Subgroups — C₇×#21951 #13199167

Order 1792.

978 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×#431	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#1939	C₂	1	8	yes	maximal, complemented
4	C₇×#1797	C₂	1	4	yes	maximal
5	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#431	—	2	8	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
12	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
14	C₄×C₇×C₄⋊C₄	C₄	1	4	yes	—
15	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₇×#95	C₄	1	2	yes	—
17	C₇×#336	C₂²	1	8	yes	—
18	#21951	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	8	yes	—
25	C₂×C₇×C₄⋊C₄	—	2	4	no	—
26	C₂×C₇×C₄⋊C₄	C₈	1	8	yes	complemented
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₇	—	4	4	no	—
30	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇	D₄	1	4	yes	—
33	C₂×C₄²×C₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₇	C₂³	1	2	yes	—
36	C₇×C₄⋊C₈	—	2	8	no	—
37	C₇×C₄⋊C₈	—	4	8	no	—
38	C₇×C₂²⋊C₈	—	2	8	no	—
39	C₄×C₇×C₈	—	2	8	no	—
40	C₇×#95	C₂×C₄	1	8	yes	—
41	C₂×#431	C₂×C₇	1	2	yes	—
42	C₂×C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
43	#1939	C₂×C₇	1	8	yes	complemented
44	#1797	C₂×C₇	1	4	yes	—
45	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	—	2	2	no	—
48	C₂²×C₄×C₇	—	2	4	no	—
49	C₂²×C₄×C₇	—	4	2	no	—
50	C₂²×C₄×C₇	—	4	2	no	—
51	C₂²×C₄×C₇	—	4	1	no	—
52	C₂²×C₄×C₇	—	4	1	no	—
53	C₂²×C₄×C₇	—	4	2	no	—
54	C₂²×C₄×C₇	—	4	2	no	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₇	M₁₆	1	4	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₇×C₈	—	2	8	no	—
69	C₂×C₇×C₈	—	2	8	no	—
70	C₂×C₇×C₈	—	2	8	no	—
71	C₇×C₄⋊C₄	—	2	8	no	—
72	C₇×C₄⋊C₄	C₂×C₈	1	16	yes	—
73	C₄²×C₇	—	2	8	no	—
74	C₄²×C₇	—	4	4	no	—
75	C₄²×C₇	C₄⋊C₄	1	8	yes	—
76	C₄²×C₇	C₂×D₄	1	4	yes	—
77	C₄²×C₇	C₂×Q₈	1	4	yes	—
78	C₂²×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
79	C₂²×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
80	C₂²×C₄²	C₂²×C₇	1	2	yes	—
81	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
82	#431	—	2	8	no	—
83	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
84	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
85	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	—
86	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
87	C₄×C₄⋊C₄	C₄×C₇	1	4	yes	—
88	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
89	C₂×#95	C₄×C₇	1	2	yes	—
90	#336	C₂²×C₇	1	8	yes	—
91	C₂³×C₇	—	2	1	no	—
92	C₂³×C₇	—	2	1	no	—
93	C₂³×C₇	—	2	1	no	—
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₇	C₂²×C₈	1	2	yes	—
97	C₂³×C₇	C₂×M₁₆	1	2	yes	—
98	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	—	2	4	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	—	2	4	no	—
110	C₂×C₄×C₇	—	4	4	no	—
111	C₂×C₄×C₇	—	4	8	no	—
112	C₂×C₄×C₇	—	4	8	no	—
113	C₂×C₄×C₇	C₄⋊C₈	1	16	yes	complemented
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
118	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
119	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
123	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
124	C₇×C₈	—	4	8	no	—
125	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
126	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
127	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
128	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
129	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₇	1	8	yes	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	C₇×C₈	1	8	yes	complemented
133	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
134	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
135	C₂×C₄²	—	4	4	no	—
136	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
138	C₂×C₄²	C₇×D₄	1	4	yes	—
139	C₂×C₄²	C₇×Q₈	1	4	yes	—
140	C₂×C₄²	C₂³×C₇	1	4	yes	—
141	C₂×C₄²	C₂³×C₇	1	2	yes	—
142	C₄⋊C₈	—	2	8	no	—
143	C₄⋊C₈	—	4	8	no	—
144	C₂²⋊C₈	—	2	8	no	—
