Subgroups — C₇×#10274 #13187490

Order 1792.

626 conjugacy classes of subgroups.

The subgroup lattice has 274 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, complemented
2	C₂×C₇×#474	C₂	1	2	yes	maximal
3	C₇×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#332	C₂	1	2	yes	maximal, complemented
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×Q₁₆	—	2	2	no	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₇×#494	—	2	4	no	—
10	C₇×#474	—	2	4	no	—
11	C₇×#474	—	2	4	no	—
12	C₂×C₇×#107	—	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×#102	—	2	2	no	—
16	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#332	—	2	4	no	—
19	#10274	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	1	no	—
23	C₂×C₇×Q₁₆	—	2	8	no	—
24	C₂×C₇×Q₁₆	—	4	4	no	—
25	C₂×C₇×Q₁₆	—	4	4	no	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	—	2	1	no	—
28	C₂²×C₇×C₈	D₄	1	2	yes	—
29	C₇×C₄⋊Q₈	—	2	2	no	—
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×C₄⋊Q₈	—	2	2	no	—
32	C₇×C₄⋊Q₈	D₄	1	4	yes	—
33	C₄×C₇×D₄	—	2	2	no	—
34	C₄×C₇×D₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₇×#107	—	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×#102	—	2	8	no	—
45	C₇×C₂²⋊C₈	—	4	4	no	—
46	C₄×C₇×C₈	—	2	2	no	—
47	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂×#474	C₂×C₇	1	2	yes	—
49	#1954	C₂×C₇	1	1	yes	characteristic
50	C₂×#332	C₂×C₇	1	2	yes	complemented
51	#1277	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂⁴×C₇	—	2	1	no	—
53	C₂×C₇×Q₈	—	2	4	no	—
54	C₂×C₇×Q₈	—	4	2	no	—
55	C₂×C₇×Q₈	—	4	4	no	—
56	C₂×C₇×Q₈	—	4	4	no	—
57	C₂×C₇×D₄	—	2	2	no	—
58	C₂×C₇×D₄	—	2	1	no	—
59	C₂×C₇×D₄	—	2	2	no	—
60	C₂×C₇×D₄	Q₁₆	1	2	yes	complemented
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	—	4	2	no	—
63	C₂²×C₄×C₇	—	4	1	no	—
64	C₂²×C₄×C₇	—	4	1	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	—	4	2	no	—
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	#41	1	1	yes	characteristic
70	C₇×Q₁₆	—	4	16	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	2	4	no	—
74	C₂×C₇×C₈	—	4	2	no	—
75	C₂×C₇×C₈	—	4	2	no	—
76	C₂×C₇×C₈	—	4	1	no	—
77	C₂×C₇×C₈	—	4	2	no	—
78	C₂×C₇×C₈	D₈	1	4	yes	—
79	C₇×C₄⋊C₄	—	2	4	no	—
80	C₇×C₄⋊C₄	—	2	2	no	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	—	4	2	no	—
83	C₇×C₄⋊C₄	—	4	2	no	—
84	C₇×C₂²⋊C₄	—	4	2	no	—
85	C₄²×C₇	C₂×D₄	1	2	yes	—
86	C₄²×C₇	C₂×D₄	1	2	yes	—
87	C₂²×Q₁₆	—	2	2	no	—
88	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
89	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
90	#494	—	2	4	no	—
91	#474	—	2	4	no	—
92	#474	—	2	4	no	—
93	C₂×#107	—	2	2	no	—
94	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
96	C₂×#102	—	2	2	no	—
97	C₂×C₂²⋊C₈	—	2	2	no	—
98	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
99	#332	—	2	4	no	—
100	C₂³×C₇	—	2	2	no	—
101	C₂³×C₇	—	4	1	no	—
102	C₂³×C₇	—	4	2	no	—
103	C₂³×C₇	—	4	1	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	#121	1	1	yes	characteristic
106	C₇×Q₈	—	4	8	no	—
107	C₇×Q₈	—	8	2	no	—
108	C₇×Q₈	—	8	2	no	—
109	C₇×D₄	—	2	4	no	—
110	C₇×D₄	—	4	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	4	1	no	—
113	C₂×C₄×C₇	—	4	4	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	8	4	no	—
116	C₂×C₄×C₇	—	8	2	no	—
117	C₂×C₄×C₇	—	8	1	no	—
118	C₂×C₄×C₇	—	8	1	no	—
119	C₂×C₄×C₇	#120	1	1	yes	characteristic
120	C₂×C₄×C₇	#120	1	2	yes	—
121	C₂×C₄×C₇	#121	1	2	yes	—
122	C₂×C₄×C₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇	#135	1	1	yes	characteristic
126	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₇	#137	1	2	yes	—
128	C₇×C₈	—	2	4	no	—
129	C₇×C₈	—	4	4	no	—
