Subgroups — #13367675

Order 1792.

846 conjugacy classes of subgroups.

The subgroup lattice has 286 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	#11740720	C₂	1	2	yes	maximal, complemented
3	#11729841	C₂	1	8	yes	maximal, complemented
4	#11729569	C₂	1	2	yes	maximal, complemented
5	#11727545	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#135	C₂²	1	4	yes	—
8	C₇×#532	C₂²	1	2	yes	—
9	C₂×#6511	C₂²	1	4	yes	—
10	C₇×#495	C₄	1	4	yes	complemented
11	C₇×#494	C₄	1	1	yes	characteristic, complemented
12	C₇×#493	C₄	1	1	yes	characteristic, complemented
13	C₇×#492	C₄	1	2	yes	complemented
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6411	C₂²	1	8	yes	—
16	#90006	C₂²	1	4	yes	—
17	#89784	—	2	2	no	—
18	#89784	—	2	2	no	—
19	D₄×C₇⋊C₈	—	2	2	no	—
20	D₄×C₇⋊C₈	—	2	2	no	—
21	C₄²×Dic₇	C₂²	1	1	yes	characteristic
22	#89696	C₂²	1	2	yes	—
23	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
24	#89347	C₂²	1	4	yes	—
25	#89345	C₂²	1	4	yes	—
26	#22280	—	7	1	no	maximal, Sylow
27	C₂×C₇×#41	C₂³	1	4	yes	—
28	C₇×#135	—	4	8	no	—
29	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
32	C₂²×C₇×C₈	C₂³	1	2	yes	—
33	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₇×#127	C₂×C₄	1	2	yes	—
35	C₇×#124	C₂×C₄	1	4	yes	—
36	C₇×#121	—	2	2	no	—
37	C₇×#121	—	2	2	no	—
38	C₄×C₇×D₄	—	2	2	no	—
39	C₄×C₇×D₄	—	2	2	no	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	#6511	—	4	8	no	—
42	C₂×C₄×Dic₇	—	2	1	no	—
43	C₂×C₄×Dic₇	—	2	1	no	—
44	C₂×C₄×Dic₇	C₂³	1	2	yes	—
45	C₂×#1136	—	2	4	no	—
46	C₂×#1136	C₂³	1	4	yes	—
47	C₂²×C₇⋊C₈	—	2	4	no	—
48	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
49	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
52	#6411	—	4	8	no	—
53	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
54	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
55	C₄⋊(C₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₇⋊C₈)	—	2	2	no	—
57	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₇⋊C₈	D₄	1	2	yes	—
59	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
60	#3147	D₇	1	1	yes	characteristic
61	#2931	—	7	2	no	—
62	#1962	—	7	2	no	—
63	#1876	—	7	8	no	—
64	#1837	—	7	1	no	—
65	#1744	—	7	1	no	—
66	C₇×#41	—	2	8	no	—
67	C₇×#41	—	2	8	no	—
68	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
69	C₂×C₇×D₄	—	2	4	no	—
70	C₂×C₇×D₄	—	2	2	no	—
71	C₂×C₇×D₄	—	2	2	no	—
72	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	—	2	4	no	—
74	C₂²×C₄×C₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₇	—	2	2	no	—
78	C₇×Q₁₆	—	4	4	no	—
79	C₇×SD₁₆	—	4	8	no	—
80	C₇×D₈	—	4	4	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₇×C₄⋊C₄	—	2	2	no	—
85	C₇×C₄⋊C₄	—	2	2	no	—
86	C₇×C₂²⋊C₄	—	2	4	no	—
87	C₇×C₂²⋊C₄	—	2	4	no	—
88	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
92	C₄×Dic₇	—	2	4	no	—
93	C₄×Dic₇	—	2	4	no	—
94	C₄×Dic₇	—	4	2	no	—
95	#1136	—	2	8	no	—
96	#1136	—	4	8	no	—
97	C₂×C₇⋊C₈	—	2	4	no	—
98	C₂×C₇⋊C₈	—	2	4	no	—
99	C₂×C₇⋊C₈	—	4	8	no	—
100	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
101	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×#135	C₂×D₇	1	4	yes	—
103	C₂×#131	—	7	4	no	—
104	#532	C₂×D₇	1	2	yes	—
105	#495	Dic₇	1	4	yes	complemented
106	#494	Dic₇	1	1	yes	characteristic, complemented
107	#493	Dic₇	1	1	yes	characteristic, complemented
108	#492	Dic₇	1	2	yes	complemented
109	#436	—	14	2	no	—
110	#436	—	14	2	no	—
111	C₈×D₄	—	14	2	no	—
112	C₈×D₄	—	14	2	no	—
113	#423	—	7	2	no	—
114	C₂×#104	—	7	8	no	—
115	#408	—	7	4	no	—
116	C₂×C₄×C₈	—	7	1	no	—
117	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
118	C₄³	—	7	1	no	—
119	#341	—	7	4	no	—
120	#336	—	7	4	no	—
121	C₂³×C₇	—	2	4	no	—
122	C₂³×C₇	C₂×#41	1	1	yes	characteristic
123	C₇×Q₈	—	2	8	no	—
124	C₇×D₄	—	2	8	no	—
125	C₇×D₄	—	4	8	no	—
126	C₇×D₄	—	4	4	no	—
127	C₇×D₄	—	4	4	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	4	4	no	—
133	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
140	C₂×Dic₇	—	2	4	no	—
141	C₂×Dic₇	—	4	2	no	—
142	C₂×Dic₇	—	4	2	no	—
143	C₇×C₈	—	2	2	no	—
144	C₇×C₈	—	2	2	no	—
145	C₇⋊C₈	—	2	2	no	—
146	C₇⋊C₈	—	2	2	no	—