145	C₄×C₈	—	2	8	no	—
146	#95	C₂×C₄×C₇	1	8	yes	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	2	2	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	2	2	no	—
151	C₂²×C₇	—	2	2	no	—
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	C₂×C₄⋊C₈	1	4	yes	—
154	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₇	#518	1	1	yes	characteristic
156	C₂²×C₇	#519	1	1	yes	characteristic
157	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
158	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
159	C₂²×C₇	#568	1	1	yes	characteristic
160	C₂²×C₇	#568	1	1	yes	characteristic
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	2	4	no	—
163	C₄×C₇	—	2	8	no	—
164	C₄×C₇	—	4	4	no	—
165	C₄×C₇	C₂×C₄⋊C₈	1	8	yes	—
166	C₄×C₇	#537	1	2	yes	—
167	C₄×C₇	#544	1	2	yes	—
168	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
179	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
180	C₂²×C₄	C₂×C₇×C₈	1	8	yes	—
181	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
182	C₂²×C₄	C₇×M₁₆	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
188	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
189	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
190	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	2	8	no	—
194	C₄⋊C₄	—	2	8	no	—
195	C₄⋊C₄	C₂×C₇×C₈	1	16	yes	—
196	C₄²	—	2	8	no	—
197	C₄²	—	4	4	no	—
198	C₄²	C₇×C₄⋊C₄	1	8	yes	—
199	C₄²	C₂×C₇×D₄	1	4	yes	—
200	C₄²	C₂×C₇×Q₈	1	4	yes	—
201	C₂×C₇	—	2	4	no	—
202	C₂×C₇	#2316	1	1	yes	characteristic
203	C₂×C₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
204	C₂×C₇	#2905	1	1	yes	characteristic
205	C₂×C₇	#2972	1	1	yes	characteristic
206	C₂×C₇	#2975	1	1	yes	characteristic
207	C₂×C₇	#2982	1	1	yes	characteristic
208	C₂×C₇	#2983	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
214	C₂³	C₂²×C₇×C₈	1	2	yes	—
215	C₂³	C₂×C₇×M₁₆	1	2	yes	—
216	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	C₇×C₄⋊C₈	1	16	yes	complemented
232	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
233	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
234	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
235	C₂×C₄	C₂²×C₇×C₈	1	8	yes	—
236	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
237	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
238	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
240	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
241	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
242	C₈	—	4	8	no	—
243	C₇	#21951	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	C₂×C₇×C₄⋊C₈	1	4	yes	—
251	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
252	C₂²	C₇×#518	1	1	yes	characteristic
253	C₂²	C₇×#519	1	1	yes	characteristic
254	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
255	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
256	C₂²	C₇×#568	1	1	yes	characteristic
257	C₂²	C₇×#568	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	8	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	C₂×C₇×C₄⋊C₈	1	8	yes	—
263	C₄	C₇×#537	1	2	yes	—
264	C₄	C₇×#544	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	C₇×#2316	1	1	yes	characteristic
267	C₂	C₂²×C₇×C₄⋊C₈	1	1	yes	characteristic
268	C₂	C₇×#2905	1	1	yes	characteristic
269	C₂	C₇×#2972	1	1	yes	characteristic
270	C₂	C₇×#2975	1	1	yes	characteristic
271	C₂	C₇×#2982	1	1	yes	characteristic
272	C₂	C₇×#2983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1939	C₂	semidirect
18. #21951	C₇	direct (N × H)
26. C₂×C₇×C₄⋊C₄	C₈	semidirect
43. #1939	C₂×C₇	semidirect
113. C₂×C₄×C₇	C₄⋊C₈	semidirect
132. C₂×C₄⋊C₄	C₇×C₈	semidirect
231. C₂×C₄	C₇×C₄⋊C₈	semidirect
243. C₇	#21951	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×#431	2
C₂×C₄×C₇×C₄⋊C₄	2
C₇×#1939	2
C₇×#1797	2
#21951	7

Subgroups — C7×#21951 #13199167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21951 #13199167