Subgroups — C₇×#17323 #13194539

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 258 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	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1452	C₂	1	2	yes	maximal, complemented
4	C₇×#1451	C₂	1	4	yes	maximal, complemented
5	C₇×#1450	C₂	1	2	yes	maximal, complemented
6	C₇×C₈×M₁₆	C₂	1	4	yes	maximal, complemented
7	C₂³×C₇×C₈	C₄	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#432	C₄	1	4	yes	—
10	C₇×#432	C₄	1	4	yes	—
11	C₇×#431	C₂²	1	4	yes	—
12	C₇×#431	C₂²	1	4	yes	—
13	C₇×#423	C₄	1	4	yes	—
14	C₇×#407	C₄	1	4	yes	—
15	C₄×C₇×M₁₆	C₂²	1	4	yes	—
16	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
17	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₇×#322	C₂²	1	4	yes	complemented
23	C₇×#322	C₂²	1	8	yes	complemented
24	C₇×C₈²	C₂²	1	4	yes	complemented
25	#17323	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
29	C₂²×C₇×C₈	—	2	8	no	—
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
42	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
43	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
44	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
45	C₇×#97	C₂×C₄	1	8	yes	—
46	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
48	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
49	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₇×C₈	C₂³	1	8	yes	—
51	C₄×C₇×C₈	C₂³	1	2	yes	—
52	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
54	#2920	C₂×C₇	1	2	yes	—
55	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
56	#1452	C₂×C₇	1	2	yes	complemented
57	#1451	C₂×C₇	1	4	yes	complemented
58	#1450	C₂×C₇	1	2	yes	complemented
59	C₈×M₁₆	C₂×C₇	1	4	yes	complemented
60	C₂⁴×C₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	2	8	no	—
63	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₇	C₄²	1	2	yes	—
66	C₂²×C₄×C₇	C₄²	1	2	yes	—
67	C₂²×C₄×C₇	C₄²	1	2	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
73	C₇×M₁₆	—	2	16	no	—
74	C₂×C₇×C₈	—	2	4	no	—
75	C₂×C₇×C₈	—	2	8	no	—
76	C₂×C₇×C₈	—	2	8	no	—
77	C₂×C₇×C₈	M₁₆	1	8	yes	complemented
78	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₇×C₈	C₂²×C₄	1	16	yes	—
80	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
82	C₄²×C₇	C₄²	1	2	yes	—
83	C₄²×C₇	C₄²	1	1	yes	characteristic
84	C₄²×C₇	C₄²	1	2	yes	—
85	C₄²×C₇	C₄²	1	1	yes	characteristic
86	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₇	C₂⁴	1	1	yes	characteristic
92	C₂³×C₈	C₄×C₇	1	2	yes	—
93	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
94	#432	C₄×C₇	1	4	yes	—
95	#432	C₄×C₇	1	4	yes	—
96	#431	C₂²×C₇	1	4	yes	—
97	#431	C₂²×C₇	1	4	yes	—
98	#423	C₄×C₇	1	4	yes	—
99	#407	C₄×C₇	1	4	yes	—
100	C₄×M₁₆	C₂²×C₇	1	4	yes	—
101	C₂×C₄×C₈	C₄×C₇	1	4	yes	—
102	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
107	#322	C₂²×C₇	1	8	yes	complemented
108	#322	C₂²×C₇	1	4	yes	complemented
109	C₈²	C₂²×C₇	1	4	yes	complemented
110	C₂³×C₇	—	2	2	no	—
111	C₂³×C₇	—	2	4	no	—
112	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
113	C₂³×C₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	2	4	no	—
116	C₂×C₄×C₇	—	2	4	no	—
117	C₂×C₄×C₇	—	2	4	no	—
118	C₂×C₄×C₇	—	2	4	no	—
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	4	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
128	C₂×C₄×C₇	#131	1	8	yes	—
129	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
131	C₇×C₈	—	2	4	no	—
132	C₇×C₈	—	2	16	no	—
133	C₇×C₈	C₂×M₁₆	1	4	yes	—
134	C₇×C₈	C₂×M₁₆	1	4	yes	—
135	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
136	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
137	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
138	C₂²×C₈	—	2	8	no	—
139	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
143	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
144	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
147	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
148	C₂×C₄²	C₂³×C₇	1	2	yes	—
149	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
150	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
151	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
152	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
153	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
154	#97	C₂×C₄×C₇	1	8	yes	—
155	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
156	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
157	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
158	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
159	C₄×C₈	C₂³×C₇	1	8	yes	—
160	C₄×C₈	C₂³×C₇	1	2	yes	—
161	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
163	C₂²×C₇	—	2	8	no	—
164	C₂²×C₇	—	2	2	no	—
165	C₂²×C₇	—	2	2	no	—
166	C₂²×C₇	—	2	2	no	—
167	C₂²×C₇	C₄×M₁₆	1	2	yes	—
168	C₂²×C₇	#405	1	4	yes	—
169	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	C₄×M₁₆	1	2	yes	—
174	C₄×C₇	#405	1	4	yes	—
175	C₄×C₇	C₂²×M₁₆	1	2	yes	—
176	C₄×C₇	C₂×#131	1	4	yes	—
177	C₂⁴	C₄²×C₇	1	1	yes	characteristic
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
181	C₂²×C₄	C₄²×C₇	1	2	yes	—
182	C₂²×C₄	C₄²×C₇	1	2	yes	—
183	C₂²×C₄	C₄²×C₇	1	2	yes	—
184	C₂²×C₄	C₄²×C₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
186	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
190	M₁₆	—	2	16	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	C₇×M₁₆	1	8	yes	complemented
195	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
196	C₂×C₈	C₂²×C₄×C₇	1	16	yes	—
197	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
199	C₄²	C₄²×C₇	1	2	yes	—
200	C₄²	C₄²×C₇	1	1	yes	characteristic
201	C₄²	C₄²×C₇	1	2	yes	—
202	C₄²	C₄²×C₇	1	1	yes	characteristic
203	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
204	C₄²	C₂²×C₄×C₇	1	2	yes	—
205	C₄²	C₂²×C₄×C₇	1	2	yes	—
206	C₄²	C₂²×C₄×C₇	1	2	yes	—
207	C₄²	C₂²×C₄×C₇	1	2	yes	—
208	C₄²	C₂⁴×C₇	1	1	yes	characteristic
209	C₂×C₇	—	2	2	no	—
210	C₂×C₇	—	2	4	no	—
211	C₂×C₇	C₂×C₄×M₁₆	1	1	yes	characteristic
212	C₂×C₇	C₂×#405	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
216	C₂³	C₂×C₄²×C₇	1	2	yes	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	C₂×C₄²×C₇	1	2	yes	—
226	C₂×C₄	C₂×C₄²×C₇	1	2	yes	—
227	C₂×C₄	C₂×C₄²×C₇	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄²×C₇	1	2	yes	—
230	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
231	C₂×C₄	C₇×#131	1	8	yes	—
232	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	2	16	no	—
236	C₈	C₂×C₇×M₁₆	1	4	yes	—
237	C₈	C₂×C₇×M₁₆	1	4	yes	—
238	C₇	#17323	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₄×C₇×M₁₆	1	2	yes	—
244	C₂²	C₇×#405	1	4	yes	—
245	C₂²	C₂²×C₄²×C₇	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	C₄×C₇×M₁₆	1	2	yes	—
250	C₄	C₇×#405	1	4	yes	—
251	C₄	C₂²×C₇×M₁₆	1	2	yes	—
252	C₄	C₂×C₇×#131	1	4	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	4	no	—
255	C₂	C₂×C₄×C₇×M₁₆	1	1	yes	characteristic
256	C₂	C₂×C₇×#405	1	2	yes	—

