Subgroups — C₇×#3248 #11742557

Order 896.

354 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#529	C₂	1	1	yes	maximal, characteristic
3	C₇×#479	C₂	1	2	yes	maximal, complemented
4	C₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#467	C₂	1	2	yes	maximal, complemented
7	C₇×#465	C₂	1	2	yes	maximal
8	C₇×#464	C₂	1	2	yes	maximal, complemented
9	C₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
14	C₂²×C₇×C₈	C₂²	1	2	yes	—
15	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#126	—	2	1	no	—
18	C₇×#125	—	2	1	no	—
19	C₇×#125	C₂²	1	1	yes	characteristic
20	C₇×#124	C₂²	1	1	yes	characteristic
21	C₇×#123	—	2	1	no	—
22	C₇×C₂²⋊Q₈	—	2	1	no	—
23	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₇×#121	C₂²	1	1	yes	characteristic
26	C₄×C₇×Q₈	—	2	1	no	—
27	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
28	C₄×C₇×Q₈	C₂²	1	2	yes	—
29	C₄×C₇×D₄	—	2	1	no	—
30	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
31	C₇×#117	—	2	1	no	—
32	C₇×#117	—	2	1	no	—
33	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
34	C₇×#107	C₂²	1	2	yes	—
35	C₇×#106	C₂²	1	2	yes	—
36	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₇×C₄⋊C₈	C₂²	1	2	yes	—
39	C₇×#103	C₂²	1	2	yes	—
40	C₇×#103	C₂²	1	2	yes	—
41	C₇×#103	C₂²	1	2	yes	—
42	C₇×#102	C₂²	1	2	yes	—
43	#3248	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×Q₈	—	2	1	no	—
45	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
46	C₂×C₇×Q₈	C₂³	1	2	yes	—
47	C₂×C₇×D₄	—	2	1	no	—
48	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	1	no	—
50	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₇×Q₁₆	—	4	2	no	—
54	C₇×SD₁₆	—	4	2	no	—
55	C₂×C₇×C₈	—	4	2	no	—
56	C₂×C₇×C₈	C₂³	1	2	yes	—
57	C₂×C₇×C₈	C₂³	1	2	yes	—
58	C₇×C₄⋊C₄	—	2	1	no	—
59	C₇×C₄⋊C₄	—	2	1	no	—
60	C₇×C₄⋊C₄	—	2	1	no	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	—	2	1	no	—
63	C₇×C₄⋊C₄	—	2	1	no	—
64	C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×C₄⋊C₄	—	2	2	no	—
66	C₇×C₄⋊C₄	—	2	1	no	—
67	C₇×C₄⋊C₄	C₂³	1	2	yes	—
68	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₇×C₂²⋊C₄	—	2	2	no	—
71	C₇×C₂²⋊C₄	—	2	1	no	—
72	C₇×C₂²⋊C₄	—	2	1	no	—
73	C₇×C₂²⋊C₄	—	2	1	no	—
74	C₇×C₂²⋊C₄	—	2	1	no	—
75	C₇×C₂²⋊C₄	—	2	1	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	D₄	1	1	yes	characteristic
79	C₄²×C₇	D₄	1	1	yes	characteristic
80	C₄²×C₇	C₂³	1	1	yes	characteristic
81	C₄²×C₇	C₂³	1	1	yes	characteristic
82	#533	C₂×C₇	1	1	yes	characteristic, complemented
83	#529	C₂×C₇	1	1	yes	characteristic
84	#479	C₂×C₇	1	2	yes	complemented
85	#475	C₂×C₇	1	1	yes	characteristic, complemented
86	#474	C₂×C₇	1	1	yes	characteristic, complemented
87	#467	C₂×C₇	1	2	yes	complemented
88	#465	C₂×C₇	1	2	yes	—
89	#464	C₂×C₇	1	2	yes	complemented
90	#462	C₂×C₇	1	1	yes	characteristic, complemented
91	#461	C₂×C₇	1	1	yes	characteristic, complemented
92	C₂×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
93	C₂³×C₇	—	2	1	no	—
94	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
95	C₇×Q₈	—	2	2	no	—
96	C₇×Q₈	—	2	1	no	—
97	C₇×Q₈	—	2	1	no	—
98	C₇×Q₈	—	2	2	no	—
99	C₇×Q₈	—	4	2	no	—
100	C₇×D₄	—	2	1	no	—
101	C₇×D₄	—	2	1	no	—
102	C₇×D₄	—	4	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	4	2	no	—
113	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
119	C₇×C₈	—	2	2	no	—
120	C₇×C₈	—	2	2	no	—
121	C₂×Q₁₆	C₂²×C₇	1	2	yes	—
122	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
123	C₂²×C₈	C₂²×C₇	1	2	yes	—
124	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
