Subgroups — C₇×#20816 #13198032

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	2	yes	maximal
3	C₇×#1844	C₂	1	2	yes	maximal, complemented
4	C₇×#1834	C₂	1	2	yes	maximal, complemented
5	C₇×#1807	C₂	1	4	yes	maximal
6	C₄×C₇×#106	C₂	1	4	yes	maximal, complemented
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₇×#117	—	2	2	no	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×#106	C₂²	1	8	yes	—
11	C₂×C₄×C₇×C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#384	C₂²	1	4	yes	—
16	C₇×#382	—	2	2	no	—
17	C₇×#380	—	2	2	no	—
18	C₇×#380	C₂²	1	4	yes	—
19	C₄×C₇×C₄⋊C₄	—	2	2	no	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
21	C₇×#340	C₂²	1	8	yes	—
22	#20816	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²×C₇×C₈	—	2	8	no	—
27	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₇×C₈	C₂³	1	2	yes	—
29	C₂²×C₇×C₈	C₂³	1	2	yes	—
30	C₇×#117	—	4	8	no	—
31	C₂×C₇×C₄⋊C₄	—	2	4	no	—
32	C₂×C₇×C₄⋊C₄	—	2	4	no	—
33	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
34	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
35	C₂×C₄²×C₇	—	2	4	no	—
36	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	Q₈	1	4	yes	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₇×#106	—	2	16	no	—
43	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
44	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
45	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
46	C₇×#95	—	2	4	no	—
47	C₇×#95	—	2	4	no	—
48	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
49	#2285	C₂×C₇	1	2	yes	—
50	#1844	C₂×C₇	1	2	yes	complemented
51	#1834	C₂×C₇	1	2	yes	complemented
52	#1807	C₂×C₇	1	4	yes	—
53	C₄×#106	C₂×C₇	1	4	yes	complemented
54	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	4	no	—
56	C₂²×C₄×C₇	—	2	4	no	—
57	C₂²×C₄×C₇	—	2	8	no	—
58	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₇	SD₁₆	1	8	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	—	2	16	no	—
71	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
73	C₂×C₇×C₈	#41	1	8	yes	—
74	C₇×C₄⋊C₄	—	2	16	no	—
75	C₇×C₄⋊C₄	—	4	8	no	—
76	C₇×C₄⋊C₄	—	4	8	no	—
77	C₇×C₂²⋊C₄	—	4	8	no	—
78	C₄²×C₇	—	4	8	no	—
79	C₄²×C₇	C₄⋊C₄	1	4	yes	—
80	C₄²×C₇	C₄⋊C₄	1	4	yes	—
81	C₄²×C₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₇	C₂×Q₈	1	2	yes	—
85	C₂³×C₈	C₂²×C₇	1	2	yes	—
86	C₂×#117	—	2	2	no	—
87	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
88	C₂×#106	C₂²×C₇	1	8	yes	—
89	C₂×C₄×C₈	C₄×C₇	1	8	yes	complemented
90	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
93	#384	C₂²×C₇	1	4	yes	—
94	#382	—	2	2	no	—
95	#380	—	2	2	no	—
96	#380	C₂²×C₇	1	4	yes	—
97	C₄×C₄⋊C₄	—	2	2	no	—
98	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
99	#340	C₂²×C₇	1	8	yes	—
100	C₂³×C₇	—	2	2	no	—
101	C₂³×C₇	—	2	2	no	—
102	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
104	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	—	2	8	no	—
110	C₂×C₄×C₇	—	4	8	no	—
111	C₂×C₄×C₇	—	4	8	no	—
112	C₂×C₄×C₇	—	4	8	no	—
113	C₂×C₄×C₇	#106	1	16	yes	complemented
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
120	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
121	C₂×C₄×C₇	#135	1	4	yes	—
122	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×#41	1	4	yes	—
125	C₇×C₈	—	2	4	no	—
126	C₇×C₈	#117	1	8	yes	complemented
127	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
128	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
129	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
130	C₂²×C₈	—	2	8	no	—
131	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
132	C₂²×C₈	C₂³×C₇	1	2	yes	—
133	C₂²×C₈	C₂³×C₇	1	2	yes	—
134	#117	—	4	8	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
138	C₂×C₂²⋊C₄	—	2	4	no	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
