Subgroups — C₇×#19654 #13196870

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 318 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, complemented
2	C₇×#3206	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₇×#1716	C₂	1	2	yes	maximal, complemented
6	C₇×#1715	C₂	1	2	yes	maximal, complemented
7	C₇×#1708	C₂	1	2	yes	maximal, complemented
8	C₇×#1705	C₂	1	2	yes	maximal, complemented
9	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#135	—	2	2	no	—
11	C₇×#532	C₂²	1	1	yes	characteristic
12	C₇×#532	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₇×#498	—	2	2	no	—
15	C₇×#498	C₂²	1	2	yes	—
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#498	C₂²	1	4	yes	—
18	C₇×#495	—	2	2	no	—
19	C₇×#494	C₂²	1	1	yes	characteristic
20	C₇×#493	C₂²	1	1	yes	characteristic
21	C₇×#492	C₂²	1	1	yes	characteristic
22	C₇×#492	C₂²	1	4	yes	—
23	C₇×#492	C₂²	1	1	yes	characteristic
24	C₇×#487	C₂²	1	4	yes	—
25	C₇×#486	C₂²	1	4	yes	—
26	C₇×#481	—	2	4	no	—
27	C₇×#476	—	2	4	no	—
28	C₇×#461	—	2	4	no	—
29	C₇×#448	—	2	4	no	—
30	C₇×#431	C₂²	1	2	yes	—
31	C₂×C₇×#106	—	2	2	no	—
32	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
33	C₇×#322	C₂²	1	2	yes	—
34	C₇×C₈²	C₂²	1	2	yes	—
35	#19654	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₇×#41	—	2	1	no	—
37	C₂×C₇×#41	—	2	1	no	—
38	C₂²×C₇×Q₈	—	2	2	no	—
39	C₇×#135	—	4	8	no	—
40	C₂×C₇×Q₁₆	—	2	2	no	—
41	C₂×C₇×SD₁₆	—	2	2	no	—
42	C₂×C₇×SD₁₆	—	2	2	no	—
43	C₂×C₇×SD₁₆	—	2	8	no	—
44	C₂×C₇×D₈	—	2	2	no	—
45	C₂²×C₇×C₈	D₄	1	2	yes	—
46	C₇×C₄⋊Q₈	—	2	2	no	—
47	C₇×C₄⋊Q₈	—	2	2	no	—
48	C₇×C₄⋊Q₈	—	2	2	no	—
49	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
50	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
52	C₇×#127	C₂³	1	1	yes	characteristic
53	C₇×#127	C₂³	1	1	yes	characteristic
54	C₇×#124	—	2	1	no	—
55	C₇×#124	—	2	1	no	—
56	C₇×#121	—	2	2	no	—
57	C₇×#121	—	2	2	no	—
58	C₄×C₇×D₄	—	2	2	no	—
59	C₄×C₇×D₄	—	2	2	no	—
60	C₂×C₇×C₄⋊C₄	—	2	2	no	—
61	C₂×C₇×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₇×#106	—	2	8	no	—
64	C₇×#106	—	2	4	no	—
65	C₇×#106	—	2	4	no	—
66	C₇×C₄⋊C₈	D₄	1	4	yes	—
67	C₇×#103	—	2	8	no	—
68	C₇×#102	—	2	8	no	—
69	C₇×C₂²⋊C₈	D₄	1	4	yes	—
70	C₄×C₇×C₈	D₄	1	2	yes	—
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂³	1	4	yes	—
73	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
74	#3227	C₂×C₇	1	2	yes	complemented
75	#3206	C₂×C₇	1	2	yes	complemented
76	C₂×#498	C₂×C₇	1	1	yes	characteristic, complemented
77	#3147	C₂×C₇	1	1	yes	characteristic
78	#1716	C₂×C₇	1	2	yes	complemented
79	#1715	C₂×C₇	1	2	yes	complemented
80	#1708	C₂×C₇	1	2	yes	complemented
81	#1705	C₂×C₇	1	2	yes	complemented
82	#1450	C₂×C₇	1	1	yes	characteristic, complemented
83	C₇×#41	—	4	4	no	—
84	C₇×#41	—	4	4	no	—
85	C₂×C₇×Q₈	—	2	4	no	—
86	C₂×C₇×Q₈	—	2	1	no	—
87	C₂×C₇×Q₈	—	2	1	no	—
88	C₂×C₇×Q₈	—	4	2	no	—
89	C₂×C₇×Q₈	—	8	4	no	—
90	C₂×C₇×D₄	—	2	2	no	—
91	C₂×C₇×D₄	—	2	2	no	—
92	C₂×C₇×D₄	—	2	1	no	—
93	C₂×C₇×D₄	—	2	1	no	—
94	C₂×C₇×D₄	—	4	1	no	—
95	C₂×C₇×D₄	—	4	1	no	—