	subgroup	G/H
center	C₄×C₇×C₈	C₂³
commutator	C₂²	C₂²×C₄²×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄²
prime radical	C₂⁴×C₇	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1452	C₂	semidirect
4. C₇×#1451	C₂	semidirect
5. C₇×#1450	C₂	semidirect
6. C₇×C₈×M₁₆	C₂	semidirect
22. C₇×#322	C₂²	semidirect
23. C₇×#322	C₂²	semidirect
24. C₇×C₈²	C₂²	semidirect
25. #17323	C₇	direct (N × H)
56. #1452	C₂×C₇	semidirect
57. #1451	C₂×C₇	semidirect
58. #1450	C₂×C₇	semidirect
59. C₈×M₁₆	C₂×C₇	semidirect
77. C₂×C₇×C₈	M₁₆	semidirect
107. #322	C₂²×C₇	semidirect
108. #322	C₂²×C₇	semidirect
109. C₈²	C₂²×C₇	semidirect
194. C₂×C₈	C₇×M₁₆	semidirect
238. C₇	#17323	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
C₂²×C₄×C₇×C₈	2
C₇×#1452	2
C₇×#1451	2
C₇×#1450	2
C₇×C₈×M₁₆	2
#17323	7

Subgroups — C7×#17323 #13194539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17323 #13194539