Subgroups — C₇×#12848 #13190064

Order 1792.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₇×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₇×#1379	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×#117	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
10	C₇×#405	—	2	1	no	—
11	C₇×#405	—	2	1	no	—
12	C₇×#405	C₄	1	2	yes	—
13	C₇×#405	C₂²	1	2	yes	—
14	C₂×C₇×#97	—	2	1	no	—
15	C₂×C₄×C₇×C₈	—	2	1	no	—
16	C₇×C₂²⋊C₁₆	—	2	1	no	—
17	C₇×C₂²⋊C₁₆	—	2	1	no	—
18	C₇×C₂²⋊C₁₆	—	2	1	no	—
19	C₇×C₂²⋊C₁₆	—	2	1	no	—
20	#12848	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×M₁₆	—	2	1	no	—
23	C₂×C₇×M₁₆	—	2	4	no	—
24	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	—	2	1	no	—
26	C₂²×C₇×C₈	—	2	4	no	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₇×#117	—	2	1	no	—
34	C₇×#117	—	2	1	no	—
35	C₇×#117	C₈	1	2	yes	—
36	C₇×#117	C₂×C₄	1	2	yes	—
37	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
39	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₄²×C₇	—	2	1	no	—
41	C₇×M₃₂	—	2	4	no	—
42	C₂×C₇×C₁₆	—	2	2	no	—
43	C₂×C₇×C₁₆	—	2	2	no	—
44	C₂×C₇×C₁₆	—	2	4	no	—
45	C₇×#97	—	2	2	no	—
46	C₇×#97	—	2	2	no	—
47	C₄×C₇×C₈	—	2	2	no	—
48	C₄×C₇×C₈	—	2	2	no	—
49	C₂×#405	C₂×C₇	1	1	yes	characteristic
50	#2112	C₂×C₇	1	1	yes	characteristic
51	C₂×C₂²⋊C₁₆	C₂×C₇	1	1	yes	characteristic
52	#1379	C₂×C₇	1	4	yes	complemented
53	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₇	—	2	2	no	—
55	C₂²×C₄×C₇	—	2	2	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₇×M₁₆	—	2	4	no	—
65	C₇×M₁₆	—	2	2	no	—
66	C₇×M₁₆	M₁₆	1	4	yes	—
67	C₂×C₇×C₈	—	2	4	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	#41	1	4	yes	—
79	C₇×C₄⋊C₄	—	2	1	no	—
80	C₇×C₄⋊C₄	—	2	1	no	—
81	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
82	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
83	C₇×C₂²⋊C₄	—	2	2	no	—
84	C₇×C₂²⋊C₄	—	2	2	no	—
85	C₄²×C₇	—	2	2	no	—
86	C₄²×C₇	—	2	2	no	—
87	C₇×C₁₆	—	4	2	no	—
88	C₇×C₁₆	—	4	2	no	—
89	C₂²×M₁₆	C₄×C₇	1	1	yes	characteristic
90	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
91	C₂×#117	C₄×C₇	1	1	yes	characteristic
92	C₂×M₃₂	C₂²×C₇	1	2	yes	—
93	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
94	#405	—	2	1	no	—
95	#405	—	2	1	no	—
96	#405	C₄×C₇	1	2	yes	—
97	#405	C₂²×C₇	1	2	yes	—
98	C₂×#97	—	2	1	no	—
99	C₂×C₄×C₈	—	2	1	no	—
100	C₂²⋊C₁₆	—	2	1	no	—
101	C₂²⋊C₁₆	—	2	1	no	—
102	C₂²⋊C₁₆	—	2	1	no	—
103	C₂²⋊C₁₆	—	2	1	no	—
104	C₂³×C₇	—	2	2	no	—
105	C₂³×C₇	—	2	2	no	—
106	C₂³×C₇	—	4	1	no	—
107	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
109	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇	#117	1	2	yes	—
125	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
127	C₇×C₈	—	2	1	no	—
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	2	4	no	—
130	C₇×C₈	#123	1	2	yes	—
131	C₇×C₈	#123	1	2	yes	—
132	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
133	C₂×M₁₆	—	2	1	no	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
139	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
140	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
141	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
142	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
144	#117	—	2	1	no	—
145	#117	—	2	1	no	—
146	#117	C₇×C₈	1	2	yes	—
147	#117	C₂×C₄×C₇	1	2	yes	—
148	C₂×C₄⋊C₄	C₇×C₈	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₇×C₈	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	M₃₂	—	2	4	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	4	no	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	2	1	no	—
162	C₂²×C₇	—	2	1	no	—
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	4	4	no	—
168	C₂²×C₇	#350	1	2	yes	—
169	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₂²×C₇	#432	1	2	yes	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	4	2	no	—
174	C₄×C₇	—	4	2	no	—
175	C₄×C₇	#381	1	1	yes	characteristic
176	C₄×C₇	#407	1	1	yes	characteristic
177	C₄×C₇	#431	1	2	yes	—
178	C₂⁴	C₇×M₁₆	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
187	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	C₇×M₁₆	1	4	yes	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
203	C₂×C₈	C₇×#41	1	4	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
207	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₁₆	—	4	2	no	—
213	C₁₆	—	4	2	no	—
214	C₂×C₇	—	2	2	no	—
215	C₂×C₇	—	2	2	no	—
216	C₂×C₇	—	4	1	no	—
217	C₂×C₇	#1817	1	1	yes	characteristic
218	C₂×C₇	C₂×#350	1	1	yes	characteristic
219	C₂×C₇	#2117	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₇×C₂²⋊C₈	1	1	yes	characteristic
224	C₂³	C₇×C₂²⋊C₈	1	1	yes	characteristic
225	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
238	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₇×#117	1	2	yes	—
241	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
242	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	C₇×#123	1	2	yes	—
247	C₈	C₇×#123	1	2	yes	—
248	C₇	#12848	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₇×#350	1	2	yes	—
258	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
259	C₂²	C₇×#432	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	C₇×#381	1	1	yes	characteristic
265	C₄	C₇×#407	1	1	yes	characteristic
266	C₄	C₇×#431	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	C₇×#1817	1	1	yes	characteristic
271	C₂	C₂×C₇×#350	1	1	yes	characteristic
272	C₂	C₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1379	C₂	semidirect
20. #12848	C₇	direct (N × H)
52. #1379	C₂×C₇	semidirect
248. C₇	#12848	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×#405	2
C₇×#2112	2
C₂×C₇×C₂²⋊C₁₆	2
C₇×#1379	2
#12848	7

Subgroups — C7×#12848 #13190064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12848 #13190064