126	#126	—	2	1	no	—
127	#125	—	2	1	no	—
128	#125	C₂²×C₇	1	1	yes	characteristic
129	#124	C₂²×C₇	1	1	yes	characteristic
130	#123	—	2	1	no	—
131	C₂²⋊Q₈	—	2	1	no	—
132	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
133	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
134	#121	C₂²×C₇	1	1	yes	characteristic
135	C₄×Q₈	—	2	1	no	—
136	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
137	C₄×Q₈	C₂²×C₇	1	2	yes	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
140	#117	—	2	1	no	—
141	#117	—	2	1	no	—
142	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
143	#107	C₂²×C₇	1	2	yes	—
144	#106	C₂²×C₇	1	2	yes	—
145	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
147	C₄⋊C₈	C₂²×C₇	1	2	yes	—
148	#103	C₂²×C₇	1	2	yes	—
149	#103	C₂²×C₇	1	2	yes	—
150	#103	C₂²×C₇	1	2	yes	—
151	#102	C₂²×C₇	1	2	yes	—
152	C₂²×C₇	—	2	1	no	—
153	C₂²×C₇	—	2	1	no	—
154	C₂²×C₇	—	4	1	no	—
155	C₂²×C₇	—	4	1	no	—
156	C₂²×C₇	—	4	1	no	—
157	C₂²×C₇	#137	1	2	yes	—
158	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₇	—	2	1	no	—
160	C₄×C₇	—	2	1	no	—
161	C₄×C₇	—	2	1	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	4	1	no	—
164	C₄×C₇	—	4	2	no	—
165	C₄×C₇	—	4	1	no	—
166	C₄×C₇	—	4	2	no	—
167	C₄×C₇	—	4	1	no	—
168	C₄×C₇	#135	1	2	yes	—
169	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
170	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
173	C₂×Q₈	C₂³×C₇	1	2	yes	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
180	Q₁₆	—	4	2	no	—
181	SD₁₆	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₂³×C₇	1	2	yes	—
184	C₂×C₈	C₂³×C₇	1	2	yes	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	C₂³×C₇	1	2	yes	—
195	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	2	no	—
205	C₄²	C₇×D₄	1	1	yes	characteristic
206	C₄²	C₇×D₄	1	1	yes	characteristic
207	C₄²	C₂³×C₇	1	1	yes	characteristic
208	C₄²	C₂³×C₇	1	1	yes	characteristic
209	C₂×C₇	—	2	2	no	—
210	C₂×C₇	—	8	1	no	—
211	C₂×C₇	#537	1	1	yes	characteristic
212	C₂×C₇	C₂×#135	1	1	yes	characteristic
213	C₂×C₇	C₂×#137	1	1	yes	characteristic
214	C₂³	—	2	1	no	—
215	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	1	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	1	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₇	#3248	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₇×#137	1	2	yes	—
249	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	C₇×#135	1	2	yes	—
260	C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
261	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₇×#537	1	1	yes	characteristic
265	C₂	C₂×C₇×#135	1	1	yes	characteristic
266	C₂	C₂×C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#533	C₂	semidirect
3. C₇×#479	C₂	semidirect
4. C₇×#475	C₂	semidirect
5. C₇×#474	C₂	semidirect
6. C₇×#467	C₂	semidirect
8. C₇×#464	C₂	semidirect
9. C₇×#462	C₂	semidirect
10. C₇×#461	C₂	semidirect
11. C₂×C₇×C₄⋊C₈	C₂	semidirect
43. #3248	C₇	direct (N × H)
82. #533	C₂×C₇	semidirect
84. #479	C₂×C₇	semidirect
85. #475	C₂×C₇	semidirect
86. #474	C₂×C₇	semidirect
87. #467	C₂×C₇	semidirect
89. #464	C₂×C₇	semidirect
90. #462	C₂×C₇	semidirect
91. #461	C₂×C₇	semidirect
92. C₂×C₄⋊C₈	C₂×C₇	semidirect
242. C₇	#3248	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×#533	2
C₇×#529	2
C₇×#479	2
C₇×#475	2
C₇×#474	2
C₇×#467	2
C₇×#465	2
C₇×#464	2
C₇×#462	2
C₇×#461	2
C₂×C₇×C₄⋊C₈	2
#3248	7

Subgroups — C7×#3248 #11742557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3248 #11742557