Subgroups — C₇×#11926 #13189142

Order 1792.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	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	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇×SD₁₆	—	2	1	no	—
7	C₂²×C₇×D₈	C₄	1	1	yes	characteristic
8	C₂×C₇×#124	—	2	1	no	—
9	C₂×C₇×M₃₂	—	2	1	no	—
10	C₇×#495	—	2	1	no	—
11	C₇×#495	—	2	1	no	—
12	C₇×#495	C₄	1	2	yes	—
13	C₇×#495	C₂²	1	2	yes	—
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	complemented
18	C₇×#335	—	2	1	no	—
19	C₇×#334	C₂²	1	2	yes	complemented
20	#11926	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₂²×C₇×D₄	—	2	1	no	—
23	C₂×C₇×Q₁₆	—	2	1	no	—
24	C₂×C₇×Q₁₆	—	4	2	no	—
25	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₇×SD₁₆	—	2	2	no	—
27	C₂×C₇×SD₁₆	—	2	2	no	—
28	C₂×C₇×SD₁₆	—	4	2	no	—
29	C₂×C₇×SD₁₆	—	4	2	no	—
30	C₂×C₇×D₈	—	2	1	no	—
31	C₂×C₇×D₈	—	4	2	no	—
32	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₇×#124	—	2	2	no	—
36	C₇×#124	—	2	2	no	—
37	C₇×#124	—	4	1	no	—
38	C₇×#124	—	4	1	no	—
39	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
40	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₇×M₃₂	—	4	2	no	—
43	C₂×C₇×C₁₆	—	2	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	C₂×#495	C₂×C₇	1	1	yes	characteristic
52	#2150	C₂×C₇	1	1	yes	characteristic, complemented
53	#1564	C₂×C₇	1	1	yes	characteristic, complemented
54	#1360	C₂×C₇	1	4	yes	complemented
55	C₂⁴×C₇	—	4	1	no	—
56	C₂×C₇×Q₈	—	2	2	no	—
57	C₂×C₇×Q₈	—	4	1	no	—
58	C₂×C₇×Q₈	—	4	2	no	—
59	C₂×C₇×Q₈	—	4	2	no	—
60	C₂×C₇×D₄	—	2	2	no	—
61	C₂×C₇×D₄	—	4	2	no	—
62	C₂×C₇×D₄	—	4	2	no	—
63	C₂×C₇×D₄	—	4	1	no	—
64	C₂²×C₄×C₇	—	4	1	no	—
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
68	C₇×Q₁₆	—	4	2	no	—
69	C₇×Q₁₆	—	4	2	no	—
70	C₇×SD₁₆	—	4	4	no	—
71	C₇×SD₁₆	—	4	2	no	—
72	C₇×SD₁₆	—	4	2	no	—
73	C₇×D₈	—	4	2	no	—
74	C₇×D₈	—	4	2	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×C₂²⋊C₄	—	4	2	no	—
84	C₇×C₂²⋊C₄	—	4	2	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	C₂²×Q₁₆	C₄×C₇	1	1	yes	characteristic
93	C₂²×SD₁₆	—	2	1	no	—
94	C₂²×D₈	C₄×C₇	1	1	yes	characteristic
95	C₂×#124	—	2	1	no	—
96	C₂×M₃₂	—	2	1	no	—
97	#495	—	2	1	no	—
98	#495	—	2	1	no	—
99	#495	C₄×C₇	1	2	yes	—
100	#495	C₂²×C₇	1	2	yes	—
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	complemented
105	#335	—	2	1	no	—
106	#334	C₂²×C₇	1	2	yes	complemented
107	C₂³×C₇	—	4	2	no	—
108	C₂³×C₇	—	8	2	no	—
109	C₂³×C₇	—	8	2	no	—
110	C₂³×C₇	—	8	2	no	—
111	C₂³×C₇	#102	1	1	yes	characteristic
112	C₇×Q₈	—	4	2	no	—
113	C₇×Q₈	—	4	2	no	—
114	C₇×Q₈	—	8	2	no	—
115	C₇×D₄	—	4	2	no	—
116	C₇×D₄	—	4	2	no	—
117	C₇×D₄	—	8	2	no	—
118	C₂×C₄×C₇	—	4	1	no	—
119	C₂×C₄×C₇	—	4	1	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	—	8	2	no	—
122	C₂×C₄×C₇	#100	1	2	yes	—
123	C₂×C₄×C₇	#102	1	1	yes	characteristic
124	C₂×C₄×C₇	#102	1	1	yes	characteristic
125	C₂×C₄×C₇	#102	1	1	yes	characteristic
126	C₂×C₄×C₇	#102	1	2	yes	—
127	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₇×C₈	—	2	1	no	—
131	C₇×C₈	—	2	1	no	—
132	C₇×C₈	—	8	2	no	—
133	C₇×C₈	#102	1	2	yes	—
134	C₇×C₈	C₂×D₈	1	1	yes	characteristic
135	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
136	C₂²×Q₈	—	2	1	no	—
137	C₂²×D₄	—	2	1	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	—	4	2	no	—
140	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
141	C₂×SD₁₆	—	2	2	no	—
142	C₂×SD₁₆	—	2	2	no	—
143	C₂×SD₁₆	—	4	2	no	—
144	C₂×SD₁₆	—	4	2	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	—	4	2	no	—
