Subgroups — C₇×#20774 #13197990

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₇×#1840	C₂	1	4	yes	maximal, complemented
3	C₇×#1839	C₂	1	4	yes	maximal, complemented
4	C₇×#1729	C₂	1	2	yes	maximal
5	C₇×#1729	C₂	1	2	yes	maximal
6	C₇×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₂×C₇×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
14	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
15	C₄×C₇×C₂²⋊C₄	C₄	1	4	yes	—
16	C₇×#376	C₂²	1	2	yes	—
17	C₄³×C₇	C₂²	1	2	yes	—
18	C₇×#336	C₂²	1	2	yes	—
19	C₇×#336	C₂²	1	2	yes	—
20	C₇×#336	C₂²	1	2	yes	—
21	C₇×#336	C₂²	1	2	yes	—
22	C₇×#336	C₂²	1	8	yes	—
23	C₇×#336	C₂²	1	8	yes	—
24	#20774	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	4	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	2	4	no	—
30	C₂²×C₇×C₈	—	2	4	no	—
31	C₂²×C₇×C₈	—	2	4	no	—
32	C₂²×C₇×C₈	—	2	4	no	—
33	C₂²×C₇×C₈	C₂³	1	4	yes	—
34	C₂²×C₇×C₈	C₂³	1	4	yes	—
35	C₇×#117	—	2	2	no	—
36	C₇×#117	—	2	4	no	—
37	C₇×#117	—	2	2	no	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇	—	2	4	no	—
44	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₇×#95	C₂×C₄	1	4	yes	—
52	C₇×#95	C₂×C₄	1	4	yes	—
53	C₄×#117	C₂×C₇	1	1	yes	characteristic
54	#1840	C₂×C₇	1	4	yes	complemented
55	#1839	C₂×C₇	1	4	yes	complemented
56	#1729	C₂×C₇	1	2	yes	—
57	#1729	C₂×C₇	1	2	yes	—
58	#1728	C₂×C₇	1	1	yes	characteristic
59	C₂×#336	C₂×C₇	1	1	yes	characteristic
60	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	#41	1	4	yes	—
71	C₂²×C₄×C₇	#41	1	2	yes	—
72	C₂²×C₄×C₇	#41	1	4	yes	—
73	C₂²×C₄×C₇	#41	1	2	yes	—
74	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
75	C₇×M₁₆	—	4	8	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	4	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	—	4	8	no	—
83	C₇×C₄⋊C₄	—	2	2	no	—
84	C₇×C₄⋊C₄	—	2	2	no	—
85	C₇×C₄⋊C₄	—	2	4	no	—
86	C₇×C₂²⋊C₄	—	2	4	no	—
87	C₇×C₂²⋊C₄	—	2	4	no	—
88	C₄²×C₇	—	2	4	no	—
89	C₄²×C₇	—	2	8	no	—
90	C₄²×C₇	—	2	2	no	—
91	C₄²×C₇	—	2	2	no	—
92	C₄²×C₇	—	2	4	no	—
93	C₄²×C₇	—	2	2	no	—
94	C₄²×C₇	—	2	2	no	—
95	C₄²×C₇	—	4	4	no	—
96	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
97	C₂³×C₈	C₂²×C₇	1	2	yes	—
98	C₂×#117	C₂²×C₇	1	1	yes	characteristic
99	C₂×#117	C₂²×C₇	1	1	yes	characteristic
100	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
101	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
102	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
103	C₄×C₂²⋊C₄	C₄×C₇	1	4	yes	—
104	#376	C₂²×C₇	1	2	yes	—
105	C₄³	C₂²×C₇	1	2	yes	—
106	#336	C₂²×C₇	1	2	yes	—
107	#336	C₂²×C₇	1	2	yes	—
108	#336	C₂²×C₇	1	8	yes	—
109	#336	C₂²×C₇	1	8	yes	—
110	#336	C₂²×C₇	1	2	yes	—
111	#336	C₂²×C₇	1	2	yes	—
112	C₂³×C₇	—	2	2	no	—
113	C₂³×C₇	—	2	2	no	—
114	C₂³×C₇	#117	1	2	yes	—
115	C₂³×C₇	#117	1	2	yes	—
116	C₂³×C₇	#117	1	2	yes	—
117	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	4	8	no	—
131	C₂×C₄×C₇	#117	1	2	yes	—
132	C₂×C₄×C₇	#117	1	2	yes	—
133	C₂×C₄×C₇	#117	1	2	yes	—
134	C₂×C₄×C₇	#126	1	8	yes	—
135	C₂×C₄×C₇	#126	1	8	yes	—
136	C₂×C₄×C₇	#131	1	4	yes	—
137	C₂×C₄×C₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
141	C₇×C₈	—	4	4	no	—
142	C₇×C₈	—	4	4	no	—
143	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
