Subgroups — C₇×#10276 #13187492

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#475	C₂	1	2	yes	maximal
3	C₇×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#332	C₂	1	2	yes	maximal
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	—	2	2	no	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×#494	—	2	4	no	—
10	C₇×#475	—	2	4	no	—
11	C₇×#475	—	2	4	no	—
12	C₂×C₇×#106	—	2	2	no	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×#103	—	2	2	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#336	—	2	2	no	—
18	C₇×#332	—	2	4	no	—
19	#10276	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
22	C₂×C₇×Q₁₆	—	2	8	no	—
23	C₂×C₇×Q₁₆	—	4	4	no	—
24	C₂×C₇×Q₁₆	—	4	4	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	—	2	1	no	—
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₇×C₄⋊Q₈	—	2	2	no	—
29	C₇×C₄⋊Q₈	—	2	2	no	—
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×C₄⋊Q₈	D₄	1	4	yes	—
32	C₄×C₇×Q₈	—	2	2	no	—
33	C₄×C₇×Q₈	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	1	no	—
37	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	—	2	1	no	—
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₇×#106	—	2	8	no	—
41	C₇×C₄⋊C₈	—	2	2	no	—
42	C₇×C₄⋊C₈	—	2	2	no	—
43	C₇×C₄⋊C₈	D₄	1	4	yes	—
44	C₇×#103	—	2	8	no	—
45	C₄×C₇×C₈	—	2	2	no	—
46	C₂×#494	C₂×C₇	1	1	yes	characteristic
47	C₂×#475	C₂×C₇	1	2	yes	—
48	#1984	C₂×C₇	1	1	yes	characteristic
49	C₂×#332	C₂×C₇	1	2	yes	—
50	#1277	C₂×C₇	1	1	yes	characteristic
51	C₂×C₇×Q₈	—	2	1	no	—
52	C₂×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×Q₈	—	2	4	no	—
54	C₂×C₇×Q₈	—	2	2	no	—
55	C₂×C₇×Q₈	—	4	2	no	—
56	C₂×C₇×Q₈	—	4	4	no	—
57	C₂×C₇×Q₈	—	4	4	no	—
58	C₂×C₇×Q₈	Q₁₆	1	2	yes	complemented
59	C₂²×C₄×C₇	—	2	1	no	—
60	C₂²×C₄×C₇	—	2	1	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	#41	1	1	yes	characteristic
65	C₇×Q₁₆	—	4	16	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	4	no	—
69	C₂×C₇×C₈	—	4	2	no	—
70	C₂×C₇×C₈	—	4	2	no	—
71	C₂×C₇×C₈	—	4	1	no	—
72	C₂×C₇×C₈	—	4	2	no	—
73	C₂×C₇×C₈	SD₁₆	1	4	yes	—
74	C₇×C₄⋊C₄	—	2	4	no	—
75	C₇×C₄⋊C₄	—	2	2	no	—
76	C₇×C₄⋊C₄	—	2	4	no	—
77	C₇×C₄⋊C₄	—	4	2	no	—
78	C₇×C₄⋊C₄	—	4	2	no	—
79	C₇×C₄⋊C₄	—	4	2	no	—
80	C₄²×C₇	—	4	1	no	—
81	C₄²×C₇	—	4	1	no	—
82	C₄²×C₇	C₂×D₄	1	2	yes	—
83	C₄²×C₇	C₂×D₄	1	2	yes	—
84	C₂²×Q₁₆	—	2	2	no	—
85	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
86	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
87	#494	—	2	4	no	—
88	#475	—	2	4	no	—
89	#475	—	2	4	no	—
90	C₂×#106	—	2	2	no	—
91	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
92	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
93	C₂×#103	—	2	2	no	—
94	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
95	#336	—	2	2	no	—
96	#332	—	2	4	no	—
97	C₂³×C₇	#121	1	1	yes	characteristic
98	C₇×Q₈	—	2	4	no	—
99	C₇×Q₈	—	4	2	no	—
100	C₇×Q₈	—	4	8	no	—
101	C₇×Q₈	—	8	2	no	—
102	C₇×Q₈	—	8	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	4	1	no	—
107	C₂×C₄×C₇	—	4	1	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	—	8	4	no	—
111	C₂×C₄×C₇	#120	1	1	yes	characteristic
112	C₂×C₄×C₇	#120	1	2	yes	—
113	C₂×C₄×C₇	#121	1	2	yes	—
114	C₂×C₄×C₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇	#135	1	1	yes	characteristic
