Subgroups — C₇×#21340 #13198556

Order 1792.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1985	C₂	1	2	yes	maximal
3	C₇×#1940	C₂	1	2	yes	maximal
4	C₇×#1917	C₂	1	4	yes	maximal
5	C₄×C₇×C₄⋊C₈	C₂	1	2	yes	maximal
6	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₇×#1744	C₂	1	2	yes	maximal
8	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₄	1	2	yes	—
10	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₇×#97	C₂²	1	2	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
19	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄³×C₇	C₂²	1	1	yes	characteristic
24	C₄³×C₇	C₂²	1	2	yes	—
25	C₇×#336	C₂²	1	4	yes	—
26	C₇×#336	C₂²	1	4	yes	—
27	C₇×#336	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#21340	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	4	yes	—
32	C₂²×C₇×C₈	C₂³	1	2	yes	—
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₄×C₇×Q₈	—	2	8	no	—
35	C₄×C₇×Q₈	—	2	4	no	—
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₇×C₄⋊C₈	—	2	8	no	—
53	C₇×C₄⋊C₈	—	2	4	no	—
54	C₇×#97	—	2	4	no	—
55	C₇×#97	—	2	4	no	—
56	C₄×C₇×C₈	—	2	2	no	—
57	C₄×C₇×C₈	—	2	4	no	—
58	C₄×C₇×C₈	Q₈	1	4	yes	complemented
59	C₇×#95	C₂×C₄	1	2	yes	—
60	C₇×#95	C₂×C₄	1	1	yes	characteristic
61	C₇×#95	C₂×C₄	1	2	yes	—
62	C₇×#95	C₂×C₄	1	1	yes	characteristic
63	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
64	#1985	C₂×C₇	1	2	yes	—
65	#1940	C₂×C₇	1	2	yes	—
66	#1917	C₂×C₇	1	4	yes	—
67	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
68	C₄×C₄⋊C₈	C₂×C₇	1	2	yes	—
69	#1744	C₂×C₇	1	2	yes	—
70	C₄×#97	C₂×C₇	1	1	yes	characteristic
71	C₂×C₇×Q₈	—	2	2	no	—
72	C₂×C₇×Q₈	—	2	4	no	—
73	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
90	C₂×C₇×C₈	#41	1	4	yes	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	2	2	no	—
93	C₇×C₄⋊C₄	—	2	4	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₄²×C₇	—	2	4	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	—	2	4	no	—
100	C₄²×C₇	—	2	4	no	—
101	C₄²×C₇	—	2	2	no	—
102	C₄²×C₇	—	2	4	no	—
103	C₄²×C₇	—	2	4	no	—
104	C₄²×C₇	M₁₆	1	8	yes	—
105	C₄²×C₇	C₂×Q₈	1	2	yes	—
106	C₄²×C₇	#41	1	4	yes	—
107	C₄²×C₇	#41	1	2	yes	—
108	C₂×C₄×Q₈	C₄×C₇	1	2	yes	—
109	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
111	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
112	C₂×#97	C₂²×C₇	1	2	yes	—
113	C₂×#97	C₂²×C₇	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
117	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
118	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
119	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
120	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
121	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
122	C₄³	C₂²×C₇	1	1	yes	characteristic
123	C₄³	C₂²×C₇	1	2	yes	—
124	#336	C₂²×C₇	1	2	yes	—
125	#336	C₂²×C₇	1	4	yes	—
126	#336	C₂²×C₇	1	4	yes	—
127	#336	C₂²×C₇	1	4	yes	—
128	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
129	C₇×Q₈	—	2	8	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	2	2	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	#117	1	2	yes	—
146	C₂×C₄×C₇	#117	1	2	yes	—
147	C₂×C₄×C₇	C₄×Q₈	1	4	yes	—
148	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
149	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
151	C₂×C₄×C₇	#131	1	4	yes	—
152	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×#41	1	2	yes	—
155	C₂×C₄×C₇	C₂×#41	1	2	yes	—
156	C₇×C₈	—	2	4	no	—
