Subgroups — C₇×#19638 #13196854

Order 1792.

754 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3206	C₂	1	2	yes	maximal, complemented
2	C₇×#3169	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3141	C₂	1	2	yes	maximal, complemented
5	C₇×#1719	C₂	1	4	yes	maximal, complemented
6	C₇×#1710	C₂	1	2	yes	maximal, complemented
7	C₇×#1706	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#532	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₇×#500	—	2	2	no	—
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	2	yes	—
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#493	C₂²	1	1	yes	characteristic
16	C₇×#492	C₂²	1	4	yes	—
17	C₇×#489	—	2	4	no	—
18	C₇×#487	C₂²	1	4	yes	—
19	C₇×#487	C₂²	1	4	yes	—
20	C₇×#486	C₂²	1	2	yes	—
21	C₇×#486	C₂²	1	4	yes	—
22	C₇×#466	—	2	2	no	—
23	C₇×#461	—	2	4	no	—
24	C₇×#448	—	2	4	no	—
25	C₄×C₇×D₈	—	2	2	no	—
26	C₇×#431	C₂²	1	1	yes	characteristic
27	C₂×C₇×#107	—	2	2	no	—
28	C₇×#417	—	2	4	no	—
29	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	2	yes	—
32	C₇×C₈²	C₂²	1	2	yes	—
33	#19638	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	2	no	—
35	C₂²×C₇×Q₈	—	2	2	no	—
36	C₂×C₇×SD₁₆	—	2	8	no	—
37	C₂×C₇×D₈	—	2	2	no	—
38	C₂×C₇×M₁₆	D₄	1	2	yes	—
39	C₂²×C₇×C₈	D₄	1	2	yes	—
40	C₇×C₄⋊Q₈	—	2	2	no	—
41	C₇×C₄⋊Q₈	—	2	2	no	—
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
45	C₇×#127	C₂³	1	2	yes	—
46	C₇×#124	—	2	2	no	—
47	C₇×#121	—	2	2	no	—
48	C₇×#121	—	2	2	no	—
49	C₄×C₇×D₄	—	2	2	no	—
50	C₄×C₇×D₄	—	2	2	no	—
51	C₂×C₇×C₄⋊C₄	—	2	2	no	—
52	C₂×C₇×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₇×#107	—	2	2	no	—
55	C₇×#107	—	2	4	no	—
56	C₇×#107	—	2	4	no	—
57	C₇×C₄⋊C₈	D₄	1	2	yes	—
58	C₇×#103	—	2	8	no	—
59	C₇×#103	—	2	8	no	—
60	C₇×#102	—	2	4	no	—
61	C₇×#102	—	2	8	no	—
62	C₇×C₂²⋊C₈	D₄	1	2	yes	—
63	C₇×#97	D₄	1	2	yes	—
64	C₄×C₇×C₈	D₄	1	2	yes	—
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	2	yes	—
67	C₄×C₇×C₈	C₂³	1	2	yes	—
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	#3206	C₂×C₇	1	2	yes	complemented
70	#3169	C₂×C₇	1	1	yes	characteristic
71	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
72	#3141	C₂×C₇	1	2	yes	complemented
73	#1719	C₂×C₇	1	4	yes	complemented
74	#1710	C₂×C₇	1	2	yes	complemented
75	#1706	C₂×C₇	1	2	yes	complemented
76	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
77	C₇×#41	—	4	8	no	—
78	C₂×C₇×Q₈	—	2	2	no	—
79	C₂×C₇×Q₈	—	2	4	no	—
80	C₂×C₇×Q₈	—	4	2	no	—
81	C₂×C₇×Q₈	—	8	4	no	—
82	C₂×C₇×D₄	—	2	2	no	—
83	C₂×C₇×D₄	—	2	2	no	—
84	C₂×C₇×D₄	—	2	2	no	—
85	C₂×C₇×D₄	—	4	2	no	—
86	C₂²×C₄×C₇	—	4	2	no	—
87	C₂²×C₄×C₇	—	4	2	no	—
88	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₇×SD₁₆	—	8	8	no	—
92	C₇×D₈	—	4	4	no	—
93	C₇×M₁₆	—	2	4	no	—
94	C₂×C₇×C₈	—	2	4	no	—
95	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
97	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
98	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
99	C₇×C₄⋊C₄	—	2	4	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	—	4	2	no	—
104	C₇×C₄⋊C₄	—	4	2	no	—
105	C₇×C₂²⋊C₄	—	4	2	no	—
106	C₇×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇	C₂⁴	1	1	yes	characteristic
111	#532	C₂²×C₇	1	2	yes	—
112	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
113	#500	—	2	2	no	—
114	#500	C₂²×C₇	1	2	yes	—
115	#500	C₂²×C₇	1	2	yes	—
116	#500	C₂²×C₇	1	1	yes	characteristic
117	#493	C₂²×C₇	1	1	yes	characteristic
118	#492	C₂²×C₇	1	4	yes	—
119	#489	—	2	4	no	—
120	#487	C₂²×C₇	1	4	yes	—
121	#487	C₂²×C₇	1	4	yes	—
122	#486	C₂²×C₇	1	4	yes	—
123	#486	C₂²×C₇	1	2	yes	—
124	#466	—	2	2	no	—
125	#461	—	2	4	no	—
126	#448	—	2	4	no	—
127	C₄×D₈	—	2	2	no	—
128	#431	C₂²×C₇	1	1	yes	characteristic
129	C₂×#107	—	2	2	no	—
130	#417	—	2	4	no	—
131	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
133	#322	C₂²×C₇	1	2	yes	—
134	C₈²	C₂²×C₇	1	2	yes	—
135	C₂³×C₇	—	4	2	no	—
136	C₂³×C₇	#127	1	1	yes	characteristic
137	C₇×Q₈	—	4	4	no	—
138	C₇×Q₈	—	4	8	no	—
139	C₇×Q₈	—	8	4	no	—
140	C₇×D₄	—	4	4	no	—
141	C₇×D₄	—	4	4	no	—
142	C₇×D₄	—	4	4	no	—
