Subgroups — C₇×#21970 #13199186

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2920	C₂	1	2	yes	maximal
2	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₇×#1950	C₂	1	8	yes	maximal, complemented
4	C₇×#1811	C₂	1	4	yes	maximal
5	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#432	—	2	2	no	—
8	C₇×#432	—	2	2	no	—
9	C₇×#431	—	2	2	no	—
10	C₇×#431	—	2	2	no	—
11	C₂×C₇×#108	C₂²	1	8	yes	—
12	C₇×#423	C₄	1	2	yes	—
13	C₇×#423	C₄	1	4	yes	—
14	C₇×#423	C₄	1	2	yes	—
15	C₇×#423	C₂²	1	4	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
17	C₇×#407	C₂²	1	4	yes	—
18	C₄×C₇×M₁₆	C₂²	1	4	yes	—
19	C₇×#341	C₂²	1	8	yes	—
20	#21970	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₇	Q₈	1	2	yes	—
22	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₇×M₁₆	—	4	4	no	—
24	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₇×M₁₆	C₂³	1	8	yes	—
27	C₂²×C₇×C₈	C₂³	1	4	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂³	1	2	yes	—
37	C₇×#108	—	4	8	no	—
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₇×C₄⋊C₈	—	2	4	no	—
40	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
41	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
42	C₇×C₂²⋊C₈	—	2	4	no	—
43	C₇×C₂²⋊C₈	—	2	4	no	—
44	C₇×#97	—	2	4	no	—
45	C₄×C₇×C₈	—	2	4	no	—
46	#2920	C₂×C₇	1	2	yes	—
47	C₂×#423	C₂×C₇	1	1	yes	characteristic
48	#1950	C₂×C₇	1	8	yes	complemented
49	#1811	C₂×C₇	1	4	yes	—
50	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	4	4	no	—
54	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
66	C₇×M₁₆	—	2	8	no	—
67	C₇×M₁₆	—	2	8	no	—
68	C₇×M₁₆	—	4	4	no	—
69	C₇×M₁₆	—	4	4	no	—
70	C₂×C₇×C₈	—	2	8	no	—
71	C₂×C₇×C₈	—	2	8	no	—
72	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
73	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
76	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
79	C₄²×C₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₇	C₂×D₄	1	2	yes	—
82	C₄²×C₇	C₂×D₄	1	2	yes	—
83	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
84	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
85	#432	—	2	2	no	—
86	#432	—	2	2	no	—
87	#431	—	2	2	no	—
88	#431	—	2	2	no	—
89	C₂×#108	C₂²×C₇	1	8	yes	—
90	#423	C₄×C₇	1	2	yes	—
91	#423	C₄×C₇	1	4	yes	—
92	#423	C₄×C₇	1	2	yes	—
93	#423	C₂²×C₇	1	4	yes	—
94	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
95	#407	C₂²×C₇	1	4	yes	—
96	C₄×M₁₆	C₂²×C₇	1	4	yes	—
97	#341	C₂²×C₇	1	8	yes	—
98	C₂³×C₇	—	2	2	no	—
99	C₂³×C₇	—	4	2	no	—
100	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	4	no	—
111	C₂×C₄×C₇	—	2	4	no	—
112	C₂×C₄×C₇	—	2	4	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
125	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
126	C₇×C₈	—	2	8	no	—
127	C₇×C₈	—	4	8	no	—
128	C₂³×C₄	C₇×Q₈	1	2	yes	—
129	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
130	C₂×M₁₆	—	4	4	no	—
131	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
132	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
133	C₂×M₁₆	C₂³×C₇	1	8	yes	—
134	C₂²×C₈	C₂³×C₇	1	4	yes	—
135	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
138	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₇×D₄	1	2	yes	—
140	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
142	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₇	1	2	yes	—
144	#108	—	4	8	no	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
148	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
149	C₂²⋊C₈	—	2	4	no	—
150	C₂²⋊C₈	—	2	4	no	—
151	#97	—	2	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	—	2	1	no	—
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	4	4	no	—
159	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₇	#544	1	2	yes	—
161	C₄×C₇	—	2	1	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	#430	1	4	yes	—
167	C₄×C₇	#518	1	1	yes	characteristic
168	C₄×C₇	#519	1	1	yes	characteristic
169	C₄×C₇	#537	1	2	yes	—
170	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
175	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
176	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
177	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
184	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
185	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
186	M₁₆	—	2	8	no	—
187	M₁₆	—	2	8	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	4	4	no	—
190	C₂×C₈	—	2	8	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
193	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
194	C₄²	C₇×C₄⋊C₄	1	2	yes	—
195	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
196	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
197	C₄²	C₇×C₄⋊C₄	1	2	yes	—
198	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
199	C₄²	C₂²×C₄×C₇	1	2	yes	—
200	C₄²	C₂²×C₄×C₇	1	2	yes	—
201	C₄²	C₂×C₇×D₄	1	2	yes	—
202	C₄²	C₂×C₇×D₄	1	2	yes	—
203	C₂×C₇	—	2	2	no	—
204	C₂×C₇	—	4	1	no	—
205	C₂×C₇	—	4	1	no	—
206	C₂×C₇	#2316	1	1	yes	characteristic
207	C₂×C₇	C₂×#430	1	2	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
211	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
212	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
229	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
231	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
232	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
234	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
235	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
236	C₈	—	2	8	no	—
237	C₈	—	4	8	no	—
238	C₇	#21970	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	4	4	no	—
245	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
246	C₂²	C₇×#544	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	C₇×#430	1	4	yes	—
253	C₄	C₇×#518	1	1	yes	characteristic
254	C₄	C₇×#519	1	1	yes	characteristic
255	C₄	C₇×#537	1	2	yes	—
256	C₂	—	2	2	no	—
257	C₂	—	4	1	no	—
258	C₂	—	4	1	no	—
259	C₂	C₇×#2316	1	1	yes	characteristic
260	C₂	C₂×C₇×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1950	C₂	semidirect
20. #21970	C₇	direct (N × H)
48. #1950	C₂×C₇	semidirect
238. C₇	#21970	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2920	2
C₂×C₇×#423	2
C₇×#1950	2
C₇×#1811	2
#21970	7

Subgroups — C7×#21970 #13199186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21970 #13199186