Subgroups — C₇×#11924 #13189140

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1360	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#135	—	2	1	no	—
6	C₂×C₇×#135	C₄	1	2	yes	—
7	C₇×#532	—	2	1	no	—
8	C₂²×C₇×C₁₆	—	2	1	no	—
9	C₇×#495	—	2	1	no	—
10	C₇×#495	C₂²	1	2	yes	—
11	C₇×#494	C₄	1	1	yes	characteristic
12	C₇×#493	C₄	1	1	yes	characteristic
13	C₇×#492	—	2	1	no	—
14	C₇×#431	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₁₆	—	2	1	no	—
17	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₇×#335	—	2	1	no	—
19	C₇×#334	C₂²	1	2	yes	—
20	#11924	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₇×#41	—	2	2	no	—
22	C₇×#135	—	4	4	no	—
23	C₇×#135	—	4	2	no	—
24	C₇×#135	—	4	2	no	—
25	C₂×C₇×Q₁₆	—	2	1	no	—
26	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₇×SD₁₆	—	2	2	no	—
28	C₂×C₇×SD₁₆	—	2	2	no	—
29	C₂×C₇×D₈	—	2	1	no	—
30	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₇×C₄⋊Q₈	—	2	1	no	—
34	C₇×#127	—	2	1	no	—
35	C₇×#124	—	2	2	no	—
36	C₇×#121	—	4	1	no	—
37	C₇×#121	—	4	1	no	—
38	C₄×C₇×D₄	—	4	1	no	—
39	C₄×C₇×D₄	—	4	1	no	—
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₇×C₁₆	—	2	2	no	—
42	C₂×C₇×C₁₆	—	4	2	no	—
43	C₇×C₄⋊C₈	—	2	2	no	—
44	C₇×C₂²⋊C₈	—	4	1	no	—
45	C₄×C₇×C₈	—	4	1	no	—
46	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	D₄	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
50	#3147	C₂×C₇	1	1	yes	characteristic
51	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
52	#1564	C₂×C₇	1	1	yes	characteristic, complemented
53	#1360	C₂×C₇	1	4	yes	complemented
54	C₇×#41	—	4	4	no	—
55	C₇×#41	—	4	4	no	—
56	C₂×C₇×Q₈	—	2	2	no	—
57	C₂×C₇×D₄	—	2	2	no	—
58	C₂×C₇×D₄	—	4	2	no	—
59	C₂×C₇×D₄	—	4	1	no	—
60	C₂×C₇×D₄	—	4	1	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
65	C₇×Q₁₆	—	4	2	no	—
66	C₇×Q₁₆	—	4	2	no	—
67	C₇×SD₁₆	—	4	4	no	—
68	C₇×SD₁₆	—	4	2	no	—
69	C₇×SD₁₆	—	4	2	no	—
70	C₇×D₈	—	4	2	no	—
71	C₇×D₈	—	4	2	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	4	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	D₈	1	2	yes	—
77	C₂×C₇×C₈	SD₁₆	1	2	yes	—
78	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
80	C₇×C₄⋊C₄	—	4	1	no	—
81	C₇×C₄⋊C₄	—	4	1	no	—
82	C₇×C₂²⋊C₄	—	4	2	no	—
83	C₇×C₂²⋊C₄	—	4	2	no	—
84	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₇	D₈	1	1	yes	characteristic
86	C₄²×C₇	SD₁₆	1	1	yes	characteristic
87	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
88	C₇×C₁₆	—	4	2	no	—
89	C₂×#135	—	2	1	no	—
90	C₂×#135	C₄×C₇	1	2	yes	—
91	#532	—	2	1	no	—
92	C₂²×C₁₆	—	2	1	no	—
93	#495	—	2	1	no	—
94	#495	C₂²×C₇	1	2	yes	—
95	#494	C₄×C₇	1	1	yes	characteristic
96	#493	C₄×C₇	1	1	yes	characteristic
97	#492	—	2	1	no	—
98	#431	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
100	C₄⋊C₁₆	—	2	1	no	—
101	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
102	#335	—	2	1	no	—
103	#334	C₂²×C₇	1	2	yes	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	#102	1	1	yes	characteristic
106	C₇×Q₈	—	4	2	no	—
107	C₇×Q₈	—	4	2	no	—
108	C₇×D₄	—	4	2	no	—
109	C₇×D₄	—	4	2	no	—
110	C₇×D₄	—	8	2	no	—
111	C₇×D₄	—	8	4	no	—
112	C₇×D₄	—	8	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	4	1	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	—	8	4	no	—
117	C₂×C₄×C₇	#100	1	2	yes	—
118	C₂×C₄×C₇	#102	1	1	yes	characteristic
119	C₂×C₄×C₇	#102	1	2	yes	—
120	C₂×C₄×C₇	#102	1	1	yes	characteristic
121	C₂×C₄×C₇	#102	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₇×C₈	—	2	1	no	—
126	C₇×C₈	—	2	1	no	—
127	C₇×C₈	—	8	2	no	—
128	C₇×C₈	#102	1	2	yes	—
