Subgroups — C₇×#11981 #13189197

Order 1792.

358 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1364	C₂	1	2	yes	maximal, complemented
5	C₇×#1361	C₂	1	2	yes	maximal, complemented
6	C₇×#532	—	2	1	no	—
7	C₂²×C₇×C₁₆	—	2	1	no	—
8	C₇×#500	C₂²	1	1	yes	characteristic
9	C₇×#493	C₂²	1	1	yes	characteristic
10	C₇×#486	—	2	1	no	—
11	C₇×#466	—	2	1	no	—
12	C₇×#466	C₄	1	2	yes	—
13	C₄×C₇×D₈	—	2	1	no	—
14	C₄×C₇×D₈	C₄	1	2	yes	—
15	C₇×#423	—	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	—	2	1	no	—
20	C₇×#335	C₂²	1	2	yes	—
21	#11981	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	—	4	1	no	—
23	C₂×C₇×D₈	—	2	1	no	—
24	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
25	C₂×C₇×M₁₆	—	4	1	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	—	2	1	no	—
29	C₇×#127	—	2	1	no	—
30	C₇×#124	—	4	1	no	—
31	C₇×#121	—	2	2	no	—
32	C₇×#121	—	4	1	no	—
33	C₄×C₇×D₄	—	2	2	no	—
34	C₄×C₇×D₄	—	4	1	no	—
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₇×C₁₆	—	2	2	no	—
37	C₂×C₇×C₁₆	—	4	2	no	—
38	C₇×#107	—	2	1	no	—
39	C₇×#107	C₂×C₄	1	2	yes	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₄⋊C₈	—	4	1	no	—
42	C₇×#102	—	2	2	no	—
43	C₇×#102	—	2	2	no	—
44	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	1	yes	characteristic
47	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
48	#3169	C₂×C₇	1	1	yes	characteristic
49	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
50	#1567	C₂×C₇	1	1	yes	characteristic, complemented
51	#1364	C₂×C₇	1	2	yes	complemented
52	#1361	C₂×C₇	1	2	yes	complemented
53	C₇×#41	—	8	2	no	—
54	C₇×#41	—	8	2	no	—
55	C₂×C₇×Q₈	—	4	1	no	—
56	C₂×C₇×D₄	—	2	2	no	—
57	C₂×C₇×D₄	—	4	2	no	—
58	C₂×C₇×D₄	—	4	1	no	—
59	C₂×C₇×D₄	—	4	1	no	—
60	C₂²×C₄×C₇	—	4	2	no	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
64	C₇×D₈	—	4	2	no	—
65	C₇×D₈	—	4	1	no	—
66	C₇×D₈	—	4	1	no	—
67	C₇×M₁₆	—	8	2	no	—
68	C₂×C₇×C₈	—	2	1	no	—
69	C₂×C₇×C₈	—	2	1	no	—
70	C₂×C₇×C₈	—	4	1	no	—
71	C₂×C₇×C₈	—	4	2	no	—
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₇×C₄⋊C₄	—	2	2	no	—
77	C₇×C₄⋊C₄	—	4	1	no	—
78	C₇×C₂²⋊C₄	—	4	2	no	—
79	C₇×C₂²⋊C₄	—	4	2	no	—
80	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₇	D₈	1	1	yes	characteristic
82	C₄²×C₇	SD₁₆	1	1	yes	characteristic
83	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
84	C₇×C₁₆	—	4	2	no	—
85	#532	—	2	1	no	—
86	C₂²×C₁₆	—	2	1	no	—
87	#500	C₂²×C₇	1	1	yes	characteristic
88	#493	C₂²×C₇	1	1	yes	characteristic
89	#486	—	2	1	no	—
90	#466	—	2	1	no	—
91	#466	C₄×C₇	1	2	yes	—
92	C₄×D₈	—	2	1	no	—
93	C₄×D₈	C₄×C₇	1	2	yes	—
94	#423	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
96	C₄⋊C₁₆	—	2	1	no	—
97	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
98	#335	—	2	1	no	—
99	#335	C₂²×C₇	1	2	yes	—
100	C₂³×C₇	—	4	2	no	—
101	C₂³×C₇	#102	1	1	yes	characteristic
102	C₇×Q₈	—	8	1	no	—
103	C₇×Q₈	—	8	1	no	—
104	C₇×D₄	—	4	2	no	—
105	C₇×D₄	—	4	2	no	—
106	C₇×D₄	—	8	2	no	—
107	C₇×D₄	—	8	4	no	—
108	C₇×D₄	—	8	1	no	—
109	C₇×D₄	—	8	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	4	1	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	—	8	4	no	—
115	C₂×C₄×C₇	#102	1	1	yes	characteristic
116	C₂×C₄×C₇	#102	1	1	yes	characteristic
117	C₂×C₄×C₇	#102	1	1	yes	characteristic
118	C₂×C₄×C₇	D₁₆	1	2	yes	—
119	C₂×C₄×C₇	SD₃₂	1	2	yes	—
120	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₇×C₈	—	2	1	no	—
124	C₇×C₈	—	2	1	no	—
125	C₇×C₈	—	2	1	no	—
126	C₇×C₈	—	8	2	no	—
127	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₇×C₈	#137	1	1	yes	characteristic
