Subgroups — C₇×#13200 #13190416

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2929	C₂	1	1	yes	maximal, characteristic
2	C₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2117	C₂	1	2	yes	maximal, complemented
4	C₇×#2112	C₂	1	2	yes	maximal, complemented
5	C₇×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#525	C₄	1	1	yes	characteristic
9	C₂×C₇×M₃₂	C₂²	1	2	yes	—
10	C₂×C₇×M₃₂	C₂²	1	2	yes	—
11	C₇×#435	—	2	2	no	—
12	C₇×C₈×D₄	—	2	2	no	—
13	C₇×#424	C₄	1	1	yes	characteristic
14	C₇×#408	C₄	1	1	yes	characteristic
15	C₇×#407	C₄	1	1	yes	characteristic
16	C₇×#405	C₂²	1	1	yes	characteristic
17	C₇×#364	—	2	2	no	—
18	C₇×#350	—	2	4	no	—
19	C₇×C₂²⋊C₁₆	—	2	4	no	—
20	C₇×#347	—	2	2	no	—
21	#13200	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₇×#131	—	2	2	no	—
25	C₇×#131	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	1	no	—
27	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	1	no	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₇×#123	—	2	1	no	—
37	C₇×C₂²⋊Q₈	—	2	1	no	—
38	C₇×#121	—	2	1	no	—
39	C₇×#120	—	2	1	no	—
40	C₄×C₇×D₄	—	2	2	no	—
41	C₇×#117	C₂×C₄	1	1	yes	characteristic
42	C₇×M₃₂	—	2	2	no	—
43	C₇×M₃₂	—	2	2	no	—
44	C₇×M₃₂	—	2	2	no	—
45	C₇×M₃₂	—	2	2	no	—
46	C₂×C₇×C₁₆	—	2	2	no	—
47	C₂×C₇×C₁₆	D₄	1	4	yes	complemented
48	C₇×C₄⋊C₈	—	2	2	no	—
49	C₇×C₂²⋊C₈	—	2	2	no	—
50	C₇×C₂²⋊C₈	—	2	2	no	—
51	C₇×#97	D₄	1	2	yes	—
52	C₄×C₇×C₈	D₄	1	2	yes	—
53	#2929	C₂×C₇	1	1	yes	characteristic
54	#2153	C₂×C₇	1	1	yes	characteristic, complemented
55	#2117	C₂×C₇	1	2	yes	complemented
56	#2112	C₂×C₇	1	2	yes	complemented
57	#2110	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
59	C₇×#41	—	2	2	no	—
60	C₇×#41	—	2	2	no	—
61	C₂×C₇×Q₈	M₁₆	1	1	yes	characteristic
62	C₂×C₇×D₄	—	2	2	no	—
63	C₂×C₇×D₄	—	2	1	no	—
64	C₂×C₇×D₄	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₇	—	2	1	no	—
66	C₂²×C₄×C₇	—	2	1	no	—
67	C₂²×C₄×C₇	—	2	2	no	—
68	C₂²×C₄×C₇	—	2	1	no	—
69	C₂²×C₄×C₇	—	2	2	no	—
70	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₇×M₁₆	—	2	2	no	—
73	C₇×M₁₆	—	2	2	no	—
74	C₇×M₁₆	—	4	2	no	—
75	C₂×C₇×C₈	—	2	1	no	—
76	C₂×C₇×C₈	—	2	1	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	4	2	no	—
79	C₂×C₇×C₈	—	4	1	no	—
80	C₂×C₇×C₈	—	4	1	no	—
81	C₂×C₇×C₈	—	4	1	no	—
82	C₂×C₇×C₈	—	4	1	no	—
83	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₇×C₈	#41	1	2	yes	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	1	no	—
90	C₇×C₂²⋊C₄	—	2	2	no	—
91	C₇×C₂²⋊C₄	—	2	2	no	—
92	C₇×C₂²⋊C₄	—	2	1	no	—
93	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
94	C₇×C₁₆	—	2	4	no	—
95	C₇×C₁₆	—	4	2	no	—
96	C₂×#131	C₂²×C₇	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
98	#525	C₄×C₇	1	1	yes	characteristic
99	C₂×M₃₂	C₂²×C₇	1	2	yes	—
100	C₂×M₃₂	C₂²×C₇	1	2	yes	—
101	#435	—	2	2	no	—
102	C₈×D₄	—	2	2	no	—
103	#424	C₄×C₇	1	1	yes	characteristic
104	#408	C₄×C₇	1	1	yes	characteristic
105	#407	C₄×C₇	1	1	yes	characteristic
106	#405	C₂²×C₇	1	1	yes	characteristic
107	#364	—	2	2	no	—
108	#350	—	2	4	no	—
109	C₂²⋊C₁₆	—	2	4	no	—
110	#347	—	2	2	no	—
111	C₂³×C₇	—	2	1	no	—
112	C₂³×C₇	—	2	2	no	—
113	C₂³×C₇	—	2	1	no	—
114	C₂³×C₇	—	4	2	no	—
115	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
116	C₇×Q₈	—	2	2	no	—
117	C₇×D₄	—	2	2	no	—
118	C₇×D₄	—	4	4	no	—
119	C₇×D₄	—	4	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	4	1	no	—
125	C₂×C₄×C₇	—	4	1	no	—
126	C₂×C₄×C₇	—	4	1	no	—
127	C₂×C₄×C₇	—	4	1	no	—
128	C₂×C₄×C₇	—	4	2	no	—