147	C₇⋊C₈	—	4	4	no	—
148	C₂×#41	C₂²×D₇	1	4	yes	—
149	#135	—	4	8	no	—
150	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
151	C₂×SD₁₆	C₂×Dic₇	1	8	yes	—
152	C₂×D₈	C₂×Dic₇	1	4	yes	—
153	#131	—	28	8	no	—
154	C₂×M₁₆	—	7	4	no	—
155	C₂×M₁₆	—	14	4	no	—
156	C₂²×C₈	—	7	2	no	—
157	C₂²×C₈	—	14	4	no	—
158	C₂²×C₈	C₂²×D₇	1	2	yes	—
159	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
160	#127	C₂×Dic₇	1	2	yes	—
161	#124	C₂×Dic₇	1	4	yes	—
162	#121	—	2	2	no	—
163	#121	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₂×C₄²	—	7	2	no	—
167	C₂×C₄²	—	14	1	no	—
168	C₂×C₄²	—	14	1	no	—
169	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
170	C₄⋊C₈	—	14	2	no	—
171	C₄⋊C₈	—	14	2	no	—
172	#104	—	28	8	no	—
173	C₂²⋊C₈	—	14	4	no	—
174	C₂²⋊C₈	—	14	4	no	—
175	C₄×C₈	—	7	1	no	—
176	C₄×C₈	—	7	2	no	—
177	C₄×C₈	—	7	1	no	—
178	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₇	1	2	yes	—
181	C₂²×C₇	—	2	1	no	—
182	C₂²×C₇	—	2	1	no	—
183	C₂²×C₇	—	4	4	no	—
184	C₂²×C₇	—	4	4	no	—
185	C₂²×C₇	—	4	4	no	—
186	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₇	#540	1	2	yes	—
188	C₄×C₇	—	2	1	no	—
189	C₄×C₇	—	2	1	no	—
190	C₄×C₇	—	4	4	no	—
191	C₄×C₇	#443	1	2	yes	—
192	C₄×C₇	#444	1	2	yes	—
193	C₄×C₇	#518	1	2	yes	—
194	C₄×C₇	#535	1	2	yes	—
195	Dic₇	—	4	4	no	—
196	#41	—	2	8	no	—
197	#41	—	2	8	no	—
198	C₂×Q₈	C₂²×Dic₇	1	4	yes	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₂²×Dic₇	1	4	yes	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	14	2	no	—
205	C₂²×C₄	#1159	1	1	yes	characteristic
206	C₂²×C₄	#1159	1	1	yes	characteristic
207	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
208	Q₁₆	—	4	4	no	—
209	SD₁₆	—	4	8	no	—
210	D₈	—	4	4	no	—
211	M₁₆	—	14	8	no	—
212	M₁₆	—	28	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	7	2	no	—
215	C₂×C₈	—	7	2	no	—
216	C₂×C₈	—	14	4	no	—
217	C₂×C₈	—	14	4	no	—
218	C₂×C₈	—	28	8	no	—
219	C₂×C₈	C₂²×Dic₇	1	2	yes	—
220	C₂×C₈	C₂²×Dic₇	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₄²	—	14	4	no	—
226	C₄²	—	14	4	no	—
227	C₄²	—	28	2	no	—
228	C₄²	D₄×D₇	1	1	yes	characteristic
229	C₄²	D₄×D₇	1	1	yes	characteristic
230	C₄²	C₂²×Dic₇	1	1	yes	characteristic
231	C₄²	C₂²×Dic₇	1	1	yes	characteristic
232	C₂×C₇	—	2	2	no	—
233	C₂×C₇	—	8	4	no	—
234	C₂×C₇	#2328	1	1	yes	characteristic
235	C₂×C₇	C₂×#443	1	1	yes	characteristic
236	C₂×C₇	C₂×#444	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	C₂×#1159	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	D₄	—	2	8	no	—
241	D₄	—	4	8	no	—
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	14	4	no	—
250	C₂×C₄	—	28	2	no	—
251	C₂×C₄	—	28	2	no	—
252	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
253	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
257	C₂×C₄	#6548	1	4	yes	—
258	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	14	2	no	—
262	C₈	—	14	2	no	—
263	C₈	—	28	4	no	—
264	C₇	#22280	1	1	yes	characteristic, complemented, Sylow
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	#90410	1	2	yes	—
271	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	28	4	no	—
276	C₄	#89968	1	2	yes	—
277	C₄	#89969	1	2	yes	—
278	C₄	#90406	1	2	yes	—
279	C₄	#90489	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	8	4	no	—
282	C₂	#11740617	1	1	yes	characteristic
283	C₂	C₂×#89968	1	1	yes	characteristic
284	C₂	C₂×#89969	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting, prime radical	C₇×#3147	C₂
socle	C₂²×C₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	#3147	D₇
hypocenter	C₇	#22280
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
2. #11740720	C₂	semidirect
3. #11729841	C₂	semidirect
4. #11729569	C₂	semidirect
5. #11727545	C₂	semidirect
6. #11727504	C₂	semidirect
10. C₇×#495	C₄	semidirect
11. C₇×#494	C₄	semidirect
12. C₇×#493	C₄	semidirect
13. C₇×#492	C₄	semidirect
105. #495	Dic₇	semidirect
106. #494	Dic₇	semidirect
107. #493	Dic₇	semidirect
108. #492	Dic₇	semidirect
264. C₇	#22280	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3147	2
#11740720	2
#11729841	2
#11729569	2
#11727545	2
#11727504	2
#22280	7