Subgroups — C₇×#19192 #13196408

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 324 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
2	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₇×#1693	C₂	1	4	yes	maximal, complemented
5	C₇×#1673	C₂	1	4	yes	maximal, complemented
6	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1549	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×#492	—	2	2	no	—
16	C₇×#487	—	2	2	no	—
17	C₇×#461	C₂²	1	8	yes	—
18	C₇×#448	—	2	4	no	—
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×#332	C₂²	1	8	yes	—
32	#19192	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	C₂×C₇×SD₁₆	—	2	8	no	—
36	C₇×#131	—	4	4	no	—
37	C₂×C₇×M₁₆	—	2	1	no	—
38	C₂×C₇×M₁₆	—	2	2	no	—
39	C₂²×C₇×C₈	—	2	2	no	—
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×C₄⋊Q₈	—	2	2	no	—
45	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
47	C₇×#127	D₄	1	1	yes	characteristic
48	C₇×#124	—	2	1	no	—
49	C₇×#121	—	2	1	no	—
50	C₇×#121	D₄	1	2	yes	—
51	C₄×C₇×D₄	—	2	1	no	—
52	C₄×C₇×D₄	C₂³	1	2	yes	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₇×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₇×#107	—	2	2	no	—
57	C₇×#107	—	2	2	no	—
58	C₇×#107	—	2	2	no	—
59	C₇×#107	—	2	2	no	—
60	C₇×C₄⋊C₈	—	2	1	no	—
61	C₇×C₄⋊C₈	C₂³	1	2	yes	—
62	C₇×C₄⋊C₈	C₂³	1	4	yes	—
63	C₇×#103	—	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	#3206	C₂×C₇	1	2	yes	—
73	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
74	#2931	C₂×C₇	1	1	yes	characteristic
75	#1693	C₂×C₇	1	4	yes	complemented
76	#1673	C₂×C₇	1	4	yes	complemented
77	#1564	C₂×C₇	1	1	yes	characteristic, complemented
78	#1549	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×SD₁₆	—	8	8	no	—
95	C₇×M₁₆	—	4	2	no	—
96	C₇×M₁₆	—	4	4	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	2	4	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	4	4	no	—
101	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
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	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
106	C₇×C₄⋊C₄	—	2	1	no	—
107	C₇×C₄⋊C₄	—	2	4	no	—
108	C₇×C₄⋊C₄	—	2	4	no	—
109	C₇×C₄⋊C₄	—	4	2	no	—
110	C₇×C₄⋊C₄	—	4	2	no	—
111	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
112	C₇×C₂²⋊C₄	—	2	2	no	—
113	C₇×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇	#41	1	1	yes	characteristic
117	C₄²×C₇	C₂⁴	1	1	yes	characteristic
118	C₂×#131	—	2	1	no	—
119	#532	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
121	#500	—	2	1	no	—
122	#500	—	2	1	no	—
123	#500	C₂²×C₇	1	2	yes	—
124	#500	C₂²×C₇	1	2	yes	—
125	#492	—	2	2	no	—
126	#487	—	2	2	no	—
127	#461	C₂²×C₇	1	8	yes	—
128	#448	—	2	4	no	—
129	#436	—	2	1	no	—
130	#436	C₂²×C₇	1	2	yes	—
131	C₈×D₄	—	2	1	no	—
132	C₈×D₄	C₂²×C₇	1	2	yes	—
133	#431	C₂²×C₇	1	2	yes	—
134	C₂×#107	—	2	1	no	—
135	C₂×#107	—	2	1	no	—
136	#423	C₂²×C₇	1	1	yes	characteristic
137	#408	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
139	#335	C₂²×C₇	1	2	yes	—
140	#334	C₂²×C₇	1	2	yes	—
141	#332	C₂²×C₇	1	8	yes	—
142	C₂³×C₇	—	2	2	no	—
143	C₂³×C₇	#121	1	1	yes	characteristic
144	C₇×Q₈	—	4	2	no	—
145	C₇×Q₈	—	4	8	no	—
146	C₇×Q₈	—	8	4	no	—
147	C₇×D₄	—	2	4	no	—
148	C₇×D₄	—	4	4	no	—
149	C₇×D₄	—	4	2	no	—
150	C₇×D₄	—	4	2	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	4	no	—
157	C₂×C₄×C₇	—	8	4	no	—
158	C₂×C₄×C₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₇	#121	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
162	C₂×C₄×C₇	#137	1	2	yes	—
163	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
166	C₇×C₈	—	2	1	no	—
167	C₇×C₈	—	2	1	no	—
168	C₇×C₈	—	4	2	no	—
169	C₇×C₈	—	4	4	no	—
170	C₇×C₈	C₂×SD₁₆	1	2	yes	—
