Subgroups — C₇×#11929 #13189145

Order 1792.

414 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×#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×#1362	C₂	1	2	yes	maximal, complemented
5	C₇×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#135	—	2	1	no	—
7	C₂×C₇×#135	C₄	1	2	yes	—
8	C₇×#532	—	2	1	no	—
9	C₂²×C₇×C₁₆	—	2	1	no	—
10	C₇×#495	—	2	1	no	—
11	C₇×#495	C₄	1	2	yes	—
12	C₇×#494	C₂²	1	1	yes	characteristic
13	C₇×#493	C₂²	1	1	yes	characteristic
14	C₇×#492	—	2	1	no	—
15	C₇×#431	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×C₄⋊C₁₆	—	2	1	no	—
18	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₇×#335	C₂²	1	2	yes	—
20	C₇×#334	—	2	1	no	—
21	#11929	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	—	2	2	no	—
23	C₇×#135	—	4	4	no	—
24	C₇×#135	—	4	2	no	—
25	C₇×#135	—	4	2	no	—
26	C₂×C₇×Q₁₆	—	2	1	no	—
27	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₇×SD₁₆	—	2	2	no	—
29	C₂×C₇×SD₁₆	—	2	2	no	—
30	C₂×C₇×D₈	—	2	1	no	—
31	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
34	C₇×C₄⋊Q₈	—	2	1	no	—
35	C₇×#127	—	2	1	no	—
36	C₇×#124	—	2	2	no	—
37	C₇×#121	—	4	1	no	—
38	C₇×#121	—	4	1	no	—
39	C₄×C₇×D₄	—	4	1	no	—
40	C₄×C₇×D₄	—	4	1	no	—
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₂×C₇×C₁₆	—	2	2	no	—
43	C₂×C₇×C₁₆	—	4	2	no	—
44	C₇×C₄⋊C₈	—	2	2	no	—
45	C₇×C₂²⋊C₈	—	4	1	no	—
46	C₄×C₇×C₈	—	4	1	no	—
47	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×C₈	D₄	1	1	yes	characteristic
49	C₄×C₇×C₈	D₄	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	#3147	C₂×C₇	1	1	yes	characteristic
52	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
53	#1564	C₂×C₇	1	1	yes	characteristic, complemented
54	#1362	C₂×C₇	1	2	yes	complemented
55	#1361	C₂×C₇	1	2	yes	complemented
56	C₇×#41	—	4	4	no	—
57	C₇×#41	—	4	4	no	—
58	C₂×C₇×Q₈	—	2	2	no	—
59	C₂×C₇×D₄	—	2	2	no	—
60	C₂×C₇×D₄	—	4	2	no	—
61	C₂×C₇×D₄	—	4	1	no	—
62	C₂×C₇×D₄	—	4	1	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
67	C₇×Q₁₆	—	4	2	no	—
68	C₇×Q₁₆	—	4	1	no	—
69	C₇×Q₁₆	—	4	1	no	—
70	C₇×SD₁₆	—	4	4	no	—
71	C₇×SD₁₆	—	4	4	no	—
72	C₇×D₈	—	4	2	no	—
73	C₇×D₈	—	4	1	no	—
74	C₇×D₈	—	4	1	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	4	2	no	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	D₈	1	2	yes	—
80	C₂×C₇×C₈	SD₁₆	1	2	yes	—
81	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
83	C₇×C₄⋊C₄	—	4	1	no	—
84	C₇×C₄⋊C₄	—	4	1	no	—
85	C₇×C₂²⋊C₄	—	4	2	no	—
86	C₇×C₂²⋊C₄	—	4	2	no	—
87	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₇	D₈	1	1	yes	characteristic
89	C₄²×C₇	SD₁₆	1	1	yes	characteristic
90	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
91	C₇×C₁₆	—	4	2	no	—
92	C₂×#135	—	2	1	no	—
93	C₂×#135	C₄×C₇	1	2	yes	—
94	#532	—	2	1	no	—
95	C₂²×C₁₆	—	2	1	no	—
96	#495	—	2	1	no	—
97	#495	C₄×C₇	1	2	yes	—
98	#494	C₂²×C₇	1	1	yes	characteristic
99	#493	C₂²×C₇	1	1	yes	characteristic
100	#492	—	2	1	no	—
101	#431	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
103	C₄⋊C₁₆	—	2	1	no	—
104	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
105	#335	C₂²×C₇	1	2	yes	—
106	#334	—	2	1	no	—
107	C₂³×C₇	—	4	2	no	—
108	C₂³×C₇	#102	1	1	yes	characteristic
109	C₇×Q₈	—	4	2	no	—
110	C₇×Q₈	—	4	2	no	—
111	C₇×D₄	—	4	2	no	—
112	C₇×D₄	—	4	2	no	—
113	C₇×D₄	—	8	2	no	—
114	C₇×D₄	—	8	4	no	—
115	C₇×D₄	—	8	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	4	1	no	—
118	C₂×C₄×C₇	—	4	2	no	—
119	C₂×C₄×C₇	—	8	4	no	—
120	C₂×C₄×C₇	#100	1	2	yes	—
121	C₂×C₄×C₇	#102	1	1	yes	characteristic
122	C₂×C₄×C₇	#102	1	1	yes	characteristic
