Subgroups — C₇×#14712 #13191928

Order 1792.

1854 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#389	C₂	1	6	yes	maximal
2	C₂×C₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#384	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
5	C₇×C₄⋊#95	C₂	1	6	yes	maximal
6	C₂²×C₇×C₄⋊C₄	C₂²	1	6	yes	—
7	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₇×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂²×C₄²×C₇	C₂²	1	3	yes	—
10	C₂²×C₄²×C₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₇×#389	—	2	24	no	—
13	C₇×#384	—	2	4	no	—
14	C₇×#384	C₄	1	8	yes	complemented
15	C₂×C₇×#95	C₂²	1	12	yes	—
16	C₄³×C₇	—	2	4	no	—
17	#14712	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂³	1	4	yes	—
19	C₂³×C₄×C₇	C₂³	1	4	yes	—
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂³	1	3	yes	—
22	C₂³×C₄×C₇	C₂³	1	3	yes	—
23	C₂×C₇×C₄⋊C₄	—	2	12	no	—
24	C₂×C₇×C₄⋊C₄	—	2	12	no	—
25	C₂×C₇×C₄⋊C₄	—	2	12	no	—
26	C₂×C₇×C₄⋊C₄	—	2	12	no	—
27	C₂×C₇×C₄⋊C₄	—	2	24	no	—
28	C₂×C₇×C₄⋊C₄	—	2	12	no	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	24	yes	—
30	C₂×C₄²×C₇	—	2	6	no	—
31	C₂×C₄²×C₇	—	2	6	no	—
32	C₂×C₄²×C₇	—	2	6	no	—
33	C₂×C₄²×C₇	—	2	6	no	—
34	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₇	D₄	1	12	yes	—
36	C₂×C₄²×C₇	D₄	1	4	yes	—
37	C₂×C₄²×C₇	Q₈	1	12	yes	—
38	C₂×C₄²×C₇	Q₈	1	4	yes	—
39	C₇×#95	—	2	24	no	—
40	C₂×#389	C₂×C₇	1	6	yes	—
41	C₂×#384	C₂×C₇	1	1	yes	characteristic
42	C₂×#384	C₂×C₇	1	1	yes	characteristic
43	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
44	C₄⋊#95	C₂×C₇	1	6	yes	—
45	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	—	2	3	no	—
48	C₂²×C₄×C₇	—	2	3	no	—
49	C₂²×C₄×C₇	—	2	3	no	—
50	C₂²×C₄×C₇	—	2	6	no	—
51	C₂²×C₄×C₇	—	2	12	no	—
52	C₂²×C₄×C₇	—	2	12	no	—
53	C₂²×C₄×C₇	—	2	12	no	—
54	C₂²×C₄×C₇	—	2	4	no	—
55	C₂²×C₄×C₇	—	2	12	no	—
56	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
59	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₇	C₂×Q₈	1	12	yes	—
63	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
64	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
65	C₂²×C₄×C₇	#41	1	6	yes	—
66	C₂²×C₄×C₇	#41	1	6	yes	—
67	C₇×C₄⋊C₄	—	2	48	no	—
68	C₇×C₄⋊C₄	—	4	24	no	—
69	C₇×C₄⋊C₄	—	4	16	no	—
70	C₄²×C₇	—	2	24	no	—
71	C₄²×C₇	—	2	24	no	—
72	C₄²×C₇	C₄⋊C₄	1	16	yes	complemented
73	C₂²×C₄⋊C₄	C₂²×C₇	1	6	yes	—
74	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
75	C₂²×C₄⋊C₄	C₂²×C₇	1	6	yes	—
76	C₂²×C₄²	C₂²×C₇	1	3	yes	—
77	C₂²×C₄²	C₂²×C₇	1	3	yes	—
78	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
79	#389	—	2	24	no	—
80	#384	—	2	4	no	—
81	#384	C₄×C₇	1	8	yes	complemented