96	C₂²×C₄×C₇	—	4	2	no	—
97	C₂²×C₄×C₇	—	4	1	no	—
98	C₂²×C₄×C₇	—	4	1	no	—
99	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₇×Q₁₆	—	4	4	no	—
102	C₇×SD₁₆	—	4	4	no	—
103	C₇×SD₁₆	—	4	4	no	—
104	C₇×SD₁₆	—	8	8	no	—
105	C₇×D₈	—	4	4	no	—
106	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
107	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
110	C₇×C₄⋊C₄	—	2	4	no	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	—	2	4	no	—
113	C₇×C₄⋊C₄	—	2	2	no	—
114	C₇×C₄⋊C₄	—	4	2	no	—
115	C₇×C₄⋊C₄	—	4	2	no	—
116	C₇×C₂²⋊C₄	—	4	2	no	—
117	C₇×C₂²⋊C₄	—	4	2	no	—
118	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇	C₂×D₄	1	2	yes	—
120	C₄²×C₇	C₂⁴	1	1	yes	characteristic
121	C₂×#135	—	2	2	no	—
122	#532	C₂²×C₇	1	1	yes	characteristic
123	#532	C₂²×C₇	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
125	#498	—	2	2	no	—
126	#498	C₂²×C₇	1	2	yes	—
127	#498	C₂²×C₇	1	2	yes	—
128	#498	C₂²×C₇	1	4	yes	—
129	#495	—	2	2	no	—
130	#494	C₂²×C₇	1	1	yes	characteristic
131	#493	C₂²×C₇	1	1	yes	characteristic
132	#492	C₂²×C₇	1	1	yes	characteristic
133	#492	C₂²×C₇	1	1	yes	characteristic
134	#492	C₂²×C₇	1	4	yes	—
135	#487	C₂²×C₇	1	4	yes	—
136	#486	C₂²×C₇	1	4	yes	—
137	#481	—	2	4	no	—
138	#476	—	2	4	no	—
139	#461	—	2	4	no	—
140	#448	—	2	4	no	—
141	#431	C₂²×C₇	1	2	yes	—
142	C₂×#106	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
144	#322	C₂²×C₇	1	2	yes	—
145	C₈²	C₂²×C₇	1	2	yes	—
146	C₂³×C₇	—	4	1	no	—
147	C₂³×C₇	—	4	1	no	—
148	C₂³×C₇	#127	1	1	yes	characteristic
149	C₇×Q₈	—	4	2	no	—
150	C₇×Q₈	—	4	2	no	—
151	C₇×Q₈	—	4	8	no	—
152	C₇×Q₈	—	8	4	no	—
153	C₇×D₄	—	4	2	no	—
154	C₇×D₄	—	4	2	no	—
155	C₇×D₄	—	4	4	no	—
156	C₇×D₄	—	4	4	no	—
157	C₇×D₄	—	8	2	no	—
158	C₇×D₄	—	8	2	no	—
159	C₂×C₄×C₇	—	2	4	no	—
160	C₂×C₄×C₇	—	4	1	no	—
161	C₂×C₄×C₇	—	4	1	no	—
162	C₂×C₄×C₇	—	4	4	no	—
163	C₂×C₄×C₇	—	8	4	no	—
164	C₂×C₄×C₇	—	8	2	no	—
165	C₂×C₄×C₇	—	8	2	no	—
166	C₂×C₄×C₇	#127	1	2	yes	—
167	C₂×C₄×C₇	#127	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
171	C₇×C₈	—	2	8	no	—
172	C₇×C₈	C₂×SD₁₆	1	4	yes	—
173	C₇×C₈	C₂×SD₁₆	1	4	yes	—
174	C₂×#41	—	2	1	no	—
175	C₂×#41	—	2	1	no	—
176	C₂²×Q₈	—	2	2	no	—
177	#135	—	4	8	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×SD₁₆	—	2	2	no	—
180	C₂×SD₁₆	—	2	2	no	—
181	C₂×SD₁₆	—	2	8	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂²×C₈	C₇×D₄	1	2	yes	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
188	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
189	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
190	#127	C₂³×C₇	1	1	yes	characteristic
191	#127	C₂³×C₇	1	1	yes	characteristic
192	#124	—	2	1	no	—
193	#124	—	2	1	no	—
194	#121	—	2	2	no	—
195	#121	—	2	2	no	—
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
201	#106	—	2	4	no	—
202	#106	—	2	4	no	—
203	#106	—	2	8	no	—
204	C₄⋊C₈	C₇×D₄	1	4	yes	—
205	#103	—	2	8	no	—
206	#102	—	2	8	no	—
207	C₂²⋊C₈	C₇×D₄	1	4	yes	—
208	C₄×C₈	C₇×D₄	1	2	yes	—
209	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₇	1	4	yes	—