157	C₇×C₈	—	2	4	no	—
158	C₇×C₈	—	4	4	no	—
159	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₇	1	4	yes	—
161	C₂²×C₈	C₂³×C₇	1	2	yes	—
162	C₂²×C₈	C₂³×C₇	1	2	yes	—
163	C₄×Q₈	—	2	8	no	—
164	C₄×Q₈	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
173	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
175	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
176	C₂×C₄²	C₂³×C₇	1	2	yes	—
177	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₇	1	2	yes	—
179	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	8	no	—
182	C₄⋊C₈	—	2	4	no	—
183	#97	—	2	4	no	—
184	#97	—	2	4	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₇×Q₈	1	4	yes	complemented
188	#95	C₂×C₄×C₇	1	2	yes	—
189	#95	C₂×C₄×C₇	1	1	yes	characteristic
190	#95	C₂×C₄×C₇	1	2	yes	—
191	#95	C₂×C₄×C₇	1	1	yes	characteristic
192	C₂²×C₇	C₂×#117	1	1	yes	characteristic
193	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
194	C₂²×C₇	C₄×#41	1	1	yes	characteristic
195	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
196	C₂²×C₇	C₂×#131	1	2	yes	—
197	C₄×C₇	—	2	2	no	—
198	C₄×C₇	—	2	4	no	—
199	C₄×C₇	—	2	2	no	—
200	C₄×C₇	—	2	4	no	—
201	C₄×C₇	—	4	2	no	—
202	C₄×C₇	—	4	2	no	—
203	C₄×C₇	—	4	2	no	—
204	C₄×C₇	#446	1	8	yes	—
205	C₄×C₇	#529	1	2	yes	—
206	C₄×C₇	#533	1	2	yes	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
226	C₂×C₈	C₇×#41	1	4	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	C₇×M₁₆	1	8	yes	—
241	C₄²	C₂×C₇×Q₈	1	2	yes	—
242	C₄²	C₇×#41	1	4	yes	—
243	C₄²	C₇×#41	1	2	yes	—
244	C₂×C₇	#2295	1	1	yes	characteristic
245	C₂×C₇	#2920	1	1	yes	characteristic
246	C₂×C₇	#2922	1	1	yes	characteristic
247	C₂×C₇	C₂×#446	1	2	yes	—
248	C₂×C₇	#2965	1	2	yes	—
249	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
250	Q₈	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	C₇×#117	1	2	yes	—
267	C₂×C₄	C₇×#117	1	2	yes	—
268	C₂×C₄	C₄×C₇×Q₈	1	4	yes	—
269	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
270	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
271	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
272	C₂×C₄	C₇×#131	1	4	yes	—
273	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
274	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×#41	1	2	yes	—
276	C₂×C₄	C₂×C₇×#41	1	2	yes	—
277	C₈	—	2	4	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	4	no	—
280	C₇	#21340	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	C₂×C₇×#117	1	1	yes	characteristic
282	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
283	C₂²	C₄×C₇×#41	1	1	yes	characteristic
284	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
285	C₂²	C₂×C₇×#131	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	C₇×#446	1	8	yes	—
294	C₄	C₇×#529	1	2	yes	—
295	C₄	C₇×#533	1	2	yes	—
296	C₂	C₇×#2295	1	1	yes	characteristic
297	C₂	C₇×#2920	1	1	yes	characteristic
298	C₂	C₇×#2922	1	1	yes	characteristic
299	C₂	C₂×C₇×#446	1	2	yes	—
300	C₂	C₇×#2965	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₇	C₂³
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
29. #21340	C₇	direct (N × H)
58. C₄×C₇×C₈	Q₈	semidirect
187. C₄×C₈	C₇×Q₈	semidirect
280. C₇	#21340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₇×Q₈	2
C₇×#1985	2
C₇×#1940	2
C₇×#1917	2
C₄×C₇×C₄⋊C₈	2
C₇×#1744	2
C₄×C₇×#97	2
#21340	7

Subgroups — C7×#21340 #13198556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21340 #13198556