82	C₂×#95	C₂²×C₇	1	12	yes	—
83	C₄³	—	2	4	no	—
84	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
85	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₇	C₂²×D₄	1	3	yes	—
87	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂³×C₇	C₂²×Q₈	1	3	yes	—
89	C₂³×C₇	C₂×#41	1	3	yes	—
90	C₂³×C₇	C₂×#41	1	3	yes	—
91	C₂×C₄×C₇	—	2	6	no	—
92	C₂×C₄×C₇	—	2	6	no	—
93	C₂×C₄×C₇	—	2	12	no	—
94	C₂×C₄×C₇	—	2	6	no	—
95	C₂×C₄×C₇	—	2	6	no	—
96	C₂×C₄×C₇	—	2	12	no	—
97	C₂×C₄×C₇	—	2	24	no	—
98	C₂×C₄×C₇	—	4	4	no	—
99	C₂×C₄×C₇	—	4	12	no	—
100	C₂×C₄×C₇	—	4	12	no	—
101	C₂×C₄×C₇	—	4	4	no	—
102	C₂×C₄×C₇	—	4	12	no	—
103	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
104	C₂×C₄×C₇	C₄×D₄	1	12	yes	—
105	C₂×C₄×C₇	C₄×Q₈	1	12	yes	—
106	C₂×C₄×C₇	#121	1	12	yes	—
107	C₂×C₄×C₇	C₂²⋊Q₈	1	12	yes	—
108	C₂×C₄×C₇	#125	1	12	yes	—
109	C₂×C₄×C₇	#127	1	4	yes	—
110	C₂×C₄×C₇	C₄⋊Q₈	1	12	yes	—
111	C₂×C₄×C₇	C₄⋊Q₈	1	12	yes	—
112	C₂³×C₄	C₂³×C₇	1	4	yes	—
113	C₂³×C₄	C₂³×C₇	1	4	yes	—
114	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
115	C₂³×C₄	C₂³×C₇	1	3	yes	—
116	C₂³×C₄	C₂³×C₇	1	3	yes	—
117	C₂×C₄⋊C₄	—	2	12	no	—
118	C₂×C₄⋊C₄	—	2	12	no	—
119	C₂×C₄⋊C₄	—	2	12	no	—
120	C₂×C₄⋊C₄	—	2	12	no	—
121	C₂×C₄⋊C₄	—	2	24	no	—
122	C₂×C₄⋊C₄	—	2	12	no	—
123	C₂×C₄⋊C₄	C₂×C₄×C₇	1	24	yes	—
124	C₂×C₄²	—	2	6	no	—
125	C₂×C₄²	—	2	6	no	—
126	C₂×C₄²	—	2	6	no	—
127	C₂×C₄²	—	2	6	no	—
128	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
129	C₂×C₄²	C₇×D₄	1	12	yes	—
130	C₂×C₄²	C₇×D₄	1	4	yes	—
131	C₂×C₄²	C₇×Q₈	1	12	yes	—
132	C₂×C₄²	C₇×Q₈	1	4	yes	—
133	#95	—	2	24	no	—
134	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
136	C₂²×C₇	C₂×C₄×Q₈	1	3	yes	—
137	C₂²×C₇	C₂×#121	1	3	yes	—
138	C₂²×C₇	C₂×C₂²⋊Q₈	1	3	yes	—
139	C₂²×C₇	C₂×#125	1	3	yes	—
140	C₂²×C₇	C₂×#127	1	1	yes	characteristic
141	C₂²×C₇	C₂×C₄⋊Q₈	1	3	yes	—
142	C₂²×C₇	C₂×C₄⋊Q₈	1	3	yes	—
143	C₂²×C₇	#532	1	3	yes	—
144	C₂²×C₇	#532	1	3	yes	—
145	C₂²×C₇	#533	1	6	yes	—
146	C₄×C₇	—	2	4	no	—
147	C₄×C₇	—	2	12	no	—
148	C₄×C₇	—	4	8	no	—
149	C₄×C₇	—	8	4	no	—
150	C₄×C₇	#389	1	24	yes	complemented
151	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	3	no	—
154	C₂²×C₄	—	2	3	no	—
155	C₂²×C₄	—	2	3	no	—
156	C₂²×C₄	—	2	6	no	—
157	C₂²×C₄	—	2	12	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	2	12	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
163	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
164	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
165	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
166	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
