Subgroups — C₇×#2447 #11741756

Order 896.

540 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#397	C₂	1	2	yes	maximal, complemented
4	C₇×#397	C₂	1	2	yes	maximal, complemented
5	C₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#390	C₂	1	1	yes	maximal, characteristic
8	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#385	C₂	1	2	yes	maximal, complemented
10	C₇×#381	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₇	C₂²	1	2	yes	—
15	C₇×#124	—	2	4	no	—
16	C₄×C₇×D₄	—	2	4	no	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	1	yes	characteristic
31	C₇×#95	C₂²	1	1	yes	characteristic
32	C₇×#95	C₂²	1	2	yes	—
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	2	yes	—
35	C₇×#95	C₂²	1	2	yes	—
36	C₇×#95	C₂²	1	2	yes	—
37	#2447	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₇	C₂³	1	2	yes	—
39	C₂×C₇×Q₈	—	2	2	no	—
40	C₂×C₇×Q₈	—	2	2	no	—
41	C₂×C₇×Q₈	—	2	2	no	—
42	C₂×C₇×D₄	—	2	2	no	—
43	C₂×C₇×D₄	—	2	2	no	—
44	C₂×C₇×D₄	D₄	1	4	yes	—
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	2	4	no	—
47	C₂²×C₄×C₇	—	4	2	no	—
48	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	2	2	no	—
59	C₇×C₄⋊C₄	—	2	2	no	—
60	C₇×C₄⋊C₄	—	2	2	no	—
61	C₇×C₄⋊C₄	—	2	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₂²⋊C₄	—	2	4	no	—
64	C₇×C₂²⋊C₄	—	2	4	no	—
65	C₇×C₂²⋊C₄	—	2	4	no	—
66	C₇×C₂²⋊C₄	—	2	4	no	—
67	C₇×C₂²⋊C₄	—	4	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	—	2	2	no	—
70	C₄²×C₇	—	2	2	no	—
71	C₂×#124	C₂×C₇	1	1	yes	characteristic
72	C₂×C₄×D₄	C₂×C₇	1	1	yes	characteristic
73	#397	C₂×C₇	1	2	yes	complemented
74	#397	C₂×C₇	1	2	yes	complemented
75	#391	C₂×C₇	1	1	yes	characteristic, complemented
76	#391	C₂×C₇	1	1	yes	characteristic, complemented
77	#390	C₂×C₇	1	1	yes	characteristic
78	#389	C₂×C₇	1	1	yes	characteristic, complemented
79	#385	C₂×C₇	1	2	yes	complemented
80	#381	C₂×C₇	1	2	yes	complemented
81	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
82	C₂³×C₇	—	2	2	no	—
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	—	2	2	no	—
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	—	2	2	no	—
88	C₂³×C₇	C₂×D₄	1	4	yes	—
89	C₂³×C₇	C₂⁴	1	1	yes	characteristic
90	C₇×Q₈	—	4	4	no	—
91	C₇×D₄	—	2	8	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	4	8	no	—
118	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₇	#41	1	2	yes	—
120	C₂×C₄×C₇	#41	1	2	yes	—
121	C₂×C₄×C₇	#41	1	2	yes	—
122	C₂×C₄×C₇	#41	1	2	yes	—
123	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
124	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₇	1	2	yes	—
126	#124	—	2	4	no	—
127	C₄×D₄	—	2	4	no	—
128	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
138	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
139	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
141	#95	C₂²×C₇	1	1	yes	characteristic
142	#95	C₂²×C₇	1	1	yes	characteristic
143	#95	C₂²×C₇	1	2	yes	—
144	#95	C₂²×C₇	1	2	yes	—
145	#95	C₂²×C₇	1	2	yes	—
146	#95	C₂²×C₇	1	2	yes	—
