Subgroups — C₇×#21484 #13198700

Order 1792.

1306 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3025	C₂	1	1	yes	maximal, characteristic
3	C₇×#2930	C₂	1	3	yes	maximal, complemented
4	C₇×#1891	C₂	1	3	yes	maximal, complemented
5	C₇×#1868	C₂	1	3	yes	maximal, complemented
6	C₇×#1791	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1757	C₂	1	3	yes	maximal, complemented
8	C₇×#585	C₂²	1	1	yes	characteristic
9	C₇×#568	—	2	6	no	—
10	C₇×#568	C₂²	1	3	yes	—
11	C₂×C₇×#131	—	2	6	no	—
12	C₂×C₇×#131	C₂²	1	3	yes	—
13	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₇×#525	C₂²	1	3	yes	—
15	C₇×#458	C₄	1	4	yes	—
16	C₇×B(4,𝔽₂)	C₄	1	4	yes	—
17	C₇×#436	—	2	6	no	—
18	C₇×#435	—	2	6	no	—
19	C₇×#424	C₂²	1	3	yes	complemented
20	C₇×#413	C₂²	1	3	yes	—
21	C₇×#410	C₂²	1	3	yes	—
22	C₇×#408	C₂²	1	3	yes	—
23	C₇×#407	C₂²	1	3	yes	complemented
24	C₇×#405	C₂²	1	3	yes	complemented
25	C₇×#344	C₂²	1	6	yes	complemented
26	#21484	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₇×2₋¹⁺⁴	—	2	3	no	—
28	C₇×H(5,𝔽₂)	—	2	3	no	—
29	C₇×H(5,𝔽₂)	C₂×C₄	1	4	yes	—
30	C₂×C₇×#41	—	2	6	no	—
31	C₂×C₇×#41	C₂³	1	3	yes	—
32	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₇×#131	—	2	4	no	—
34	C₇×#131	—	2	12	no	—
35	C₇×#131	—	2	6	no	—
36	C₇×#131	—	2	12	no	—
37	C₇×#131	—	2	6	no	—
38	C₂×C₇×M₁₆	—	2	12	no	—
39	C₂×C₇×M₁₆	—	2	6	no	—
40	C₂×C₇×M₁₆	—	2	3	no	—
41	C₂×C₇×M₁₆	—	2	3	no	—
42	C₂×C₇×M₁₆	D₄	1	6	yes	complemented
43	C₂×C₇×M₁₆	C₂³	1	3	yes	—
44	C₂×C₇×M₁₆	C₂³	1	3	yes	—
45	C₂²×C₇×C₈	—	2	4	no	—
46	C₂²×C₇×C₈	D₄	1	6	yes	complemented
47	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₇×#123	C₂×C₄	1	6	yes	—
50	C₇×C₂²⋊Q₈	—	2	3	no	—
51	C₇×#121	—	2	3	no	—
52	C₇×#120	C₂×C₄	1	6	yes	—
53	C₄×C₇×D₄	—	2	6	no	—
54	C₇×#117	C₂³	1	3	yes	—
55	C₇×C₄⋊C₈	—	2	6	no	—
56	C₇×#101	—	2	3	no	—
57	C₇×#100	—	2	3	no	—
58	C₇×#99	C₂×C₄	1	12	yes	—
59	C₇×C₂²⋊C₈	—	2	6	no	—
60	C₇×C₂²⋊C₈	—	2	6	no	—
61	C₇×#97	—	2	3	no	—
62	C₄×C₇×C₈	—	2	3	no	—
63	#3573	C₂×C₇	1	1	yes	characteristic, complemented
64	#3025	C₂×C₇	1	1	yes	characteristic
65	#2930	C₂×C₇	1	3	yes	complemented
66	#1891	C₂×C₇	1	3	yes	complemented
67	#1868	C₂×C₇	1	3	yes	complemented
68	#1791	C₂×C₇	1	1	yes	characteristic, complemented
69	#1757	C₂×C₇	1	3	yes	complemented
70	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₇×#41	—	2	4	no	—
