Subgroups — C₇×#17813 #13195029

Order 1792.

730 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	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#423	C₂	1	2	yes	maximal
3	C₇×#1570	C₂	1	8	yes	maximal, complemented
4	C₇×#1566	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	2	no	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#431	C₄	1	4	yes	—
8	C₇×#431	C₂²	1	4	yes	—
9	C₇×#423	—	2	2	no	—
10	C₇×#423	—	2	2	no	—
11	C₇×#423	C₂²	1	4	yes	—
12	C₇×#423	C₂²	1	4	yes	—
13	C₂×C₇×C₄⋊C₈	—	2	2	no	—
14	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₇×#334	C₂²	1	16	yes	complemented
19	#17813	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
23	C₂×C₇×M₁₆	—	2	4	no	—
24	C₂×C₇×M₁₆	—	2	4	no	—
25	C₂×C₇×M₁₆	—	4	8	no	—
26	C₂²×C₇×C₈	—	2	4	no	—
27	C₂²×C₇×C₈	C₂×C₄	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	D₄	1	2	yes	—
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂³	1	2	yes	—
36	C₇×C₄⋊C₈	—	2	4	no	—
37	C₇×C₄⋊C₈	—	2	4	no	—
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
40	C₇×C₄⋊C₈	C₂³	1	8	yes	—
41	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₇×C₈	C₂³	1	4	yes	—
44	C₂×#431	C₂×C₇	1	1	yes	characteristic
45	C₂×#423	C₂×C₇	1	2	yes	—
46	#1570	C₂×C₇	1	8	yes	complemented
47	#1566	C₂×C₇	1	4	yes	complemented
48	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	4	2	no	—
50	C₂²×C₄×C₇	—	4	2	no	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	SD₁₆	1	8	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
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	C₇×M₁₆	—	4	8	no	—
65	C₇×M₁₆	—	4	8	no	—
66	C₂×C₇×C₈	—	2	8	no	—
67	C₂×C₇×C₈	—	4	4	no	—
68	C₂×C₇×C₈	—	4	8	no	—
69	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
70	C₄²×C₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₇	C₂×D₄	1	2	yes	—
76	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₇	C₂×Q₈	1	2	yes	—
79	C₄²×C₇	C₂⁴	1	1	yes	characteristic
80	C₂²×M₁₆	—	2	2	no	—
81	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
82	#431	C₄×C₇	1	4	yes	—
83	#431	C₂²×C₇	1	4	yes	—
84	#423	—	2	2	no	—
85	#423	—	2	2	no	—
86	#423	C₂²×C₇	1	4	yes	—
87	#423	C₂²×C₇	1	4	yes	—
88	C₂×C₄⋊C₈	—	2	2	no	—
89	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
90	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
91	C₂×C₂²⋊C₈	C₄×C₇	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
93	#334	C₂²×C₇	1	16	yes	complemented
94	C₂³×C₇	—	4	1	no	—
95	C₂³×C₇	—	4	1	no	—
96	C₂³×C₇	#106	1	4	yes	—
97	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	4	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	4	2	no	—
104	C₂×C₄×C₇	—	4	1	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	#106	1	4	yes	—
108	C₂×C₄×C₇	#106	1	8	yes	—
109	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
116	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
117	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
118	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
121	C₇×C₈	—	2	8	no	—
122	C₇×C₈	—	4	8	no	—
123	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
124	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
125	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	—	4	8	no	—
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
131	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
133	C₂×C₄²	C₇×D₄	1	2	yes	—
134	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
135	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
136	C₂×C₄²	C₇×Q₈	1	2	yes	—
137	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
138	C₂×C₄²	C₂³×C₇	1	2	yes	—
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
143	C₄⋊C₈	C₂³×C₇	1	8	yes	—
144	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
145	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
146	C₄×C₈	C₂³×C₇	1	4	yes	—
147	C₂²×C₇	—	2	1	no	—
148	C₂²×C₇	—	2	2	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	—	4	4	no	—
152	C₂²×C₇	C₂×#106	1	4	yes	—
153	C₂²×C₇	C₂×#106	1	2	yes	—
154	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	—	2	2	no	—
158	C₄×C₇	—	2	1	no	—
159	C₄×C₇	C₂×#106	1	4	yes	—
160	C₄×C₇	C₂×#106	1	2	yes	—
161	C₄×C₇	#430	1	2	yes	—
162	C₄×C₇	#568	1	2	yes	—
163	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
164	C₂⁴	C₇×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
168	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
170	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
173	C₂²×C₄	C₇×SD₁₆	1	8	yes	—
174	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
175	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
177	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
180	M₁₆	—	4	8	no	—
181	M₁₆	—	4	8	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	4	8	no	—
185	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
186	C₄²	C₇×C₄⋊C₄	1	2	yes	—
187	C₄²	C₇×C₄⋊C₄	1	2	yes	—
188	C₄²	C₇×C₄⋊C₄	1	2	yes	—
189	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
190	C₄²	C₂²×C₄×C₇	1	2	yes	—
191	C₄²	C₂×C₇×D₄	1	2	yes	—
192	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
193	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
194	C₄²	C₂×C₇×Q₈	1	2	yes	—
195	C₄²	C₂⁴×C₇	1	1	yes	characteristic
196	C₂×C₇	—	2	2	no	—
197	C₂×C₇	—	2	4	no	—
198	C₂×C₇	#2905	1	1	yes	characteristic
199	C₂×C₇	C₂²×#106	1	1	yes	characteristic
200	C₂×C₇	C₂×#430	1	1	yes	characteristic
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	C₇×#106	1	4	yes	—
204	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
205	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₇×#106	1	4	yes	—
215	C₂×C₄	C₇×#106	1	8	yes	—
216	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
217	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
218	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
221	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
223	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
224	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
225	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
228	C₈	—	2	8	no	—
229	C₈	—	4	8	no	—
230	C₇	#17813	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂×C₇×#106	1	4	yes	—
237	C₂²	C₂×C₇×#106	1	2	yes	—
238	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	C₂×C₇×#106	1	4	yes	—
244	C₄	C₂×C₇×#106	1	2	yes	—
245	C₄	C₇×#430	1	2	yes	—
246	C₄	C₇×#568	1	2	yes	—
247	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	4	no	—
250	C₂	C₇×#2905	1	1	yes	characteristic
251	C₂	C₂²×C₇×#106	1	1	yes	characteristic
252	C₂	C₂×C₇×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
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×#1570	C₂	semidirect
4. C₇×#1566	C₂	semidirect
18. C₇×#334	C₂²	semidirect
19. #17813	C₇	direct (N × H)
46. #1570	C₂×C₇	semidirect
47. #1566	C₂×C₇	semidirect
93. #334	C₂²×C₇	semidirect
230. C₇	#17813	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×C₇×#423	2
C₇×#1570	2
C₇×#1566	2
#17813	7

Subgroups — C7×#17813 #13195029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17813 #13195029