129	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₇	#131	1	2	yes	—
133	C₇×C₈	—	2	2	no	—
134	C₇×C₈	—	4	1	no	—
135	C₇×C₈	—	4	1	no	—
136	C₇×C₈	#121	1	2	yes	—
137	C₇×C₈	#124	1	1	yes	characteristic
138	C₇×C₈	#127	1	1	yes	characteristic
139	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
140	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
141	#131	—	2	2	no	—
142	#131	—	2	2	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
153	#123	—	2	1	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	#121	—	2	1	no	—
156	#120	—	2	1	no	—
157	C₄×D₄	—	2	2	no	—
158	#117	C₂×C₄×C₇	1	1	yes	characteristic
159	M₃₂	—	2	2	no	—
160	M₃₂	—	2	2	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	2	2	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	C₇×D₄	1	4	yes	complemented
165	C₄⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	2	no	—
168	#97	C₇×D₄	1	2	yes	—
169	C₄×C₈	C₇×D₄	1	2	yes	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	4	no	—
172	C₂²×C₇	—	4	1	no	—
173	C₂²×C₇	—	4	1	no	—
174	C₂²×C₇	—	4	2	no	—
175	C₂²×C₇	—	4	2	no	—
176	C₂²×C₇	—	4	1	no	—
177	C₂²×C₇	—	4	1	no	—
178	C₂²×C₇	#407	1	1	yes	characteristic
179	C₂²×C₇	#435	1	2	yes	—
180	C₄×C₇	—	4	2	no	—
181	C₄×C₇	—	4	1	no	—
182	C₄×C₇	—	4	1	no	—
183	C₄×C₇	#390	1	1	yes	characteristic
184	C₄×C₇	#408	1	1	yes	characteristic
185	C₄×C₇	#435	1	2	yes	—
186	C₂⁴	C₇×M₁₆	1	1	yes	characteristic
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	C₂×Q₈	C₇×M₁₆	1	1	yes	characteristic
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₇×M₁₆	1	1	yes	characteristic
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	4	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
212	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
214	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
215	C₂×C₈	C₇×#41	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
222	C₁₆	—	2	4	no	—
223	C₁₆	—	4	2	no	—
224	C₂×C₇	—	2	1	no	—
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	—	4	2	no	—
227	C₂×C₇	—	8	1	no	—
228	C₂×C₇	#1955	1	1	yes	characteristic
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
249	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
250	C₂×C₄	C₇×#131	1	2	yes	—
251	C₈	—	2	2	no	—
252	C₈	—	4	1	no	—
253	C₈	—	4	1	no	—
254	C₈	C₇×#121	1	2	yes	—
255	C₈	C₇×#124	1	1	yes	characteristic
256	C₈	C₇×#127	1	1	yes	characteristic
257	C₇	#13200	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₇×#407	1	1	yes	characteristic
267	C₂²	C₇×#435	1	2	yes	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	C₇×#390	1	1	yes	characteristic
272	C₄	C₇×#408	1	1	yes	characteristic
273	C₄	C₇×#435	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₇×#1955	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	#127
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1955
prime radical	C₇×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2153	C₂	semidirect
3. C₇×#2117	C₂	semidirect
4. C₇×#2112	C₂	semidirect
5. C₇×#2110	C₂	semidirect
21. #13200	C₇	direct (N × H)
47. C₂×C₇×C₁₆	D₄	semidirect
54. #2153	C₂×C₇	semidirect
55. #2117	C₂×C₇	semidirect
56. #2112	C₂×C₇	semidirect
57. #2110	C₂×C₇	semidirect
164. C₂×C₁₆	C₇×D₄	semidirect
257. C₇	#13200	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×#2929	2
C₇×#2153	2
C₇×#2117	2
C₇×#2112	2
C₇×#2110	2
#13200	7

Subgroups — C7×#13200 #13190416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13200 #13190416