Subgroups — C₇×#17819 #13195035

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#431	C₂	1	1	yes	maximal, characteristic
3	C₇×#1574	C₂	1	8	yes	maximal, complemented
4	C₇×#1568	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₇×#432	—	2	2	no	—
7	C₇×#432	C₂²	1	4	yes	—
8	C₇×#431	—	2	2	no	—
9	C₇×#431	C₄	1	4	yes	—
10	C₇×#431	C₂²	1	4	yes	—
11	C₇×#431	C₂²	1	4	yes	—
12	C₇×#423	C₂²	1	4	yes	—
13	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
14	C₇×#407	—	2	2	no	—
15	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
16	C₄×C₇×M₁₆	—	2	2	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₇×#335	C₂²	1	8	yes	complemented
19	C₇×#334	C₂²	1	8	yes	complemented
20	#17819	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
24	C₂×C₇×M₁₆	—	2	4	no	—
25	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₇×C₄⋊C₈	—	2	4	no	—
35	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₇×C₄⋊C₈	C₂³	1	8	yes	—
37	C₇×C₂²⋊C₈	—	2	4	no	—
38	C₇×C₂²⋊C₈	—	2	4	no	—
39	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
40	C₇×#97	—	2	4	no	—
41	C₄×C₇×C₈	—	2	4	no	—
42	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₇×C₈	C₂³	1	4	yes	—
44	#2920	C₂×C₇	1	2	yes	—
45	C₂×#431	C₂×C₇	1	1	yes	characteristic
46	#1574	C₂×C₇	1	8	yes	complemented
47	#1568	C₂×C₇	1	4	yes	complemented
48	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	4	no	—
50	C₂²×C₄×C₇	—	4	2	no	—
51	C₂²×C₄×C₇	—	4	2	no	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
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	2	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
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	C₂×C₇×C₈	—	2	8	no	—
66	C₂×C₇×C₈	—	4	4	no	—
67	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
68	C₄²×C₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₇	C₂×D₄	1	2	yes	—
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₇	C₂×Q₈	1	2	yes	—
77	C₄²×C₇	C₂⁴	1	1	yes	characteristic
78	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
79	#432	—	2	2	no	—
80	#432	C₂²×C₇	1	4	yes	—
81	#431	—	2	2	no	—
82	#431	C₄×C₇	1	4	yes	—
83	#431	C₂²×C₇	1	4	yes	—
84	#431	C₂²×C₇	1	4	yes	—
85	#423	C₂²×C₇	1	4	yes	—
86	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
87	#407	—	2	2	no	—
88	C₂×C₂²⋊C₈	C₄×C₇	1	2	yes	—
89	C₄×M₁₆	—	2	2	no	—
90	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
91	#335	C₂²×C₇	1	8	yes	complemented
92	#334	C₂²×C₇	1	8	yes	complemented
93	C₂³×C₇	—	2	2	no	—
94	C₂³×C₇	—	4	1	no	—
95	C₂³×C₇	—	4	1	no	—
96	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	—	2	2	no	—
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	—	2	4	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	—	4	1	no	—
108	C₂×C₄×C₇	—	4	2	no	—
109	C₂×C₄×C₇	#108	1	4	yes	—
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
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×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
119	C₇×C₈	—	2	8	no	—
120	C₇×C₈	—	4	8	no	—
121	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
122	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
123	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
126	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
127	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
128	C₂×C₄²	C₇×D₄	1	2	yes	—
129	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
130	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
131	C₂×C₄²	C₇×Q₈	1	2	yes	—
132	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₇	1	2	yes	—
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
136	C₄⋊C₈	C₂³×C₇	1	8	yes	—
137	C₂²⋊C₈	—	2	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
140	#97	—	2	4	no	—
141	C₄×C₈	—	2	4	no	—
142	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
143	C₄×C₈	C₂³×C₇	1	4	yes	—
144	C₂²×C₇	—	2	1	no	—
145	C₂²×C₇	—	2	1	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	2	2	no	—
149	C₂²×C₇	—	4	4	no	—
150	C₂²×C₇	C₂×#108	1	2	yes	—
151	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₇	—	2	1	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	C₂×#108	1	4	yes	—
159	C₄×C₇	#568	1	2	yes	—
160	C₄×C₇	#575	1	2	yes	—
161	C₂⁴	C₇×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
166	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
167	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
169	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
175	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
181	C₄²	C₇×C₄⋊C₄	1	2	yes	—
182	C₄²	C₇×C₄⋊C₄	1	2	yes	—
183	C₄²	C₇×C₄⋊C₄	1	2	yes	—
184	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
185	C₄²	C₂²×C₄×C₇	1	2	yes	—
186	C₄²	C₂×C₇×D₄	1	2	yes	—
187	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
188	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
189	C₄²	C₂×C₇×Q₈	1	2	yes	—
190	C₄²	C₂⁴×C₇	1	1	yes	characteristic
191	C₂×C₇	—	2	2	no	—
192	C₂×C₇	—	4	2	no	—
193	C₂×C₇	#2905	1	1	yes	characteristic
194	C₂×C₇	#2911	1	1	yes	characteristic
195	C₂×C₇	C₂²×#108	1	1	yes	characteristic
196	C₂³	—	2	2	no	—
197	C₂³	—	4	1	no	—
198	C₂³	—	4	1	no	—
199	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
200	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₇×#108	1	4	yes	—
213	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
217	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
219	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
222	C₈	—	2	8	no	—
223	C₈	—	4	8	no	—
224	C₇	#17819	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×C₇×#108	1	2	yes	—
232	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	C₂×C₇×#108	1	4	yes	—
240	C₄	C₇×#568	1	2	yes	—
241	C₄	C₇×#575	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₇×#2905	1	1	yes	characteristic
245	C₂	C₇×#2911	1	1	yes	characteristic
246	C₂	C₂²×C₇×#108	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×#1574	C₂	semidirect
4. C₇×#1568	C₂	semidirect
18. C₇×#335	C₂²	semidirect
19. C₇×#334	C₂²	semidirect
20. #17819	C₇	direct (N × H)
46. #1574	C₂×C₇	semidirect
47. #1568	C₂×C₇	semidirect
91. #335	C₂²×C₇	semidirect
92. #334	C₂²×C₇	semidirect
224. C₇	#17819	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×#431	2
C₇×#1574	2
C₇×#1568	2
#17819	7

Subgroups — C7×#17819 #13195035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17819 #13195035