129	C₂×#41	—	4	1	no	—
130	C₂×D₈	—	2	1	no	—
131	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
132	C₂×M₁₆	—	4	1	no	—
133	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
134	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	1	no	—
136	#127	—	2	1	no	—
137	#124	—	4	1	no	—
138	#121	—	2	2	no	—
139	#121	—	4	1	no	—
140	C₄×D₄	—	2	2	no	—
141	C₄×D₄	—	4	1	no	—
142	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	4	2	no	—
145	#107	—	2	1	no	—
146	#107	C₂×C₄×C₇	1	2	yes	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	4	1	no	—
149	#102	—	2	2	no	—
150	#102	—	2	2	no	—
151	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₄×C₈	C₇×D₄	1	1	yes	characteristic
153	C₄×C₈	C₇×D₄	1	1	yes	characteristic
154	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	8	2	no	—
158	C₂²×C₇	—	8	2	no	—
159	C₂²×C₇	—	8	2	no	—
160	C₂²×C₇	#357	1	2	yes	—
161	C₂²×C₇	C₂×#102	1	1	yes	characteristic
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	8	2	no	—
166	C₄×C₇	#357	1	2	yes	—
167	C₄×C₇	#413	1	1	yes	characteristic
168	C₄×C₇	#417	1	1	yes	characteristic
169	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
170	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
171	#41	—	8	2	no	—
172	#41	—	8	2	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
181	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
182	D₈	—	4	2	no	—
183	D₈	—	4	1	no	—
184	D₈	—	4	1	no	—
185	M₁₆	—	8	2	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
193	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
199	C₄²	C₇×D₈	1	1	yes	characteristic
200	C₄²	C₇×SD₁₆	1	1	yes	characteristic
201	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
202	C₁₆	—	4	2	no	—
203	C₂×C₇	—	2	2	no	—
204	C₂×C₇	—	16	2	no	—
205	C₂×C₇	#1547	1	1	yes	characteristic
206	C₂×C₇	C₂×#357	1	1	yes	characteristic
207	C₂×C₇	#2148	1	1	yes	characteristic
208	C₂³	—	4	2	no	—
209	C₂³	C₇×#102	1	1	yes	characteristic
210	Q₈	—	8	1	no	—
211	Q₈	—	8	1	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	1	no	—
217	D₄	—	8	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	C₇×#102	1	1	yes	characteristic
224	C₂×C₄	C₇×#102	1	1	yes	characteristic
225	C₂×C₄	C₇×#102	1	1	yes	characteristic
226	C₂×C₄	C₇×D₁₆	1	2	yes	—
227	C₂×C₄	C₇×SD₃₂	1	2	yes	—
228	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
230	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	8	2	no	—
235	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₈	C₇×#137	1	1	yes	characteristic
237	C₇	#11981	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₇×#357	1	2	yes	—
244	C₂²	C₂×C₇×#102	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	C₇×#357	1	2	yes	—
250	C₄	C₇×#413	1	1	yes	characteristic
251	C₄	C₇×#417	1	1	yes	characteristic
252	C₄	C₂×C₇×D₁₆	1	1	yes	characteristic
253	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	16	2	no	—
256	C₂	C₇×#1547	1	1	yes	characteristic
257	C₂	C₂×C₇×#357	1	1	yes	characteristic
258	C₂	C₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#1567	C₂	semidirect
4. C₇×#1364	C₂	semidirect
5. C₇×#1361	C₂	semidirect
21. #11981	C₇	direct (N × H)
49. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
50. #1567	C₂×C₇	semidirect
51. #1364	C₂×C₇	semidirect
52. #1361	C₂×C₇	semidirect
237. C₇	#11981	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×#3169	2
C₂×C₇×C₄⋊C₁₆	2
C₇×#1567	2
C₇×#1364	2
C₇×#1361	2
#11981	7

Subgroups — C7×#11981 #13189197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11981 #13189197