72	C₇×#41	—	2	12	no	—
73	C₇×#41	—	2	12	no	—
74	C₇×#41	—	2	6	no	—
75	C₇×#41	—	2	6	no	—
76	C₂×C₇×Q₈	—	2	6	no	—
77	C₂×C₇×Q₈	#41	1	3	yes	—
78	C₂×C₇×D₄	—	2	6	no	—
79	C₂×C₇×D₄	—	2	12	no	—
80	C₂×C₇×D₄	—	2	6	no	—
81	C₂×C₇×D₄	C₂²×C₄	1	6	yes	—
82	C₂×C₇×D₄	#41	1	3	yes	—
83	C₂²×C₄×C₇	—	2	4	no	—
84	C₂²×C₄×C₇	—	2	3	no	—
85	C₂²×C₄×C₇	—	2	3	no	—
86	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₇×M₁₆	—	2	12	no	—
90	C₇×M₁₆	—	2	6	no	—
91	C₇×M₁₆	—	2	12	no	—
92	C₇×M₁₆	—	2	6	no	—
93	C₇×M₁₆	—	4	1	no	—
94	C₇×M₁₆	—	4	3	no	—
95	C₂×C₇×C₈	—	2	12	no	—
96	C₂×C₇×C₈	—	2	6	no	—
97	C₂×C₇×C₈	—	2	6	no	—
98	C₂×C₇×C₈	—	2	3	no	—
99	C₂×C₇×C₈	—	2	3	no	—
100	C₂×C₇×C₈	C₂×D₄	1	6	yes	—
101	C₂×C₇×C₈	C₂×D₄	1	3	yes	—
102	C₂×C₇×C₈	C₂×D₄	1	3	yes	—
103	C₇×C₄⋊C₄	—	2	6	no	—
104	C₇×C₄⋊C₄	—	2	3	no	—
105	C₇×C₂²⋊C₄	—	2	6	no	—
106	C₇×C₂²⋊C₄	—	2	6	no	—
107	C₇×C₂²⋊C₄	C₂²×C₄	1	6	yes	—
108	C₄²×C₇	—	2	3	no	—
109	#585	C₂²×C₇	1	1	yes	characteristic
110	#568	—	2	6	no	—
111	#568	C₂²×C₇	1	3	yes	—
112	C₂×#131	—	2	6	no	—
113	C₂×#131	C₂²×C₇	1	3	yes	—
114	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
115	#525	C₂²×C₇	1	3	yes	—
116	#458	C₄×C₇	1	4	yes	—
117	B(4,𝔽₂)	C₄×C₇	1	4	yes	—
118	#436	—	2	6	no	—
119	#435	—	2	6	no	—
120	#424	C₂²×C₇	1	3	yes	complemented
121	#413	C₂²×C₇	1	3	yes	—
122	#410	C₂²×C₇	1	3	yes	—
123	#408	C₂²×C₇	1	3	yes	—
124	#407	C₂²×C₇	1	3	yes	complemented
125	#405	C₂²×C₇	1	3	yes	complemented
126	#344	C₂²×C₇	1	6	yes	complemented
127	C₂³×C₇	—	2	4	no	—
128	C₂³×C₇	—	2	3	no	—
129	C₂³×C₇	—	8	1	no	—
130	C₂³×C₇	C₄×D₄	1	6	yes	—
131	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
132	C₇×Q₈	—	2	4	no	—
133	C₇×Q₈	—	2	6	no	—
134	C₇×D₄	—	2	12	no	—
135	C₇×D₄	—	2	12	no	—
136	C₇×D₄	—	2	6	no	—
137	C₇×D₄	—	8	6	no	—
138	C₂×C₄×C₇	—	2	12	no	—
139	C₂×C₄×C₇	—	2	6	no	—
140	C₂×C₄×C₇	—	2	6	no	—
141	C₂×C₄×C₇	—	2	6	no	—
142	C₂×C₄×C₇	—	2	3	no	—
143	C₂×C₄×C₇	—	4	1	no	—
144	C₂×C₄×C₇	—	4	3	no	—
145	C₂×C₄×C₇	C₄×D₄	1	6	yes	—
146	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
147	C₂×C₄×C₇	C₂×#41	1	3	yes	—
148	C₇×C₈	—	2	6	no	—
149	C₇×C₈	—	4	1	no	—
150	C₇×C₈	—	4	3	no	—
151	C₇×C₈	#120	1	1	yes	characteristic
152	C₇×C₈	#120	1	3	yes	—
153	2₋¹⁺⁴	—	2	3	no	—
154	H(5,𝔽₂)	—	2	3	no	—
155	H(5,𝔽₂)	C₂×C₄×C₇	1	4	yes	—
156	C₂×#41	—	2	6	no	—
157	C₂×#41	C₂³×C₇	1	3	yes	—