144	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	2	4	no	—
147	C₂×M₁₆	—	2	4	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	C₂³×C₇	1	4	yes	—
152	C₂²×C₈	C₂³×C₇	1	4	yes	—
153	#117	—	2	2	no	—
154	#117	—	2	4	no	—
155	#117	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
165	C₂×C₄²	C₂³×C₇	1	2	yes	—
166	C₂×C₄²	C₂³×C₇	1	2	yes	—
167	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
169	#95	C₂×C₄×C₇	1	4	yes	—
170	#95	C₂×C₄×C₇	1	4	yes	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	4	no	—
174	C₂²×C₇	—	2	4	no	—
175	C₂²×C₇	#383	1	4	yes	—
176	C₂²×C₇	C₂×#117	1	1	yes	characteristic
177	C₂²×C₇	C₂×#117	1	1	yes	characteristic
178	C₂²×C₇	C₂×#117	1	1	yes	characteristic
179	C₂²×C₇	C₂×#131	1	2	yes	—
180	C₂²×C₇	#568	1	2	yes	—
181	C₄×C₇	—	2	4	no	—
182	C₄×C₇	—	4	2	no	—
183	C₄×C₇	—	4	2	no	—
184	C₄×C₇	—	4	2	no	—
185	C₄×C₇	—	4	4	no	—
186	C₄×C₇	#383	1	4	yes	—
187	C₄×C₇	C₂×#126	1	2	yes	—
188	C₄×C₇	C₂×#126	1	2	yes	—
189	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
199	C₂²×C₄	C₇×#41	1	4	yes	—
200	C₂²×C₄	C₇×#41	1	2	yes	—
201	C₂²×C₄	C₇×#41	1	4	yes	—
202	C₂²×C₄	C₇×#41	1	2	yes	—
203	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
204	M₁₆	—	4	8	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	8	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	8	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	4	no	—
225	C₂×C₇	—	2	4	no	—
226	C₂×C₇	C₂×#383	1	1	yes	characteristic
227	C₂×C₇	#2922	1	1	yes	characteristic
228	C₂×C₇	#2923	1	1	yes	characteristic
229	C₂×C₇	#2924	1	2	yes	—
230	C₂×C₇	#2925	1	2	yes	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₇×#117	1	2	yes	—
234	C₂³	C₇×#117	1	2	yes	—
235	C₂³	C₇×#117	1	2	yes	—
236	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	C₇×#117	1	2	yes	—
251	C₂×C₄	C₇×#117	1	2	yes	—
252	C₂×C₄	C₇×#117	1	2	yes	—
253	C₂×C₄	C₇×#126	1	8	yes	—
254	C₂×C₄	C₇×#126	1	8	yes	—
255	C₂×C₄	C₇×#131	1	4	yes	—
256	C₂×C₄	C₂×C₇×#41	1	2	yes	—
257	C₂×C₄	C₂×C₇×#41	1	2	yes	—
258	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
259	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
260	C₈	—	4	4	no	—
261	C₈	—	4	4	no	—
262	C₇	#20774	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	C₇×#383	1	4	yes	—
268	C₂²	C₂×C₇×#117	1	1	yes	characteristic
269	C₂²	C₂×C₇×#117	1	1	yes	characteristic
270	C₂²	C₂×C₇×#117	1	1	yes	characteristic
271	C₂²	C₂×C₇×#131	1	2	yes	—
272	C₂²	C₇×#568	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	4	no	—
278	C₄	C₇×#383	1	4	yes	—
279	C₄	C₂×C₇×#126	1	2	yes	—
280	C₄	C₂×C₇×#126	1	2	yes	—
281	C₂	—	2	4	no	—
282	C₂	C₂×C₇×#383	1	1	yes	characteristic
283	C₂	C₇×#2922	1	1	yes	characteristic
284	C₂	C₇×#2923	1	1	yes	characteristic
285	C₂	C₇×#2924	1	2	yes	—
286	C₂	C₇×#2925	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₇×#1840	C₂	semidirect
3. C₇×#1839	C₂	semidirect
24. #20774	C₇	direct (N × H)
54. #1840	C₂×C₇	semidirect
55. #1839	C₂×C₇	semidirect
262. C₇	#20774	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#117	2
C₇×#1840	2
C₇×#1839	2
C₇×#1729	2
C₇×#1728	2
C₂×C₇×#336	2
#20774	7

Subgroups — C7×#20774 #13197990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20774 #13197990