129	C₇×C₈	C₂×D₈	1	1	yes	characteristic
130	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×#41	—	2	2	no	—
132	#135	—	4	4	no	—
133	#135	—	4	2	no	—
134	#135	—	4	2	no	—
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
137	C₂×SD₁₆	—	2	2	no	—
138	C₂×SD₁₆	—	2	2	no	—
139	C₂×D₈	—	2	1	no	—
140	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
141	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
142	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	1	no	—
144	#127	—	2	1	no	—
145	#124	—	2	2	no	—
146	#121	—	4	1	no	—
147	#121	—	4	1	no	—
148	C₄×D₄	—	4	1	no	—
149	C₄×D₄	—	4	1	no	—
150	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	4	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₂²⋊C₈	—	4	1	no	—
155	C₄×C₈	—	4	1	no	—
156	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
157	C₄×C₈	C₇×D₄	1	1	yes	characteristic
158	C₄×C₈	C₇×D₄	1	1	yes	characteristic
159	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	2	1	no	—
162	C₂²×C₇	—	8	2	no	—
163	C₂²×C₇	—	8	2	no	—
164	C₂²×C₇	—	8	2	no	—
165	C₂²×C₇	#331	1	2	yes	—
166	C₂²×C₇	C₂×#102	1	1	yes	characteristic
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	8	2	no	—
171	C₄×C₇	#331	1	2	yes	—
172	C₄×C₇	#359	1	2	yes	—
173	C₄×C₇	C₂×#100	1	1	yes	characteristic
174	C₄×C₇	C₂×#102	1	1	yes	characteristic
175	#41	—	4	4	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
185	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	Q₁₆	—	4	2	no	—
188	SD₁₆	—	4	4	no	—
189	SD₁₆	—	4	2	no	—
190	SD₁₆	—	4	2	no	—
191	D₈	—	4	2	no	—
192	D₈	—	4	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₇×D₈	1	2	yes	—
198	C₂×C₈	C₇×SD₁₆	1	2	yes	—
199	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
206	C₄²	C₇×D₈	1	1	yes	characteristic
207	C₄²	C₇×SD₁₆	1	1	yes	characteristic
208	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₂×C₇	—	2	2	no	—
211	C₂×C₇	—	16	2	no	—
212	C₂×C₇	C₂×#331	1	1	yes	characteristic
213	C₂×C₇	C₂×#359	1	1	yes	characteristic
214	C₂×C₇	#2145	1	1	yes	characteristic
215	C₂³	—	4	2	no	—
216	C₂³	C₇×#102	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	2	no	—
222	D₄	—	8	4	no	—
223	D₄	—	8	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	4	no	—
228	C₂×C₄	C₇×#100	1	2	yes	—
229	C₂×C₄	C₇×#102	1	1	yes	characteristic
230	C₂×C₄	C₇×#102	1	1	yes	characteristic
231	C₂×C₄	C₇×#102	1	2	yes	—
232	C₂×C₄	C₇×#102	1	1	yes	characteristic
233	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	C₇×#102	1	2	yes	—
240	C₈	C₂×C₇×D₈	1	1	yes	characteristic
241	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
242	C₇	#11924	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₇×#331	1	2	yes	—
249	C₂²	C₂×C₇×#102	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	C₇×#331	1	2	yes	—
255	C₄	C₇×#359	1	2	yes	—
256	C₄	C₂×C₇×#100	1	1	yes	characteristic
257	C₄	C₂×C₇×#102	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×C₇×#331	1	1	yes	characteristic
261	C₂	C₂×C₇×#359	1	1	yes	characteristic
262	C₂	C₇×#2145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#102
prime radical	C₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
3. C₇×#1564	C₂	semidirect
4. C₇×#1360	C₂	semidirect
20. #11924	C₇	direct (N × H)
51. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
52. #1564	C₂×C₇	semidirect
53. #1360	C₂×C₇	semidirect
242. C₇	#11924	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×#3147	2
C₂×C₇×C₄⋊C₁₆	2
C₇×#1564	2
C₇×#1360	2
#11924	7

Subgroups — C7×#11924 #13189140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11924 #13189140