158	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
159	#131	—	2	4	no	—
160	#131	—	2	12	no	—
161	#131	—	2	6	no	—
162	#131	—	2	12	no	—
163	#131	—	2	6	no	—
164	C₂×M₁₆	—	2	12	no	—
165	C₂×M₁₆	—	2	6	no	—
166	C₂×M₁₆	—	2	3	no	—
167	C₂×M₁₆	—	2	3	no	—
168	C₂×M₁₆	C₇×D₄	1	6	yes	complemented
169	C₂×M₁₆	C₂³×C₇	1	3	yes	—
170	C₂×M₁₆	C₂³×C₇	1	3	yes	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	C₇×D₄	1	6	yes	complemented
173	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
175	#123	C₂×C₄×C₇	1	6	yes	—
176	C₂²⋊Q₈	—	2	3	no	—
177	#121	—	2	3	no	—
178	#120	C₂×C₄×C₇	1	6	yes	—
179	C₄×D₄	—	2	6	no	—
180	#117	C₂³×C₇	1	3	yes	—
181	C₄⋊C₈	—	2	6	no	—
182	#101	—	2	3	no	—
183	#100	—	2	3	no	—
184	#99	C₂×C₄×C₇	1	12	yes	—
185	C₂²⋊C₈	—	2	6	no	—
186	C₂²⋊C₈	—	2	6	no	—
187	#97	—	2	3	no	—
188	C₄×C₈	—	2	3	no	—
189	C₂²×C₇	—	2	6	no	—
190	C₂²×C₇	—	4	1	no	—
191	C₂²×C₇	—	4	6	no	—
192	C₂²×C₇	—	4	3	no	—
193	C₂²×C₇	—	4	3	no	—
194	C₂²×C₇	—	4	1	no	—
195	C₂²×C₇	—	8	4	no	—
196	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
197	C₄×C₇	—	2	6	no	—
198	C₄×C₇	—	4	6	no	—
199	C₄×C₇	—	4	1	no	—
200	C₄×C₇	C₂×#120	1	1	yes	characteristic
201	C₄×C₇	#525	1	3	yes	—
202	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
203	#41	—	2	4	no	—
204	#41	—	2	12	no	—
205	#41	—	2	12	no	—
206	#41	—	2	6	no	—
207	#41	—	2	6	no	—
208	C₂×Q₈	—	2	6	no	—
209	C₂×Q₈	C₇×#41	1	3	yes	—
210	C₂×D₄	—	2	6	no	—
211	C₂×D₄	—	2	12	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	C₂²×C₄×C₇	1	6	yes	—
214	C₂×D₄	C₇×#41	1	3	yes	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	3	no	—
217	C₂²×C₄	—	2	3	no	—
218	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
220	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
221	M₁₆	—	2	12	no	—
222	M₁₆	—	2	6	no	—
223	M₁₆	—	2	12	no	—
224	M₁₆	—	2	6	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	3	no	—
227	C₂×C₈	—	2	12	no	—
228	C₂×C₈	—	2	6	no	—
229	C₂×C₈	—	2	6	no	—
230	C₂×C₈	—	2	3	no	—
231	C₂×C₈	—	2	3	no	—
232	C₂×C₈	C₂×C₇×D₄	1	6	yes	—
233	C₂×C₈	C₂×C₇×D₄	1	3	yes	—
234	C₂×C₈	C₂×C₇×D₄	1	3	yes	—
235	C₄⋊C₄	—	2	6	no	—
236	C₄⋊C₄	—	2	3	no	—
237	C₂²⋊C₄	—	2	6	no	—
238	C₂²⋊C₄	—	2	6	no	—
239	C₂²⋊C₄	C₂²×C₄×C₇	1	6	yes	—
240	C₄²	—	2	3	no	—
241	C₂×C₇	—	2	3	no	—
242	C₂×C₇	—	4	6	no	—
243	C₂×C₇	—	8	1	no	—