123	C₂×C₄×C₇	#102	1	2	yes	—
124	C₂×C₄×C₇	#102	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	2	1	no	—
130	C₇×C₈	—	8	2	no	—
131	C₇×C₈	#102	1	2	yes	—
132	C₇×C₈	C₂×D₈	1	1	yes	characteristic
133	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×#41	—	2	2	no	—
135	#135	—	4	4	no	—
136	#135	—	4	2	no	—
137	#135	—	4	2	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
140	C₂×SD₁₆	—	2	2	no	—
141	C₂×SD₁₆	—	2	2	no	—
142	C₂×D₈	—	2	1	no	—
143	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	1	no	—
147	#127	—	2	1	no	—
148	#124	—	2	2	no	—
149	#121	—	4	1	no	—
150	#121	—	4	1	no	—
151	C₄×D₄	—	4	1	no	—
152	C₄×D₄	—	4	1	no	—
153	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
160	C₄×C₈	C₇×D₄	1	1	yes	characteristic
161	C₄×C₈	C₇×D₄	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	8	2	no	—
166	C₂²×C₇	—	8	2	no	—
167	C₂²×C₇	—	8	2	no	—
168	C₂²×C₇	#331	1	2	yes	—
169	C₂²×C₇	C₂×#102	1	1	yes	characteristic
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	8	2	no	—
174	C₄×C₇	#331	1	2	yes	—
175	C₄×C₇	C₂×#100	1	1	yes	characteristic
176	C₄×C₇	C₂×#102	1	1	yes	characteristic
177	C₄×C₇	#508	1	2	yes	—
178	#41	—	4	4	no	—
179	#41	—	4	4	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
188	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
189	Q₁₆	—	4	2	no	—
190	Q₁₆	—	4	1	no	—
191	Q₁₆	—	4	1	no	—
192	SD₁₆	—	4	4	no	—
193	SD₁₆	—	4	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	1	no	—
196	D₈	—	4	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	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	C₇×D₈	1	2	yes	—
202	C₂×C₈	C₇×SD₁₆	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
204	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
210	C₄²	C₇×D₈	1	1	yes	characteristic
211	C₄²	C₇×SD₁₆	1	1	yes	characteristic
212	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
213	C₁₆	—	4	2	no	—
214	C₂×C₇	—	2	2	no	—
215	C₂×C₇	—	16	2	no	—
216	C₂×C₇	C₂×#331	1	1	yes	characteristic
217	C₂×C₇	#2138	1	1	yes	characteristic
218	C₂×C₇	#2148	1	1	yes	characteristic
219	C₂³	—	4	2	no	—
220	C₂³	C₇×#102	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	4	no	—
227	D₄	—	8	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	C₇×#100	1	2	yes	—
233	C₂×C₄	C₇×#102	1	1	yes	characteristic
234	C₂×C₄	C₇×#102	1	1	yes	characteristic
235	C₂×C₄	C₇×#102	1	1	yes	characteristic
236	C₂×C₄	C₇×#102	1	2	yes	—
237	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	8	2	no	—
243	C₈	C₇×#102	1	2	yes	—
244	C₈	C₂×C₇×D₈	1	1	yes	characteristic
245	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
246	C₇	#11929	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₇×#331	1	2	yes	—
253	C₂²	C₂×C₇×#102	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	C₇×#331	1	2	yes	—
259	C₄	C₂×C₇×#100	1	1	yes	characteristic
260	C₄	C₂×C₇×#102	1	1	yes	characteristic
261	C₄	C₇×#508	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	16	2	no	—
264	C₂	C₂×C₇×#331	1	1	yes	characteristic
265	C₂	C₇×#2138	1	1	yes	characteristic
266	C₂	C₇×#2148	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×#1362	C₂	semidirect
5. C₇×#1361	C₂	semidirect
21. #11929	C₇	direct (N × H)
52. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
53. #1564	C₂×C₇	semidirect
54. #1362	C₂×C₇	semidirect
55. #1361	C₂×C₇	semidirect
246. C₇	#11929	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×#1362	2
C₇×#1361	2
#11929	7

Subgroups — C7×#11929 #13189145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11929 #13189145