171	C₇×C₈	C₂×SD₁₆	1	2	yes	—
172	C₂×#41	—	2	1	no	—
173	C₂²×Q₈	—	2	2	no	—
174	C₂×SD₁₆	—	2	8	no	—
175	#131	—	4	4	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
180	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
185	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
186	#127	C₇×D₄	1	1	yes	characteristic
187	#124	—	2	1	no	—
188	#121	—	2	1	no	—
189	#121	C₇×D₄	1	2	yes	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	C₂³×C₇	1	2	yes	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
195	#107	—	2	2	no	—
196	#107	—	2	2	no	—
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	C₂³×C₇	1	2	yes	—
201	C₄⋊C₈	C₂³×C₇	1	4	yes	—
202	#103	—	2	8	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	C₇×D₄	1	1	yes	characteristic
208	C₄×C₈	C₇×D₄	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	—	2	1	no	—
213	C₂²×C₇	—	4	2	no	—
214	C₂²×C₇	—	4	2	no	—
215	C₂²×C₇	—	4	2	no	—
216	C₂²×C₇	#461	1	2	yes	—
217	C₂²×C₇	C₂×#121	1	1	yes	characteristic
218	C₄×C₇	—	2	1	no	—
219	C₄×C₇	—	2	1	no	—
220	C₄×C₇	—	4	2	no	—
221	C₄×C₇	—	8	4	no	—
222	C₄×C₇	#461	1	2	yes	—
223	C₄×C₇	#473	1	2	yes	—
224	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
225	C₄×C₇	C₂×#137	1	1	yes	characteristic
226	C₄×C₇	#576	1	2	yes	—
227	#41	—	4	2	no	—
228	#41	—	4	2	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	8	4	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
241	C₂²×C₄	C₇×#41	1	1	yes	characteristic
242	SD₁₆	—	8	8	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	4	no	—
249	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
250	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
252	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
263	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
264	C₄²	C₇×#41	1	1	yes	characteristic
265	C₄²	C₂⁴×C₇	1	1	yes	characteristic
266	C₂×C₇	—	2	2	no	—
267	C₂×C₇	—	8	2	no	—
268	C₂×C₇	C₂×#461	1	1	yes	characteristic
269	C₂×C₇	#3058	1	1	yes	characteristic
270	C₂×C₇	C₂×#473	1	1	yes	characteristic
271	C₂³	—	2	2	no	—
272	C₂³	C₇×#121	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	8	no	—
275	Q₈	—	8	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	C₇×#121	1	1	yes	characteristic
288	C₂×C₄	C₇×#121	1	1	yes	characteristic
289	C₂×C₄	C₇×#121	1	1	yes	characteristic
290	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
291	C₂×C₄	C₇×#137	1	2	yes	—
292	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	4	2	no	—
298	C₈	—	4	4	no	—
299	C₈	C₂×C₇×SD₁₆	1	2	yes	—
300	C₈	C₂×C₇×SD₁₆	1	2	yes	—
301	C₇	#19192	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₇×#461	1	2	yes	—
308	C₂²	C₂×C₇×#121	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₇×#461	1	2	yes	—
314	C₄	C₇×#473	1	2	yes	—
315	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
316	C₄	C₂×C₇×#137	1	1	yes	characteristic
317	C₄	C₇×#576	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×C₇×#461	1	1	yes	characteristic
321	C₂	C₇×#3058	1	1	yes	characteristic
322	C₂	C₂×C₇×#473	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×#1693	C₂	semidirect
5. C₇×#1673	C₂	semidirect
6. C₇×#1564	C₂	semidirect
7. C₇×#1549	C₂	semidirect
32. #19192	C₇	direct (N × H)
73. C₂×#500	C₂×C₇	semidirect
75. #1693	C₂×C₇	semidirect
76. #1673	C₂×C₇	semidirect
77. #1564	C₂×C₇	semidirect
78. #1549	C₂×C₇	semidirect
101. C₂×C₇×C₈	SD₁₆	semidirect
249. C₂×C₈	C₇×SD₁₆	semidirect
301. C₇	#19192	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
C₂×C₇×#500	2
C₇×#2931	2
C₇×#1693	2
C₇×#1673	2
C₇×#1564	2
C₇×#1549	2
#19192	7

Subgroups — C7×#19192 #13196408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19192 #13196408