244	C₂×C₇	C₄×#120	1	1	yes	characteristic
245	C₂³	—	2	4	no	—
246	C₂³	—	2	3	no	—
247	C₂³	—	8	1	no	—
248	C₂³	C₄×C₇×D₄	1	6	yes	—
249	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	6	no	—
252	D₄	—	2	12	no	—
253	D₄	—	2	12	no	—
254	D₄	—	2	6	no	—
255	D₄	—	8	6	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	2	6	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	C₄×C₇×D₄	1	6	yes	—
264	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
265	C₂×C₄	C₂×C₇×#41	1	3	yes	—
266	C₈	—	2	6	no	—
267	C₈	—	4	1	no	—
268	C₈	—	4	3	no	—
269	C₈	C₇×#120	1	1	yes	characteristic
270	C₈	C₇×#120	1	3	yes	—
271	C₇	#21484	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	—	2	6	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	6	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	3	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	8	4	no	—
279	C₂²	C₂×C₄×C₇×D₄	1	3	yes	—
280	C₄	—	2	6	no	—
281	C₄	—	4	6	no	—
282	C₄	—	4	1	no	—
283	C₄	C₂×C₇×#120	1	1	yes	characteristic
284	C₄	C₇×#525	1	3	yes	—
285	C₂	—	2	3	no	—
286	C₂	—	4	6	no	—
287	C₂	—	8	1	no	—
288	C₂	C₄×C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂×#120
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	C₄×#120
prime radical	C₇×#3025	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3573	C₂	semidirect
3. C₇×#2930	C₂	semidirect
4. C₇×#1891	C₂	semidirect
5. C₇×#1868	C₂	semidirect
6. C₇×#1791	C₂	semidirect
7. C₇×#1757	C₂	semidirect
13. C₂²×C₇×M₁₆	C₂²	semidirect
19. C₇×#424	C₂²	semidirect
23. C₇×#407	C₂²	semidirect
24. C₇×#405	C₂²	semidirect
25. C₇×#344	C₂²	semidirect
26. #21484	C₇	direct (N × H)
42. C₂×C₇×M₁₆	D₄	semidirect
46. C₂²×C₇×C₈	D₄	semidirect
63. #3573	C₂×C₇	semidirect
65. #2930	C₂×C₇	semidirect
66. #1891	C₂×C₇	semidirect
67. #1868	C₂×C₇	semidirect
68. #1791	C₂×C₇	semidirect
69. #1757	C₂×C₇	semidirect
114. C₂²×M₁₆	C₂²×C₇	semidirect
120. #424	C₂²×C₇	semidirect
124. #407	C₂²×C₇	semidirect
125. #405	C₂²×C₇	semidirect
126. #344	C₂²×C₇	semidirect
168. C₂×M₁₆	C₇×D₄	semidirect
172. C₂²×C₈	C₇×D₄	semidirect
271. C₇	#21484	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×#3573	2
C₇×#3025	2
C₇×#2930	2
C₇×#1891	2
C₇×#1868	2
C₇×#1791	2
C₇×#1757	2
#21484	7

Subgroups — C7×#21484 #13198700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21484 #13198700