211	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
212	C₂²×C₇	—	2	2	no	—
213	C₂²×C₇	—	8	1	no	—
214	C₂²×C₇	—	8	1	no	—
215	C₂²×C₇	—	8	2	no	—
216	C₂²×C₇	—	8	2	no	—
217	C₂²×C₇	#492	1	2	yes	—
218	C₂²×C₇	C₂×#127	1	1	yes	characteristic
219	C₄×C₇	—	2	2	no	—
220	C₄×C₇	—	8	1	no	—
221	C₄×C₇	—	8	1	no	—
222	C₄×C₇	—	8	4	no	—
223	C₄×C₇	#492	1	2	yes	—
224	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
225	C₄×C₇	#577	1	4	yes	—
226	#41	—	4	4	no	—
227	#41	—	4	4	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	8	4	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
243	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
244	Q₁₆	—	4	4	no	—
245	SD₁₆	—	4	4	no	—
246	SD₁₆	—	4	4	no	—
247	SD₁₆	—	8	8	no	—
248	D₈	—	4	4	no	—
249	C₂×C₈	C₇×SD₁₆	1	8	yes	complemented
250	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
251	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
252	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
262	C₄²	C₂×C₇×D₄	1	2	yes	—
263	C₄²	C₂⁴×C₇	1	1	yes	characteristic
264	C₂×C₇	—	2	2	no	—
265	C₂×C₇	—	16	1	no	—
266	C₂×C₇	—	16	1	no	—
267	C₂×C₇	C₂×#492	1	1	yes	characteristic
268	C₂×C₇	#3154	1	2	yes	—
269	C₂³	—	4	1	no	—
270	C₂³	—	4	1	no	—
271	C₂³	C₇×#127	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	8	no	—
275	Q₈	—	8	4	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	8	2	no	—
281	D₄	—	8	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₇×#127	1	2	yes	—
290	C₂×C₄	C₇×#127	1	1	yes	characteristic
291	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
292	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₈	C₂×C₇×SD₁₆	1	4	yes	—
296	C₈	C₂×C₇×SD₁₆	1	4	yes	—
297	C₇	#19654	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₇×#492	1	2	yes	—
304	C₂²	C₂×C₇×#127	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	4	no	—
309	C₄	C₇×#492	1	2	yes	—
310	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
311	C₄	C₇×#577	1	4	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	—	16	1	no	—
315	C₂	C₂×C₇×#492	1	1	yes	characteristic
316	C₂	C₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3147	C₂
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
1. C₇×#3227	C₂	semidirect
2. C₇×#3206	C₂	semidirect
3. C₂×C₇×#498	C₂	semidirect
5. C₇×#1716	C₂	semidirect
6. C₇×#1715	C₂	semidirect
7. C₇×#1708	C₂	semidirect
8. C₇×#1705	C₂	semidirect
9. C₇×#1450	C₂	semidirect
35. #19654	C₇	direct (N × H)
74. #3227	C₂×C₇	semidirect
75. #3206	C₂×C₇	semidirect
76. C₂×#498	C₂×C₇	semidirect
78. #1716	C₂×C₇	semidirect
79. #1715	C₂×C₇	semidirect
80. #1708	C₂×C₇	semidirect
81. #1705	C₂×C₇	semidirect
82. #1450	C₂×C₇	semidirect
106. C₂×C₇×C₈	SD₁₆	semidirect
249. C₂×C₈	C₇×SD₁₆	semidirect
297. C₇	#19654	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
C₇×#3206	2
C₂×C₇×#498	2
C₇×#3147	2
C₇×#1716	2
C₇×#1715	2
C₇×#1708	2
C₇×#1705	2
C₇×#1450	2
#19654	7

Subgroups — C7×#19654 #13196870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19654 #13196870