Subgroups — C₇×#22023 #13199239

Order 1792.

606 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₇×#1984	C₂	1	2	yes	maximal
3	C₇×#1939	C₂	1	2	yes	maximal
4	C₇×#1917	C₂	1	4	yes	maximal
5	C₇×#1840	C₂	1	2	yes	maximal
6	C₇×#1832	C₂	1	2	yes	maximal
7	C₇×#1767	C₂	1	1	yes	maximal, characteristic
8	C₇×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₇×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
14	C₇×#391	C₄	1	2	yes	—
15	C₇×#389	C₄	1	2	yes	—
16	C₇×#387	C₄	1	4	yes	—
17	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₇×#376	C₂²	1	2	yes	—
20	C₄³×C₇	C₂²	1	1	yes	characteristic
21	C₇×#336	C₂²	1	4	yes	—
22	C₇×#336	C₂²	1	4	yes	—
23	C₇×#336	C₂²	1	1	yes	characteristic
24	C₇×#336	C₂²	1	1	yes	characteristic
25	C₇×#336	C₂²	1	2	yes	—
26	C₇×#336	C₂²	1	2	yes	—
27	C₇×#336	C₂²	1	4	yes	—
28	#22023	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	2	yes	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	4	yes	—
33	C₄×C₇×Q₈	—	2	4	no	—
34	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×C₄⋊C₈	—	2	4	no	—
46	C₇×C₄⋊C₈	—	4	4	no	—
47	C₇×#97	—	4	2	no	—
48	C₄×C₇×C₈	—	2	4	no	—
49	C₇×#95	C₂×C₄	1	4	yes	—
50	C₇×#95	C₂×C₄	1	4	yes	—
51	C₇×#95	C₂×C₄	1	2	yes	—
52	C₇×#95	C₂×C₄	1	4	yes	—
53	#2295	C₂×C₇	1	1	yes	characteristic
54	#1984	C₂×C₇	1	2	yes	—
55	#1939	C₂×C₇	1	2	yes	—
56	#1917	C₂×C₇	1	4	yes	—
57	#1840	C₂×C₇	1	2	yes	—
58	#1832	C₂×C₇	1	2	yes	—
59	#1767	C₂×C₇	1	1	yes	characteristic
60	#1744	C₂×C₇	1	1	yes	characteristic
61	C₂×C₇×Q₈	—	2	2	no	—
62	C₂×C₇×Q₈	—	2	4	no	—
63	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
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	—	2	4	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₇×C₄⋊C₄	—	2	2	no	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	—	2	4	no	—
83	C₇×C₄⋊C₄	—	4	4	no	—
84	C₄²×C₇	—	2	4	no	—
85	C₄²×C₇	—	2	4	no	—
86	C₄²×C₇	—	2	1	no	—
87	C₄²×C₇	—	2	2	no	—
88	C₄²×C₇	—	2	2	no	—
89	C₄²×C₇	—	2	1	no	—
90	C₄²×C₇	—	2	2	no	—
91	C₄²×C₇	—	4	2	no	—
92	C₄²×C₇	#41	1	4	yes	—
93	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
94	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
96	C₂×#97	C₂²×C₇	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
98	#391	C₄×C₇	1	2	yes	—
99	#389	C₄×C₇	1	2	yes	—
100	#387	C₄×C₇	1	4	yes	—
101	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
102	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
103	#376	C₂²×C₇	1	2	yes	—
104	C₄³	C₂²×C₇	1	1	yes	characteristic
105	#336	C₂²×C₇	1	1	yes	characteristic
106	#336	C₂²×C₇	1	2	yes	—
107	#336	C₂²×C₇	1	4	yes	—
108	#336	C₂²×C₇	1	1	yes	characteristic
109	#336	C₂²×C₇	1	4	yes	—
110	#336	C₂²×C₇	1	4	yes	—
111	#336	C₂²×C₇	1	2	yes	—
112	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
113	C₇×Q₈	—	4	4	no	—
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	#117	1	2	yes	—