141	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₇×Q₈	1	4	yes	—
144	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
146	#106	—	2	16	no	—
147	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
148	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
149	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
150	#95	—	2	4	no	—
151	#95	—	2	4	no	—
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	—	2	8	no	—
154	C₂²×C₇	—	2	2	no	—
155	C₂²×C₇	C₂×#106	1	4	yes	—
156	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₇	#532	1	2	yes	—
158	C₂²×C₇	C₂²×SD₁₆	1	2	yes	—
159	C₂²×C₇	C₂×#135	1	2	yes	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	8	8	no	—
163	C₄×C₇	C₂×#106	1	4	yes	—
164	C₄×C₇	C₂×#106	1	4	yes	—
165	C₄×C₇	#427	1	4	yes	—
166	C₄×C₇	C₂×#117	1	2	yes	—
167	C₄×C₇	#533	1	2	yes	—
168	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
173	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
174	C₂²×C₄	C₇×SD₁₆	1	8	yes	—
175	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
176	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	16	no	—
185	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
186	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
187	C₂×C₈	C₇×#41	1	8	yes	—
188	C₄⋊C₄	—	2	16	no	—
189	C₄⋊C₄	—	4	8	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₄²	—	4	8	no	—
193	C₄²	C₇×C₄⋊C₄	1	4	yes	—
194	C₄²	C₇×C₄⋊C₄	1	4	yes	—
195	C₄²	C₂²×C₄×C₇	1	2	yes	—
196	C₄²	C₂²×C₄×C₇	1	2	yes	—
197	C₄²	C₂×C₇×Q₈	1	2	yes	—
198	C₄²	C₂×C₇×Q₈	1	2	yes	—
199	C₂×C₇	—	2	4	no	—
200	C₂×C₇	#2285	1	1	yes	characteristic
201	C₂×C₇	C₂²×#106	1	1	yes	characteristic
202	C₂×C₇	C₂×#427	1	1	yes	characteristic
203	C₂×C₇	#3167	1	4	yes	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
207	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
208	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₇×#106	1	16	yes	complemented
218	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
219	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
220	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
221	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
222	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
223	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
224	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
225	C₂×C₄	C₇×#135	1	4	yes	—
226	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
228	C₂×C₄	C₂×C₇×#41	1	4	yes	—
229	C₈	—	2	4	no	—
230	C₈	C₇×#117	1	8	yes	complemented
231	C₇	#20816	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	C₂×C₇×#106	1	4	yes	—
236	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
237	C₂²	C₇×#532	1	2	yes	—
238	C₂²	C₂²×C₇×SD₁₆	1	2	yes	—
239	C₂²	C₂×C₇×#135	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	8	no	—
243	C₄	C₂×C₇×#106	1	4	yes	—
244	C₄	C₂×C₇×#106	1	4	yes	—
245	C₄	C₇×#427	1	4	yes	—
246	C₄	C₂×C₇×#117	1	2	yes	—
247	C₄	C₇×#533	1	2	yes	—
248	C₂	—	2	4	no	—
249	C₂	C₇×#2285	1	1	yes	characteristic
250	C₂	C₂²×C₇×#106	1	1	yes	characteristic
251	C₂	C₂×C₇×#427	1	1	yes	characteristic
252	C₂	C₇×#3167	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1844	C₂	semidirect
4. C₇×#1834	C₂	semidirect
6. C₄×C₇×#106	C₂	semidirect
11. C₂×C₄×C₇×C₈	C₄	semidirect
22. #20816	C₇	direct (N × H)
50. #1844	C₂×C₇	semidirect
51. #1834	C₂×C₇	semidirect
53. C₄×#106	C₂×C₇	semidirect
89. C₂×C₄×C₈	C₄×C₇	semidirect
113. C₂×C₄×C₇	#106	semidirect
126. C₇×C₈	#117	semidirect
217. C₂×C₄	C₇×#106	semidirect
230. C₈	C₇×#117	semidirect
231. C₇	#20816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₇×#2285	2
C₇×#1844	2
C₇×#1834	2
C₇×#1807	2
C₄×C₇×#106	2
#20816	7

Subgroups — C7×#20816 #13198032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20816 #13198032