Subgroups — #13506363

Order 1792.

830 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	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	#11740501	C₂	1	2	yes	maximal
3	C₂×#89836	C₂	1	2	yes	maximal
4	C₂×#89455	C₂	1	1	yes	maximal, characteristic
5	#11728361	C₂	1	2	yes	maximal
6	#11728360	C₂	1	2	yes	maximal
7	#11728309	C₂	1	2	yes	maximal
8	#11728308	C₂	1	2	yes	maximal
9	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#125	C₂²	1	2	yes	—
11	C₂×#6433	C₂²	1	2	yes	—
12	C₇×#499	—	2	4	no	—
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×#6392	C₂²	1	2	yes	—
17	C₂×#6391	C₂²	1	2	yes	—
18	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
22	#89836	—	2	8	no	—
23	#89749	C₂²	1	4	yes	—
24	#89748	C₂²	1	4	yes	—
25	#89744	—	2	8	no	—
26	C₂×#6382	C₂²	1	4	yes	—
27	C₂×#6381	C₂²	1	4	yes	—
28	C₄²×Dic₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
30	#89455	—	2	4	no	—
31	#33311	—	7	1	no	maximal, Sylow
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₇×#125	—	2	4	no	—
35	C₇×#125	—	2	4	no	—
36	C₂×C₇×C₄⋊C₄	—	2	4	no	—
37	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₇	C₂³	1	4	yes	—
40	C₂×Dic₇⋊C₄	—	2	4	no	—
41	C₂×Dic₇⋊C₄	—	2	8	no	—
42	C₂×C₄×Dic₇	D₄	1	4	yes	—
43	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
45	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
47	#6433	—	2	4	no	—
48	#6433	—	2	4	no	—
49	C₇×#107	—	2	4	no	—
50	C₇×#106	—	2	4	no	—
51	C₄×C₇×C₈	D₄	1	4	yes	—
52	C₇⋊#95	—	2	8	no	—
53	#6392	—	2	4	no	—
54	#6391	—	2	4	no	—
55	C₈×Dic₇	Q₈	1	4	yes	—
56	C₈×Dic₇	Q₈	1	4	yes	—
57	C₈×Dic₇	Q₈	1	4	yes	—
58	C₈×Dic₇	Q₈	1	4	yes	—
59	#6382	—	2	8	no	—
60	#6381	—	2	8	no	—
61	C₄×C₇⋊C₈	D₄	1	4	yes	—
62	C₂×#499	—	7	1	no	—
63	C₂×#499	—	7	2	no	—
64	C₂×#499	D₇	1	1	yes	characteristic
65	#2859	—	7	2	no	—
66	#1846	—	7	2	no	—
67	#1846	—	7	2	no	—
68	#1845	—	7	2	no	—
69	#1844	—	7	2	no	—
70	C₄²×C₈	—	7	1	no	—
71	C₂²×C₄×C₇	—	2	4	no	—
72	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₇	—	2	4	no	—
75	C₂²×Dic₇	C₂×D₄	1	2	yes	—
76	C₂²×Dic₇	C₂×D₄	1	2	yes	—
77	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
79	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
81	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
82	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	2	4	no	—
85	C₇×C₄⋊C₄	—	4	8	no	—
86	C₄²×C₇	C₂×D₄	1	2	yes	—
87	C₄²×C₇	C₂×D₄	1	2	yes	—
88	C₄⋊Dic₇	—	2	4	no	—
89	C₄⋊Dic₇	—	2	4	no	—
90	Dic₇⋊C₄	—	4	8	no	—
91	Dic₇⋊C₄	—	4	16	no	—
92	C₄×Dic₇	—	2	8	no	—
93	C₄×Dic₇	C₂×Q₈	1	2	yes	—
94	C₄×Dic₇	C₂×Q₈	1	2	yes	—
95	C₄×Dic₇	C₂×Q₈	1	2	yes	—
96	C₄×Dic₇	C₂×Q₈	1	2	yes	—
97	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
98	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
101	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
102	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
103	C₂×#125	—	7	2	no	—
104	C₂×#125	C₂×D₇	1	2	yes	—
105	#499	—	2	4	no	—
106	#499	—	14	8	no	—
107	#499	—	14	4	no	—
108	C₂×#107	—	7	2	no	—
109	C₂×#107	—	7	4	no	—
110	C₂×#107	C₂×D₇	1	2	yes	—
111	C₂×#106	—	7	2	no	—
112	C₂×#106	—	7	4	no	—
113	C₂×#106	C₂×D₇	1	2	yes	—
114	C₂×C₄×C₈	—	7	1	no	—
115	C₂×C₄×C₈	—	7	1	no	—
116	C₂×C₄×C₈	—	7	1	no	—
117	C₂×C₄×C₈	—	7	1	no	—
118	C₂×C₄×C₈	—	7	1	no	—
119	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
120	#389	—	14	8	no	—
121	#384	—	7	4	no	—
122	#382	—	7	4	no	—
123	C₄³	—	7	1	no	—
124	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	4	4	no	—
127	C₂×C₄×C₇	—	4	8	no	—
128	C₂×C₄×C₇	#127	1	4	yes	—
129	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×Dic₇	—	2	4	no	—
136	C₂×Dic₇	—	4	4	no	—
137	C₂×Dic₇	—	4	8	no	—
138	C₂×Dic₇	C₄⋊Q₈	1	4	yes	—
139	C₂×Dic₇	C₄⋊Q₈	1	4	yes	—
140	C₂×Dic₇	#135	1	8	yes	—