147	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
148	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
150	#124	—	2	2	no	—
151	#124	—	2	2	no	—
152	#124	—	4	1	no	—
153	#124	—	4	1	no	—
154	C₂×C₂²⋊C₄	—	4	1	no	—
155	C₂×C₂²⋊C₄	—	4	1	no	—
156	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
157	M₃₂	—	4	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	4	1	no	—
161	C₄×C₈	—	4	1	no	—
162	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
163	C₄×C₈	C₇×D₄	1	1	yes	characteristic
164	C₄×C₈	C₇×D₄	1	1	yes	characteristic
165	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	8	2	no	—
169	C₂²×C₇	—	8	2	no	—
170	C₂²×C₇	—	8	2	no	—
171	C₂²×C₇	—	16	4	no	—
172	C₂²×C₇	#331	1	2	yes	—
173	C₂²×C₇	C₂×#102	1	1	yes	characteristic
174	C₄×C₇	—	2	1	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	—	2	1	no	—
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	—	8	2	no	—
179	C₄×C₇	#331	1	2	yes	—
180	C₄×C₇	C₂×#100	1	1	yes	characteristic
181	C₄×C₇	C₂×#102	1	1	yes	characteristic
182	C₂⁴	—	4	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
194	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	2	no	—
197	SD₁₆	—	4	4	no	—
198	SD₁₆	—	4	2	no	—
199	SD₁₆	—	4	2	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₇×D₈	1	2	yes	—
207	C₂×C₈	C₇×SD₁₆	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₂²⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
215	C₄²	C₇×D₈	1	1	yes	characteristic
216	C₄²	C₇×SD₁₆	1	1	yes	characteristic
217	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
218	C₁₆	—	4	2	no	—
219	C₂×C₇	—	2	2	no	—
220	C₂×C₇	—	16	2	no	—
221	C₂×C₇	C₂×#331	1	1	yes	characteristic
222	C₂×C₇	#2143	1	1	yes	characteristic
223	C₂×C₇	#2145	1	1	yes	characteristic
224	C₂³	—	4	2	no	—
225	C₂³	—	8	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	C₇×#102	1	1	yes	characteristic
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	8	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₇×#100	1	2	yes	—
240	C₂×C₄	C₇×#102	1	1	yes	characteristic
241	C₂×C₄	C₇×#102	1	1	yes	characteristic
242	C₂×C₄	C₇×#102	1	1	yes	characteristic
243	C₂×C₄	C₇×#102	1	2	yes	—
244	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	8	2	no	—
250	C₈	C₇×#102	1	2	yes	—
251	C₈	C₂×C₇×D₈	1	1	yes	characteristic
252	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
253	C₇	#11926	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	16	4	no	—
260	C₂²	C₇×#331	1	2	yes	—
261	C₂²	C₂×C₇×#102	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	2	no	—
267	C₄	C₇×#331	1	2	yes	—
268	C₄	C₂×C₇×#100	1	1	yes	characteristic
269	C₄	C₂×C₇×#102	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₂×C₇×#331	1	1	yes	characteristic
273	C₂	C₇×#2143	1	1	yes	characteristic
274	C₂	C₇×#2145	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	C₂²×C₇×D₈	C₄
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. C₇×#2150	C₂	semidirect
3. C₇×#1564	C₂	semidirect
4. C₇×#1360	C₂	semidirect
17. C₇×C₄⋊C₁₆	C₂²	semidirect
19. C₇×#334	C₂²	semidirect
20. #11926	C₇	direct (N × H)
52. #2150	C₂×C₇	semidirect
53. #1564	C₂×C₇	semidirect
54. #1360	C₂×C₇	semidirect
104. C₄⋊C₁₆	C₂²×C₇	semidirect
106. #334	C₂²×C₇	semidirect
253. C₇	#11926	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×#495	2
C₇×#2150	2
C₇×#1564	2
C₇×#1360	2
#11926	7

Subgroups — C7×#11926 #13189142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11926 #13189142