118	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₇	#137	1	2	yes	—
120	C₇×C₈	—	2	4	no	—
121	C₇×C₈	—	4	4	no	—
122	C₇×C₈	—	8	1	no	—
123	C₂²×Q₈	—	2	2	no	—
124	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
125	C₂×Q₁₆	—	2	8	no	—
126	C₂×Q₁₆	—	4	4	no	—
127	C₂×Q₁₆	—	4	4	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	C₇×D₄	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₇×D₄	1	4	yes	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
143	#106	—	2	8	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	C₇×D₄	1	4	yes	—
147	#103	—	2	8	no	—
148	C₄×C₈	—	2	2	no	—
149	C₂²×C₇	#392	1	1	yes	characteristic
150	C₂²×C₇	#460	1	2	yes	—
151	C₂²×C₇	#467	1	1	yes	characteristic
152	C₂²×C₇	#470	1	1	yes	characteristic
153	C₂²×C₇	#473	1	2	yes	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	4	2	no	—
156	C₄×C₇	—	8	4	no	—
157	C₄×C₇	—	8	1	no	—
158	C₄×C₇	#450	1	2	yes	—
159	C₄×C₇	#451	1	2	yes	—
160	C₄×C₇	#452	1	2	yes	—
161	C₄×C₇	#492	1	1	yes	characteristic
162	C₄×C₇	#496	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	4	4	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	C₇×Q₁₆	1	2	yes	complemented
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
176	C₂²×C₄	C₇×#41	1	1	yes	characteristic
177	Q₁₆	—	4	16	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₇×SD₁₆	1	4	yes	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	C₂×C₇×D₄	1	2	yes	—
195	C₄²	C₂×C₇×D₄	1	2	yes	—
196	C₂×C₇	#1670	1	2	yes	—
197	C₂×C₇	#1690	1	2	yes	—
198	C₂×C₇	#2001	1	1	yes	characteristic
199	C₂×C₇	#2014	1	1	yes	characteristic
200	C₂×C₇	#2020	1	1	yes	characteristic
201	C₂³	C₇×#121	1	1	yes	characteristic
202	Q₈	—	2	4	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	4	8	no	—
205	Q₈	—	8	2	no	—
206	Q₈	—	8	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	C₇×#120	1	1	yes	characteristic
216	C₂×C₄	C₇×#120	1	2	yes	—
217	C₂×C₄	C₇×#121	1	2	yes	—
218	C₂×C₄	C₇×#127	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
220	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
221	C₂×C₄	C₇×#135	1	1	yes	characteristic
222	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
223	C₂×C₄	C₇×#137	1	2	yes	—
224	C₈	—	2	4	no	—
225	C₈	—	4	4	no	—
226	C₈	—	8	1	no	—
227	C₇	#10276	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	C₇×#392	1	1	yes	characteristic
229	C₂²	C₇×#460	1	2	yes	—
230	C₂²	C₇×#467	1	1	yes	characteristic
231	C₂²	C₇×#470	1	1	yes	characteristic
232	C₂²	C₇×#473	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	4	no	—
236	C₄	—	8	1	no	—
237	C₄	C₇×#450	1	2	yes	—
238	C₄	C₇×#451	1	2	yes	—
239	C₄	C₇×#452	1	2	yes	—
240	C₄	C₇×#492	1	1	yes	characteristic
241	C₄	C₇×#496	1	1	yes	characteristic
242	C₂	C₇×#1670	1	2	yes	—
243	C₂	C₇×#1690	1	2	yes	—
244	C₂	C₇×#2001	1	1	yes	characteristic
245	C₂	C₇×#2014	1	1	yes	characteristic
246	C₂	C₇×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
19. #10276	C₇	direct (N × H)
58. C₂×C₇×Q₈	Q₁₆	semidirect
170. C₂×Q₈	C₇×Q₁₆	semidirect
227. C₇	#10276	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
C₂×C₇×#475	2
C₇×#1984	2
C₂×C₇×#332	2
C₇×#1277	2
#10276	7

Subgroups — C7×#10276 #13187492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10276 #13187492