130	C₇×C₈	—	8	1	no	—
131	C₂²×Q₈	—	2	2	no	—
132	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
133	C₂³×C₄	—	2	1	no	—
134	C₂×Q₁₆	—	2	8	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	C₇×D₄	1	2	yes	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	C₇×D₄	1	4	yes	—
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
151	#107	—	2	8	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	C₇×D₄	1	4	yes	—
155	#102	—	2	8	no	—
156	C₂²⋊C₈	—	4	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₂²×C₇	—	4	4	no	—
159	C₂²×C₇	—	4	2	no	—
160	C₂²×C₇	—	8	2	no	—
161	C₂²×C₇	—	8	1	no	—
162	C₂²×C₇	—	8	1	no	—
163	C₂²×C₇	#392	1	1	yes	characteristic
164	C₂²×C₇	#459	1	2	yes	—
165	C₂²×C₇	#467	1	1	yes	characteristic
166	C₂²×C₇	#470	1	1	yes	characteristic
167	C₂²×C₇	#473	1	2	yes	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	8	1	no	—
170	C₄×C₇	—	8	4	no	—
171	C₄×C₇	#447	1	2	yes	—
172	C₄×C₇	#451	1	2	yes	—
173	C₄×C₇	#452	1	2	yes	—
174	C₄×C₇	#493	1	1	yes	characteristic
175	C₄×C₇	#496	1	1	yes	characteristic
176	C₂⁴	—	2	1	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	4	4	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	C₇×Q₁₆	1	2	yes	complemented
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
193	C₂²×C₄	C₇×#41	1	1	yes	characteristic
194	Q₁₆	—	4	16	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	C₇×D₈	1	4	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	C₂×C₇×D₄	1	2	yes	—
210	C₄²	C₂×C₇×D₄	1	2	yes	—
211	C₂×C₇	—	8	2	no	—
212	C₂×C₇	#1669	1	2	yes	—
213	C₂×C₇	#1689	1	2	yes	—
214	C₂×C₇	#2001	1	1	yes	characteristic
215	C₂×C₇	#2011	1	1	yes	characteristic
216	C₂×C₇	#2020	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	4	1	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₇×#121	1	1	yes	characteristic
223	Q₈	—	4	8	no	—
224	Q₈	—	8	2	no	—
225	Q₈	—	8	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₇×#120	1	1	yes	characteristic
237	C₂×C₄	C₇×#120	1	2	yes	—
238	C₂×C₄	C₇×#121	1	2	yes	—
239	C₂×C₄	C₇×#127	1	1	yes	characteristic
240	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
241	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
242	C₂×C₄	C₇×#135	1	1	yes	characteristic
243	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
244	C₂×C₄	C₇×#137	1	2	yes	—
245	C₈	—	2	4	no	—
246	C₈	—	4	4	no	—
247	C₈	—	8	1	no	—
248	C₇	#10274	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	2	no	—
254	C₂²	C₇×#392	1	1	yes	characteristic
255	C₂²	C₇×#459	1	2	yes	—
256	C₂²	C₇×#467	1	1	yes	characteristic
257	C₂²	C₇×#470	1	1	yes	characteristic
258	C₂²	C₇×#473	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	8	1	no	—
261	C₄	—	8	4	no	—
262	C₄	C₇×#447	1	2	yes	—
263	C₄	C₇×#451	1	2	yes	—
264	C₄	C₇×#452	1	2	yes	—
265	C₄	C₇×#493	1	1	yes	characteristic
266	C₄	C₇×#496	1	1	yes	characteristic
267	C₂	—	8	2	no	—
268	C₂	C₇×#1669	1	2	yes	—
269	C₂	C₇×#1689	1	2	yes	—
270	C₂	C₇×#2001	1	1	yes	characteristic
271	C₂	C₇×#2011	1	1	yes	characteristic
272	C₂	C₇×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂²×C₇×D₄	D₄
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
1. C₂×C₇×#494	C₂	semidirect
4. C₂×C₇×#332	C₂	semidirect
5. C₇×#1277	C₂	semidirect
19. #10274	C₇	direct (N × H)
47. C₂×#494	C₂×C₇	semidirect
50. C₂×#332	C₂×C₇	semidirect
51. #1277	C₂×C₇	semidirect
60. C₂×C₇×D₄	Q₁₆	semidirect
184. C₂×D₄	C₇×Q₁₆	semidirect
248. C₇	#10274	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×#474	2
C₇×#1954	2
C₂×C₇×#332	2
C₇×#1277	2
#10274	7

Subgroups — C7×#10274 #13187490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10274 #13187490