141	C₇×C₈	C₄⋊Q₈	1	4	yes	—
142	C₇×C₈	C₄⋊Q₈	1	4	yes	—
143	C₇⋊C₈	C₄⋊Q₈	1	4	yes	—
144	C₇⋊C₈	C₄⋊Q₈	1	4	yes	—
145	C₂²×C₈	—	7	1	no	—
146	C₂²×C₈	—	7	1	no	—
147	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
149	#125	—	2	4	no	—
150	#125	—	2	4	no	—
151	#125	—	14	4	no	—
152	#125	—	14	4	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	7	4	no	—
155	C₂×C₄⋊C₄	—	14	8	no	—
156	C₂×C₄⋊C₄	—	14	4	no	—
157	C₂×C₄⋊C₄	C₂²×D₇	1	4	yes	—
158	C₂×C₄²	—	7	1	no	—
159	C₂×C₄²	—	7	1	no	—
160	C₂×C₄²	—	7	4	no	—
161	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
162	#107	—	2	4	no	—
163	#107	—	14	8	no	—
164	#107	—	14	4	no	—
165	#106	—	2	4	no	—
166	#106	—	14	8	no	—
167	#106	—	14	4	no	—
168	C₄×C₈	—	7	4	no	—
169	C₄×C₈	—	7	4	no	—
170	C₄×C₈	—	7	4	no	—
171	C₄×C₈	—	7	4	no	—
172	C₄×C₈	—	7	4	no	—
173	C₄×C₈	C₂²⋊D₇	1	4	yes	—
174	#95	—	14	8	no	—
175	C₂²×C₇	C₂×#127	1	1	yes	characteristic
176	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
178	C₂²×C₇	C₂×#135	1	4	yes	—
179	C₄×C₇	—	2	4	no	—
180	C₄×C₇	—	8	4	no	—
181	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
182	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
183	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
184	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
185	Dic₇	—	2	4	no	—
186	Dic₇	—	8	4	no	—
187	Dic₇	#499	1	8	yes	complemented
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₂²×C₄	—	14	4	no	—
192	C₂²×C₄	D₄×D₇	1	2	yes	—
193	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
194	C₂×C₈	—	7	2	no	—
195	C₂×C₈	—	7	2	no	—
196	C₂×C₈	—	7	2	no	—
197	C₂×C₈	—	7	2	no	—
198	C₂×C₈	—	7	2	no	—
199	C₂×C₈	—	7	2	no	—
200	C₂×C₈	Q₈×D₇	1	2	yes	—
201	C₂×C₈	Q₈×D₇	1	2	yes	—
202	C₂×C₈	Q₈×D₇	1	2	yes	—
203	C₂×C₈	Q₈×D₇	1	2	yes	—
204	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	8	no	—
209	C₄⋊C₄	—	14	4	no	—
210	C₄⋊C₄	—	14	4	no	—
211	C₄⋊C₄	—	28	16	no	—
212	C₄⋊C₄	—	28	8	no	—
213	C₄²	—	7	2	no	—
214	C₄²	—	7	2	no	—
215	C₄²	—	7	2	no	—
216	C₄²	—	7	2	no	—
217	C₄²	—	14	8	no	—
218	C₄²	D₄×D₇	1	2	yes	—
219	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
220	C₂×C₇	#2868	1	1	yes	characteristic
221	C₂×C₇	#3147	1	2	yes	—
222	C₂×C₇	C₂×#499	1	2	yes	—
223	C₂×C₇	#3160	1	2	yes	—
224	C₂³	C₂×D₄×D₇	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	7	4	no	—
229	C₂×C₄	—	7	8	no	—
230	C₂×C₄	—	7	4	no	—
231	C₂×C₄	—	14	4	no	—
232	C₂×C₄	—	28	8	no	—
233	C₂×C₄	—	28	4	no	—
234	C₂×C₄	#6503	1	4	yes	—
235	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
237	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
241	C₈	—	7	4	no	—
242	C₈	—	7	4	no	—
243	C₈	#6507	1	4	yes	—
244	C₈	#6507	1	4	yes	—
245	C₇	#33311	1	1	yes	characteristic, complemented, Sylow
246	C₂²	D₇×C₄⋊Q₈	1	1	yes	characteristic
247	C₂²	D₇×C₄⋊Q₈	1	1	yes	characteristic
248	C₂²	D₇×#135	1	4	yes	—
249	C₂²	C₂×#6503	1	1	yes	characteristic
250	C₄	—	2	4	no	—
251	C₄	—	7	8	no	—
252	C₄	—	8	4	no	—
253	C₄	—	14	4	no	—
254	C₄	—	56	4	no	—
255	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
256	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
257	C₄	C₂×#6507	1	1	yes	characteristic
258	C₄	C₂×#6507	1	1	yes	characteristic
259	C₂	D₇×#499	1	2	yes	—
260	C₂	#11739361	1	2	yes	—
261	C₂	#11740698	1	1	yes	characteristic
262	C₂	#11741348	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂×C₇×#499	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	C₂×#499	D₇
hypocenter	C₇	#33311
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
187. Dic₇	#499	semidirect
245. C₇	#33311	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#499	2
#11740501	2
C₂×#89836	2
C₂×#89455	2
#11728361	2
#11728360	2
#11728309	2
#11728308	2
C₄×C₈×Dic₇	2
#33311	7