Subgroups — C₇×#17814 #13195030

Order 1792.

730 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	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#423	C₂	1	2	yes	maximal
3	C₇×#1571	C₂	1	8	yes	maximal, complemented
4	C₇×#1567	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×#335	C₂²	1	16	yes	complemented
19	#17814	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	#1571	C₂×C₇	1	8	yes	complemented
47	#1567	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	D₈	1	4	yes	—
58	C₂²×C₄×C₇	Q₁₆	1	4	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
65	C₇×M₁₆	—	4	8	no	—
66	C₇×M₁₆	—	4	8	no	—
67	C₂×C₇×C₈	—	2	8	no	—
68	C₂×C₇×C₈	—	4	4	no	—
69	C₂×C₇×C₈	—	4	8	no	—
70	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
71	C₄²×C₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₇	C₂×D₄	1	2	yes	—
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₇	C₂×Q₈	1	2	yes	—
80	C₄²×C₇	C₂⁴	1	1	yes	characteristic
81	C₂²×M₁₆	—	2	2	no	—
82	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
83	#431	C₄×C₇	1	4	yes	—
84	#431	C₂²×C₇	1	4	yes	—
85	#423	—	2	2	no	—
86	#423	—	2	2	no	—
87	#423	C₂²×C₇	1	4	yes	—
88	#423	C₂²×C₇	1	4	yes	—
89	C₂×C₄⋊C₈	—	2	2	no	—
90	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
91	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
92	C₂×C₂²⋊C₈	C₄×C₇	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
94	#335	C₂²×C₇	1	16	yes	complemented
95	C₂³×C₇	—	4	1	no	—
96	C₂³×C₇	—	4	1	no	—
97	C₂³×C₇	#107	1	4	yes	—
98	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	4	2	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	4	1	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	#107	1	4	yes	—
109	C₂×C₄×C₇	#107	1	8	yes	—
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
117	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
123	C₇×C₈	—	2	8	no	—
124	C₇×C₈	—	4	8	no	—
125	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
126	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
127	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	4	8	no	—
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
133	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
135	C₂×C₄²	C₇×D₄	1	2	yes	—
136	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
138	C₂×C₄²	C₇×Q₈	1	2	yes	—
139	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₇	1	2	yes	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
145	C₄⋊C₈	C₂³×C₇	1	8	yes	—
146	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
147	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
148	C₄×C₈	C₂³×C₇	1	4	yes	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	—	2	2	no	—
151	C₂²×C₇	—	2	2	no	—
152	C₂²×C₇	—	2	1	no	—
153	C₂²×C₇	—	4	4	no	—
154	C₂²×C₇	C₂×#107	1	4	yes	—
155	C₂²×C₇	C₂×#107	1	2	yes	—
156	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	2	1	no	—
161	C₄×C₇	C₂×#107	1	4	yes	—
162	C₄×C₇	C₂×#107	1	2	yes	—
163	C₄×C₇	#430	1	2	yes	—
164	C₄×C₇	#568	1	2	yes	—
165	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
167	C₂⁴	C₇×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
171	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
173	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
176	C₂²×C₄	C₇×D₈	1	4	yes	—
177	C₂²×C₄	C₇×Q₁₆	1	4	yes	—
178	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
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	M₁₆	—	4	8	no	—
185	M₁₆	—	4	8	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	8	no	—
189	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
190	C₄²	C₇×C₄⋊C₄	1	2	yes	—
191	C₄²	C₇×C₄⋊C₄	1	2	yes	—
192	C₄²	C₇×C₄⋊C₄	1	2	yes	—
193	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
194	C₄²	C₂²×C₄×C₇	1	2	yes	—
195	C₄²	C₂×C₇×D₄	1	2	yes	—
196	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
197	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
198	C₄²	C₂×C₇×Q₈	1	2	yes	—
199	C₄²	C₂⁴×C₇	1	1	yes	characteristic
200	C₂×C₇	—	2	2	no	—
201	C₂×C₇	—	2	4	no	—
202	C₂×C₇	#2905	1	1	yes	characteristic
203	C₂×C₇	C₂²×#107	1	1	yes	characteristic
204	C₂×C₇	C₂×#430	1	1	yes	characteristic
205	C₂³	—	4	1	no	—
206	C₂³	—	4	1	no	—
207	C₂³	C₇×#107	1	4	yes	—
208	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
209	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	C₇×#107	1	4	yes	—
219	C₂×C₄	C₇×#107	1	8	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	1	yes	characteristic
223	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
225	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
227	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
228	C₂×C₄	C₂×C₇×Q₁₆	1	4	yes	—
229	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
233	C₈	—	2	8	no	—
234	C₈	—	4	8	no	—
235	C₇	#17814	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×C₇×#107	1	4	yes	—
242	C₂²	C₂×C₇×#107	1	2	yes	—
243	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	C₂×C₇×#107	1	4	yes	—
249	C₄	C₂×C₇×#107	1	2	yes	—
250	C₄	C₇×#430	1	2	yes	—
251	C₄	C₇×#568	1	2	yes	—
252	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
253	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	4	no	—
256	C₂	C₇×#2905	1	1	yes	characteristic
257	C₂	C₂²×C₇×#107	1	1	yes	characteristic
258	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×#1571	C₂	semidirect
4. C₇×#1567	C₂	semidirect
18. C₇×#335	C₂²	semidirect
19. #17814	C₇	direct (N × H)
46. #1571	C₂×C₇	semidirect
47. #1567	C₂×C₇	semidirect
94. #335	C₂²×C₇	semidirect
235. C₇	#17814	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×#1571	2
C₇×#1567	2
#17814	7

Subgroups — C7×#17814 #13195030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17814 #13195030