Subgroups — #13272049

Order 1792.

494 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×#89361	C₂	1	4	yes	maximal, complemented
3	#11727520	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#135	—	2	1	no	—
6	C₂×C₇×#135	C₂²	1	2	yes	—
7	C₇×#532	—	2	1	no	—
8	C₇×#495	—	2	1	no	—
9	C₇×#495	C₄	1	2	yes	complemented
10	C₇×#494	C₄	1	1	yes	characteristic, complemented
11	C₇×#493	C₄	1	1	yes	characteristic, complemented
12	C₇×#492	—	2	1	no	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6392	—	2	1	no	—
15	C₂×#6391	C₂²	1	2	yes	—
16	C₂×#6386	C₂²	1	2	yes	—
17	#89708	—	2	1	no	—
18	#89361	—	2	8	no	—
19	#89256	—	2	1	no	—
20	C₄×C₇⋊C₁₆	—	2	1	no	—
21	#13211	—	7	1	no	maximal, Sylow
22	C₂×C₇×#41	—	2	2	no	—
23	C₇×#135	—	2	4	no	—
24	C₇×#135	—	2	4	no	—
25	C₇×#135	—	4	2	no	—
26	C₇×#135	—	4	2	no	—
27	C₂×C₇×Q₁₆	—	2	1	no	—
28	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₇×SD₁₆	—	2	2	no	—
30	C₂×C₇×SD₁₆	—	2	2	no	—
31	C₂×C₇×D₈	—	2	1	no	—
32	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₇×C₄⋊Q₈	—	2	1	no	—
36	C₇×#127	—	2	1	no	—
37	C₇×#124	—	2	2	no	—
38	C₇×#121	—	4	2	no	—
39	C₄×C₇×D₄	—	4	2	no	—
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₇	—	2	2	no	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₇⋊#95	—	4	2	no	—
47	#6392	—	4	2	no	—
48	#6391	—	2	4	no	—
49	#6386	—	2	2	no	—
50	#6386	D₄	1	4	yes	complemented
51	C₂×C₇⋊C₁₆	—	2	2	no	—
52	#3147	D₇	1	1	yes	characteristic
53	C₂×#360	—	7	4	no	—
54	C₄×M₃₂	—	7	1	no	—
55	#1846	—	7	1	no	—
56	C₇×#41	—	4	2	no	—
57	C₇×#41	—	4	2	no	—
58	C₇×#41	—	4	2	no	—
59	C₇×#41	—	4	2	no	—
60	C₂×C₇×Q₈	—	2	2	no	—
61	C₂×C₇×D₄	—	2	2	no	—
62	C₂×C₇×D₄	—	4	2	no	—
63	C₂×C₇×D₄	—	4	2	no	—
64	C₂²×C₄×C₇	—	4	2	no	—
65	C₂²×C₄×C₇	D₈	1	2	yes	—
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₇	—	4	2	no	—
68	C₇×Q₁₆	—	2	4	no	—
69	C₇×Q₁₆	—	4	2	no	—
70	C₇×SD₁₆	—	4	4	no	—
71	C₇×SD₁₆	—	4	4	no	—
72	C₇×D₈	—	2	4	no	—
73	C₇×D₈	—	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	2	yes	—
80	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₇×C₈	#41	1	2	yes	—
82	C₇×C₄⋊C₄	—	4	2	no	—
83	C₇×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₇	SD₁₆	1	2	yes	—
87	C₄⋊Dic₇	—	4	2	no	—
88	C₄⋊Dic₇	—	4	2	no	—
89	C₇⋊C₁₆	—	2	4	no	—
90	C₂×#135	—	2	1	no	—
91	C₂×#135	C₂×D₇	1	2	yes	—
92	#532	—	2	1	no	—
93	C₂×M₃₂	—	7	2	no	—
94	#495	—	2	1	no	—
95	#495	Dic₇	1	2	yes	complemented
96	#494	Dic₇	1	1	yes	characteristic, complemented
97	#493	Dic₇	1	1	yes	characteristic, complemented
98	#492	—	2	1	no	—
99	C₂×#107	—	14	1	no	—
100	C₂×#106	—	7	2	no	—
101	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
102	#382	—	14	1	no	—
103	#360	—	14	8	no	—
104	#346	—	14	1	no	—
105	C₄×C₁₆	—	14	1	no	—
106	C₂³×C₇	—	4	2	no	—
107	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
108	C₇×Q₈	—	4	2	no	—
109	C₇×Q₈	—	4	2	no	—
110	C₇×D₄	—	4	2	no	—
111	C₇×D₄	—	4	2	no	—
112	C₇×D₄	—	8	4	no	—
113	C₇×D₄	—	8	2	no	—
114	C₇×D₄	—	8	2	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	8	2	no	—
118	C₂×C₄×C₇	—	8	2	no	—
119	C₂×C₄×C₇	#102	1	2	yes	—
120	C₂×C₄×C₇	#102	1	2	yes	—
121	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₇	#127	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×Dic₇	—	8	4	no	—
128	C₂×Dic₇	—	8	2	no	—
129	C₇×C₈	#102	1	2	yes	—
130	C₇×C₈	#102	1	2	yes	—
131	C₇×C₈	C₄×D₄	1	2	yes	—
132	C₇×C₈	#121	1	2	yes	—
133	C₂×#41	—	2	2	no	—
134	#135	—	2	4	no	—
135	#135	—	2	4	no	—
136	#135	—	4	2	no	—
137	#135	—	4	2	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	C₂×Dic₇	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×Dic₇	1	2	yes	—
