
141	C₂³×C₇	—	2	2	no	—
142	C₂³×C₇	#121	1	1	yes	characteristic
143	C₇×Q₈	—	4	2	no	—
144	C₇×Q₈	—	8	4	no	—
145	C₇×Q₈	—	8	4	no	—
146	C₇×D₄	—	2	4	no	—
147	C₇×D₄	—	4	4	no	—
148	C₇×D₄	—	4	2	no	—
149	C₇×D₄	—	4	2	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	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	4	no	—
156	C₂×C₄×C₇	—	8	4	no	—
157	C₂×C₄×C₇	#121	1	1	yes	characteristic
158	C₂×C₄×C₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₇	#137	1	2	yes	—
162	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
165	C₇×C₈	—	2	1	no	—
166	C₇×C₈	—	2	1	no	—
167	C₇×C₈	—	4	2	no	—
168	C₇×C₈	—	4	4	no	—
169	C₇×C₈	C₂×SD₁₆	1	2	yes	—
170	C₇×C₈	C₂×SD₁₆	1	2	yes	—
171	C₂×#41	—	2	1	no	—
172	C₂²×Q₈	—	2	2	no	—
173	#131	—	4	4	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
183	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
184	#127	C₇×D₄	1	1	yes	characteristic
185	#124	—	2	1	no	—
186	#121	—	2	1	no	—
187	#121	C₇×D₄	1	2	yes	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	C₂³×C₇	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
193	#107	—	2	2	no	—
194	#107	—	2	2	no	—
195	#107	—	2	2	no	—
196	#107	—	2	2	no	—
197	#106	—	2	8	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	C₂³×C₇	1	2	yes	—
200	C₄⋊C₈	C₂³×C₇	1	4	yes	—
201	#102	—	2	8	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	C₇×D₄	1	1	yes	characteristic
207	C₄×C₈	C₇×D₄	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
210	C₂²×C₇	—	2	1	no	—
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	—	4	2	no	—
213	C₂²×C₇	—	4	2	no	—
214	C₂²×C₇	—	4	2	no	—
215	C₂²×C₇	#461	1	2	yes	—
216	C₂²×C₇	C₂×#121	1	1	yes	characteristic
217	C₄×C₇	—	2	1	no	—
218	C₄×C₇	—	2	1	no	—
219	C₄×C₇	—	4	2	no	—
220	C₄×C₇	—	8	4	no	—
221	C₄×C₇	#461	1	2	yes	—
222	C₄×C₇	#472	1	2	yes	—
223	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
224	C₄×C₇	C₂×#137	1	1	yes	characteristic
225	C₄×C₇	#578	1	2	yes	—
226	#41	—	4	2	no	—
227	#41	—	4	2	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	8	4	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₇×#41	1	1	yes	characteristic
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	4	no	—
247	C₂×C₈	C₇×SD₁₆	1	4	yes	—
248	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
249	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
250	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
261	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
262	C₄²	C₇×#41	1	1	yes	characteristic
263	C₄²	C₂⁴×C₇	1	1	yes	characteristic
264	C₂×C₇	—	2	2	no	—
265	C₂×C₇	—	8	2	no	—
266	C₂×C₇	C₂×#461	1	1	yes	characteristic
267	C₂×C₇	#3059	1	1	yes	characteristic
268	C₂×C₇	C₂×#472	1	1	yes	characteristic
269	C₂³	—	2	2	no	—
270	C₂³	C₇×#121	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	8	4	no	—
273	Q₈	—	8	4	no	—
274	D₄	—	2	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	C₇×#121	1	1	yes	characteristic
286	C₂×C₄	C₇×#121	1	1	yes	characteristic
287	C₂×C₄	C₇×#121	1	1	yes	characteristic
288	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
289	C₂×C₄	C₇×#137	1	2	yes	—
290	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	4	2	no	—
296	C₈	—	4	4	no	—
297	C₈	C₂×C₇×SD₁₆	1	2	yes	—
298	C₈	C₂×C₇×SD₁₆	1	2	yes	—
299	C₇	#19193	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₇×#461	1	2	yes	—
306	C₂²	C₂×C₇×#121	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	4	no	—
311	C₄	C₇×#461	1	2	yes	—
312	C₄	C₇×#472	1	2	yes	—
313	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
314	C₄	C₂×C₇×#137	1	1	yes	characteristic
315	C₄	C₇×#578	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	8	2	no	—
318	C₂	C₂×C₇×#461	1	1	yes	characteristic
319	C₂	C₇×#3059	1	1	yes	characteristic
320	C₂	C₂×C₇×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#500	C₂	semidirect
4. C₇×#1695	C₂	semidirect
5. C₇×#1675	C₂	semidirect
6. C₇×#1564	C₂	semidirect
7. C₇×#1548	C₂	semidirect
32. #19193	C₇	direct (N × H)
73. C₂×#500	C₂×C₇	semidirect
75. #1695	C₂×C₇	semidirect
76. #1675	C₂×C₇	semidirect
77. #1564	C₂×C₇	semidirect
78. #1548	C₂×C₇	semidirect
299. C₇	#19193	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×#3227	2
C₂×C₇×#500	2
C₇×#2931	2
C₇×#1695	2
C₇×#1675	2
C₇×#1564	2
C₇×#1548	2
#19193	7

Subgroups — C7×#19193 #13196409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19193 #13196409