143	C₇×D₄	—	8	4	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	4	2	no	—
147	C₂×C₄×C₇	—	4	4	no	—
148	C₂×C₄×C₇	—	8	4	no	—
149	C₂×C₄×C₇	—	8	4	no	—
150	C₂×C₄×C₇	#127	1	1	yes	characteristic
151	C₂×C₄×C₇	#127	1	1	yes	characteristic
152	C₂×C₄×C₇	#127	1	1	yes	characteristic
153	C₂×C₄×C₇	#135	1	4	yes	—
154	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
157	C₇×C₈	—	2	4	no	—
158	C₇×C₈	—	2	2	no	—
159	C₇×C₈	—	2	4	no	—
160	C₇×C₈	#137	1	4	yes	—
161	C₂×#41	—	2	2	no	—
162	C₂²×Q₈	—	2	2	no	—
163	C₂×SD₁₆	—	2	8	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×M₁₆	C₇×D₄	1	2	yes	—
166	C₂²×C₈	C₇×D₄	1	2	yes	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
171	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
172	#127	C₂³×C₇	1	2	yes	—
173	#124	—	2	2	no	—
174	#121	—	2	2	no	—
175	#121	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
181	#107	—	2	4	no	—
182	#107	—	2	4	no	—
183	#107	—	2	2	no	—
184	C₄⋊C₈	C₇×D₄	1	2	yes	—
185	#103	—	2	8	no	—
186	#103	—	2	8	no	—
187	#102	—	2	8	no	—
188	#102	—	2	4	no	—
189	C₂²⋊C₈	C₇×D₄	1	2	yes	—
190	#97	C₇×D₄	1	2	yes	—
191	C₄×C₈	C₇×D₄	1	2	yes	—
192	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₇	1	2	yes	—
194	C₄×C₈	C₂³×C₇	1	2	yes	—
195	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
196	C₂²×C₇	—	2	1	no	—
197	C₂²×C₇	—	2	1	no	—
198	C₂²×C₇	—	8	2	no	—
199	C₂²×C₇	—	8	2	no	—
200	C₂²×C₇	—	8	2	no	—
201	C₂²×C₇	#495	1	2	yes	—
202	C₂²×C₇	C₂×#127	1	1	yes	characteristic
203	C₄×C₇	—	2	1	no	—
204	C₄×C₇	—	2	1	no	—
205	C₄×C₇	—	8	2	no	—
206	C₄×C₇	—	8	4	no	—
207	C₄×C₇	#495	1	2	yes	—
208	C₄×C₇	C₂×#135	1	2	yes	—
209	C₄×C₇	C₂×#137	1	2	yes	—
210	C₄×C₇	#576	1	2	yes	—
211	#41	—	4	8	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
225	SD₁₆	—	8	8	no	—
226	D₈	—	4	4	no	—
227	M₁₆	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
230	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
231	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
232	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
242	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
243	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
244	C₄²	C₂⁴×C₇	1	1	yes	characteristic
245	C₂×C₇	—	2	2	no	—
246	C₂×C₇	—	16	2	no	—
247	C₂×C₇	C₂×#495	1	1	yes	characteristic
248	C₂×C₇	#3152	1	1	yes	characteristic
249	C₂×C₇	#3155	1	1	yes	characteristic
250	C₂³	—	4	2	no	—
251	C₂³	C₇×#127	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	8	no	—
254	Q₈	—	8	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	8	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₇×#127	1	1	yes	characteristic
266	C₂×C₄	C₇×#127	1	1	yes	characteristic
267	C₂×C₄	C₇×#127	1	1	yes	characteristic
268	C₂×C₄	C₇×#135	1	4	yes	—
269	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	4	no	—
275	C₈	C₇×#137	1	4	yes	—
276	C₇	#19638	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₇×#495	1	2	yes	—
283	C₂²	C₂×C₇×#127	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	4	no	—
287	C₄	—	8	2	no	—
288	C₄	C₇×#495	1	2	yes	—
289	C₄	C₂×C₇×#135	1	2	yes	—
290	C₄	C₂×C₇×#137	1	2	yes	—
291	C₄	C₇×#576	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	16	2	no	—
294	C₂	C₂×C₇×#495	1	1	yes	characteristic
295	C₂	C₇×#3152	1	1	yes	characteristic
296	C₂	C₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3169	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×#3206	C₂	semidirect
3. C₂×C₇×#500	C₂	semidirect
4. C₇×#3141	C₂	semidirect
5. C₇×#1719	C₂	semidirect
6. C₇×#1710	C₂	semidirect
7. C₇×#1706	C₂	semidirect
8. C₇×C₈×M₁₆	C₂	semidirect
33. #19638	C₇	direct (N × H)
69. #3206	C₂×C₇	semidirect
71. C₂×#500	C₂×C₇	semidirect
72. #3141	C₂×C₇	semidirect
73. #1719	C₂×C₇	semidirect
74. #1710	C₂×C₇	semidirect
75. #1706	C₂×C₇	semidirect
76. C₈×M₁₆	C₂×C₇	semidirect
276. C₇	#19638	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×#3206	2
C₇×#3169	2
C₂×C₇×#500	2
C₇×#3141	2
C₇×#1719	2
C₇×#1710	2
C₇×#1706	2
C₇×C₈×M₁₆	2
#19638	7

Subgroups — C7×#19638 #13196854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19638 #13196854