147	#95	C₂²×C₇	1	2	yes	—
148	C₂²×C₇	—	2	4	no	—
149	C₂²×C₇	—	2	4	no	—
150	C₂²×C₇	—	2	4	no	—
151	C₂²×C₇	—	4	2	no	—
152	C₂²×C₇	—	4	2	no	—
153	C₂²×C₇	—	4	2	no	—
154	C₂²×C₇	—	4	2	no	—
155	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
156	C₂²×C₇	C₂×#41	1	1	yes	characteristic
157	C₂²×C₇	C₂×#41	1	1	yes	characteristic
158	C₂²×C₇	C₂×#41	1	1	yes	characteristic
159	C₂²×C₇	C₂×#41	1	1	yes	characteristic
160	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	4	2	no	—
165	C₄×C₇	—	4	1	no	—
166	C₄×C₇	—	4	1	no	—
167	C₄×C₇	—	4	2	no	—
168	C₄×C₇	—	4	1	no	—
169	C₄×C₇	—	4	1	no	—
170	C₄×C₇	—	4	2	no	—
171	C₄×C₇	#123	1	4	yes	—
172	C₂⁴	C₂³×C₇	1	2	yes	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₇×D₄	1	4	yes	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₇	1	2	yes	—
184	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₇	1	2	yes	—
187	C₂²×C₄	C₂³×C₇	1	2	yes	—
188	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₂×C₇	—	4	4	no	—
206	C₂×C₇	C₂×#123	1	1	yes	characteristic
207	C₂×C₇	#529	1	1	yes	characteristic
208	C₂×C₇	#542	1	1	yes	characteristic
209	C₂×C₇	#546	1	1	yes	characteristic
210	C₂×C₇	#547	1	1	yes	characteristic
211	C₂×C₇	#556	1	1	yes	characteristic
212	C₂×C₇	#559	1	1	yes	characteristic
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	C₂×C₇×D₄	1	4	yes	—
220	C₂³	C₂⁴×C₇	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	D₄	—	2	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
250	C₂×C₄	C₇×#41	1	2	yes	—
251	C₂×C₄	C₇×#41	1	2	yes	—
252	C₂×C₄	C₇×#41	1	2	yes	—
253	C₂×C₄	C₇×#41	1	2	yes	—
254	C₇	#2447	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
263	C₂²	C₂×C₇×#41	1	1	yes	characteristic
264	C₂²	C₂×C₇×#41	1	1	yes	characteristic
265	C₂²	C₂×C₇×#41	1	1	yes	characteristic
266	C₂²	C₂×C₇×#41	1	1	yes	characteristic
267	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	C₇×#123	1	4	yes	—
279	C₂	—	4	4	no	—
280	C₂	C₂×C₇×#123	1	1	yes	characteristic
281	C₂	C₇×#529	1	1	yes	characteristic
282	C₂	C₇×#542	1	1	yes	characteristic
283	C₂	C₇×#546	1	1	yes	characteristic
284	C₂	C₇×#547	1	1	yes	characteristic
285	C₂	C₇×#556	1	1	yes	characteristic
286	C₂	C₇×#559	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#397	C₂	semidirect
4. C₇×#397	C₂	semidirect
5. C₇×#391	C₂	semidirect
6. C₇×#391	C₂	semidirect
8. C₇×#389	C₂	semidirect
9. C₇×#385	C₂	semidirect
10. C₇×#381	C₂	semidirect
11. C₄×C₇×C₄⋊C₄	C₂	semidirect
37. #2447	C₇	direct (N × H)
73. #397	C₂×C₇	semidirect
74. #397	C₂×C₇	semidirect
75. #391	C₂×C₇	semidirect
76. #391	C₂×C₇	semidirect
78. #389	C₂×C₇	semidirect
79. #385	C₂×C₇	semidirect
80. #381	C₂×C₇	semidirect
81. C₄×C₄⋊C₄	C₂×C₇	semidirect
254. C₇	#2447	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×#124	2
C₂×C₄×C₇×D₄	2
C₇×#397	2
C₇×#391	2
C₇×#390	2
C₇×#389	2
C₇×#385	2
C₇×#381	2
C₄×C₇×C₄⋊C₄	2
#2447	7

Subgroups — C7×#2447 #11741756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2447 #11741756