167	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
168	C₂²×C₄	C₂×C₇×Q₈	1	12	yes	—
169	C₂²×C₄	C₂×C₇×Q₈	1	6	yes	—
170	C₂²×C₄	C₂×C₇×Q₈	1	6	yes	—
171	C₂²×C₄	C₇×#41	1	6	yes	—
172	C₂²×C₄	C₇×#41	1	6	yes	—
173	C₄⋊C₄	—	2	48	no	—
174	C₄⋊C₄	—	4	24	no	—
175	C₄⋊C₄	—	4	16	no	—
176	C₄²	—	2	24	no	—
177	C₄²	—	2	24	no	—
178	C₄²	C₇×C₄⋊C₄	1	16	yes	complemented
179	C₂×C₇	C₂×#389	1	3	yes	—
180	C₂×C₇	C₄×#127	1	1	yes	characteristic
181	C₂×C₇	C₄×C₄⋊Q₈	1	3	yes	—
182	C₂×C₇	#2856	1	1	yes	characteristic
183	C₂×C₇	#2857	1	3	yes	—
184	C₂×C₇	#2859	1	3	yes	—
185	C₂×C₇	#2868	1	1	yes	characteristic
186	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
187	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
188	C₂³	C₂²×C₇×D₄	1	3	yes	—
189	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
190	C₂³	C₂²×C₇×Q₈	1	3	yes	—
191	C₂³	C₂×C₇×#41	1	3	yes	—
192	C₂³	C₂×C₇×#41	1	3	yes	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	24	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	12	no	—
202	C₂×C₄	—	4	12	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	12	no	—
205	C₂×C₄	C₂×C₇×C₄⋊C₄	1	12	yes	—
206	C₂×C₄	C₄×C₇×D₄	1	12	yes	—
207	C₂×C₄	C₄×C₇×Q₈	1	12	yes	—
208	C₂×C₄	C₇×#121	1	12	yes	—
209	C₂×C₄	C₇×C₂²⋊Q₈	1	12	yes	—
210	C₂×C₄	C₇×#125	1	12	yes	—
211	C₂×C₄	C₇×#127	1	4	yes	—
212	C₂×C₄	C₇×C₄⋊Q₈	1	12	yes	—
213	C₂×C₄	C₇×C₄⋊Q₈	1	12	yes	—
214	C₇	#14712	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
216	C₂²	C₂×C₄×C₇×D₄	1	3	yes	—
217	C₂²	C₂×C₄×C₇×Q₈	1	3	yes	—
218	C₂²	C₂×C₇×#121	1	3	yes	—
219	C₂²	C₂×C₇×C₂²⋊Q₈	1	3	yes	—
220	C₂²	C₂×C₇×#125	1	3	yes	—
221	C₂²	C₂×C₇×#127	1	1	yes	characteristic
222	C₂²	C₂×C₇×C₄⋊Q₈	1	3	yes	—
223	C₂²	C₂×C₇×C₄⋊Q₈	1	3	yes	—
224	C₂²	C₇×#532	1	3	yes	—
225	C₂²	C₇×#532	1	3	yes	—
226	C₂²	C₇×#533	1	6	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	2	12	no	—
229	C₄	—	4	8	no	—
230	C₄	—	8	4	no	—
231	C₄	C₇×#389	1	24	yes	complemented
232	C₂	C₂×C₇×#389	1	3	yes	—
233	C₂	C₄×C₇×#127	1	1	yes	characteristic
234	C₂	C₄×C₇×C₄⋊Q₈	1	3	yes	—
235	C₂	C₇×#2856	1	1	yes	characteristic
236	C₂	C₇×#2857	1	3	yes	—
237	C₂	C₇×#2859	1	3	yes	—
238	C₂	C₇×#2868	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
14. C₇×#384	C₄	semidirect
17. #14712	C₇	direct (N × H)
72. C₄²×C₇	C₄⋊C₄	semidirect
81. #384	C₄×C₇	semidirect
150. C₄×C₇	#389	semidirect
178. C₄²	C₇×C₄⋊C₄	semidirect
214. C₇	#14712	direct (N × H)
231. C₄	C₇×#389	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#389	2
C₂×C₇×#384	2
C₂×C₄³×C₇	2
C₇×C₄⋊#95	2
#14712	7

Subgroups — C7×#14712 #13191928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#14712 #13191928