129	C₂×C₄×C₇	#117	1	2	yes	—
130	C₂×C₄×C₇	#131	1	4	yes	—
131	C₂×C₄×C₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
135	C₇×C₈	—	4	2	no	—
136	C₇×C₈	—	4	2	no	—
137	C₇×C₈	—	4	4	no	—
138	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₇	1	2	yes	—
140	C₂²×C₈	C₂³×C₇	1	2	yes	—
141	C₂²×C₈	C₂³×C₇	1	4	yes	—
142	C₄×Q₈	—	2	4	no	—
143	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
147	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
149	C₂×C₄²	C₂³×C₇	1	2	yes	—
150	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
151	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₇	1	2	yes	—
153	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	4	4	no	—
156	#97	—	4	2	no	—
157	C₄×C₈	—	2	4	no	—
158	#95	C₂×C₄×C₇	1	4	yes	—
159	#95	C₂×C₄×C₇	1	4	yes	—
160	#95	C₂×C₄×C₇	1	2	yes	—
161	#95	C₂×C₄×C₇	1	4	yes	—
162	C₂²×C₇	C₂×#117	1	1	yes	characteristic
163	C₂²×C₇	#519	1	1	yes	characteristic
164	C₂²×C₇	#520	1	1	yes	characteristic
165	C₂²×C₇	C₂×#131	1	2	yes	—
166	C₂²×C₇	#568	1	2	yes	—
167	C₄×C₇	—	2	2	no	—
168	C₄×C₇	—	4	2	no	—
169	C₄×C₇	—	4	2	no	—
170	C₄×C₇	—	4	2	no	—
171	C₄×C₇	—	4	2	no	—
172	C₄×C₇	—	4	4	no	—
173	C₄×C₇	#433	1	4	yes	—
174	C₄×C₇	#541	1	2	yes	—
175	C₄×C₇	#542	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
180	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
185	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	C₇×#41	1	4	yes	—
208	C₂×C₇	#2320	1	1	yes	characteristic
209	C₂×C₇	C₂×#433	1	1	yes	characteristic
210	C₂×C₇	#2923	1	1	yes	characteristic
211	C₂×C₇	#2985	1	2	yes	—
212	C₂×C₇	#2993	1	2	yes	—
213	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	C₇×#117	1	2	yes	—
230	C₂×C₄	C₇×#117	1	2	yes	—
231	C₂×C₄	C₇×#131	1	4	yes	—
232	C₂×C₄	C₂×C₇×#41	1	2	yes	—
233	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
236	C₈	—	4	2	no	—
237	C₈	—	4	2	no	—
238	C₈	—	4	4	no	—
239	C₇	#22023	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	C₂×C₇×#117	1	1	yes	characteristic
241	C₂²	C₇×#519	1	1	yes	characteristic
242	C₂²	C₇×#520	1	1	yes	characteristic
243	C₂²	C₂×C₇×#131	1	2	yes	—
244	C₂²	C₇×#568	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	4	no	—
251	C₄	C₇×#433	1	4	yes	—
252	C₄	C₇×#541	1	2	yes	—
253	C₄	C₇×#542	1	2	yes	—
254	C₂	C₇×#2320	1	1	yes	characteristic
255	C₂	C₂×C₇×#433	1	1	yes	characteristic
256	C₂	C₇×#2923	1	1	yes	characteristic
257	C₂	C₇×#2985	1	2	yes	—
258	C₂	C₇×#2993	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
28. #22023	C₇	direct (N × H)
239. C₇	#22023	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×#2295	2
C₇×#1984	2
C₇×#1939	2
C₇×#1917	2
C₇×#1840	2
C₇×#1832	2
C₇×#1767	2
C₇×#1744	2
#22023	7

Subgroups — C7×#22023 #13199239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22023 #13199239