144	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	1	no	—
147	#127	—	2	1	no	—
148	#124	—	2	2	no	—
149	#121	—	4	2	no	—
150	C₄×D₄	—	4	2	no	—
151	C₂×C₄⋊C₄	—	14	2	no	—
152	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
153	M₃₂	—	7	4	no	—
154	M₃₂	—	14	2	no	—
155	C₂×C₁₆	—	14	2	no	—
156	#107	—	28	2	no	—
157	#106	—	14	4	no	—
158	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
159	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
162	#95	—	28	2	no	—
163	C₂²×C₇	—	2	2	no	—
164	C₂²×C₇	—	8	4	no	—
165	C₂²×C₇	—	8	2	no	—
166	C₂²×C₇	C₂×#102	1	1	yes	characteristic
167	C₂²×C₇	#486	1	1	yes	characteristic
168	C₂²×C₇	#493	1	1	yes	characteristic
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	8	2	no	—
172	C₄×C₇	#390	1	1	yes	characteristic
173	C₄×C₇	C₂×#102	1	1	yes	characteristic
174	C₄×C₇	#486	1	1	yes	characteristic
175	C₄×C₇	#492	1	1	yes	characteristic
176	Dic₇	—	16	2	no	—
177	#41	—	4	2	no	—
178	#41	—	4	2	no	—
179	#41	—	4	2	no	—
180	#41	—	4	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	28	2	no	—
187	C₂²×C₄	#1141	1	2	yes	—
188	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
189	Q₁₆	—	2	4	no	—
190	Q₁₆	—	4	2	no	—
191	SD₁₆	—	4	4	no	—
192	SD₁₆	—	4	4	no	—
193	D₈	—	2	4	no	—
194	D₈	—	4	2	no	—
195	C₂×C₈	#1141	1	2	yes	—
196	C₂×C₈	#1142	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
199	C₂×C₈	D₄×D₇	1	2	yes	—
200	C₂×C₈	#1159	1	2	yes	—
201	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	28	2	no	—
205	C₄⋊C₄	—	28	2	no	—
206	C₂²⋊C₄	—	4	4	no	—
207	C₄²	#1142	1	2	yes	—
208	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
209	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
210	C₁₆	—	14	4	no	—
211	C₂×C₇	—	2	1	no	—
212	C₂×C₇	—	2	1	no	—
213	C₂×C₇	—	16	2	no	—
214	C₂×C₇	#1957	1	1	yes	characteristic
215	C₂×C₇	#2145	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	C₂×#1141	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	4	no	—
223	D₄	—	8	2	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	56	4	no	—
230	C₂×C₄	—	56	2	no	—
231	C₂×C₄	#6406	1	2	yes	—
232	C₂×C₄	#6406	1	2	yes	—
233	C₂×C₄	C₂×#1141	1	1	yes	characteristic
234	C₂×C₄	C₂×#1142	1	1	yes	characteristic
235	C₂×C₄	C₂×#1142	1	1	yes	characteristic
236	C₂×C₄	#6499	1	1	yes	characteristic
237	C₂×C₄	#6503	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
239	C₈	#6406	1	2	yes	—
240	C₈	#6406	1	2	yes	—
241	C₈	D₄×Dic₇	1	2	yes	—
242	C₈	#6501	1	2	yes	—
243	C₇	#13211	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	#89842	1	1	yes	characteristic
248	C₂²	#89845	1	1	yes	characteristic
249	C₂²	C₂×#6406	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	112	2	no	—
254	C₄	#89842	1	1	yes	characteristic
255	C₄	#89848	1	1	yes	characteristic
256	C₄	C₂×#6406	1	1	yes	characteristic
257	C₄	#89993	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	2	no	—
261	C₂	#11729518	1	1	yes	characteristic
262	C₂	#11729830	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting, prime radical	C₇×#3147	C₂
socle	C₂²×C₇	C₂×#102
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#3147	D₇
hypocenter	C₇	#13211
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
2. C₂×#89361	C₂	semidirect
3. #11727520	C₂	semidirect
4. C₄×#6386	C₂	semidirect
9. C₇×#495	C₄	semidirect
10. C₇×#494	C₄	semidirect
11. C₇×#493	C₄	semidirect
50. #6386	D₄	semidirect
95. #495	Dic₇	semidirect
96. #494	Dic₇	semidirect
97. #493	Dic₇	semidirect
243. C₇	#13211	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
C₂×#89361	2
#11727520	2
C₄×#6386	2
#13211	7