Subgroups — #13506366

Order 1792.

830 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	C₂×#89650	C₂	1	2	yes	maximal
2	C₂×#89630	C₂	1	2	yes	maximal
3	#11731665	C₂	1	2	yes	maximal
4	C₇×#1846	C₂	1	1	yes	maximal, characteristic
5	#11728360	C₂	1	2	yes	maximal
6	#11728308	C₂	1	2	yes	maximal
7	#11727520	C₂	1	1	yes	maximal, characteristic
8	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
9	#11727303	C₂	1	2	yes	maximal
10	C₂×#6453	C₂²	1	4	yes	—
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₇×#106	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×#382	C₂²	1	2	yes	—
15	C₂×#6392	C₂²	1	2	yes	—
16	C₂×#6391	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
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	#89748	C₂²	1	4	yes	—
22	C₂×#6382	C₂²	1	4	yes	—
23	C₂×#6381	C₂²	1	4	yes	—
24	#89708	C₂²	1	2	yes	—
25	C₄²×Dic₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
27	#89650	—	2	8	no	—
28	#89630	—	2	8	no	—
29	#89419	—	2	8	no	—
30	#33311	—	7	1	no	maximal, Sylow
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇	—	2	4	no	—
36	C₂×C₄⋊Dic₇	C₂³	1	4	yes	—
37	C₂×Dic₇⋊C₄	—	2	4	no	—
38	C₂×Dic₇⋊C₄	—	2	8	no	—
39	C₂×C₄×Dic₇	D₄	1	4	yes	—
40	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
42	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
44	#6453	—	2	8	no	—
45	#6453	—	2	8	no	—
46	C₇×#107	—	2	4	no	—
47	C₇×#106	—	2	4	no	—
48	C₄×C₇×C₈	Q₈	1	4	yes	—
49	C₇×#95	—	2	4	no	—
50	C₇⋊#95	—	2	4	no	—
51	#6392	—	2	4	no	—
52	#6391	—	2	4	no	—
53	C₈×Dic₇	D₄	1	4	yes	—
54	C₈×Dic₇	D₄	1	4	yes	—
55	C₈×Dic₇	Q₈	1	4	yes	—
56	C₈×Dic₇	Q₈	1	4	yes	—
57	#6382	—	2	8	no	—
58	#6381	—	2	8	no	—
59	C₄×C₇⋊C₈	Q₈	1	4	yes	—
60	C₂×#499	—	7	2	no	—
61	C₂×#499	—	7	2	no	—
62	#2859	—	7	2	no	—
63	#1846	—	7	1	no	—
64	#1846	—	7	2	no	—
65	#1846	D₇	1	1	yes	characteristic
66	#1845	—	7	2	no	—
67	#1844	—	7	2	no	—
68	C₄²×C₈	—	7	1	no	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	4	no	—
73	C₂²×Dic₇	C₂×D₄	1	2	yes	—
74	C₂²×Dic₇	C₂×D₄	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
78	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
79	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	—	2	4	no	—
83	C₄²×C₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₇	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₇	—	2	4	no	—
86	C₄⋊Dic₇	—	2	4	no	—
87	C₄⋊Dic₇	—	4	8	no	—
88	Dic₇⋊C₄	—	4	8	no	—
89	Dic₇⋊C₄	—	4	16	no	—
90	C₄×Dic₇	—	2	8	no	—
91	C₄×Dic₇	C₂×D₄	1	2	yes	—
92	C₄×Dic₇	C₂×D₄	1	2	yes	—
93	C₄×Dic₇	C₂×Q₈	1	2	yes	—
94	C₄×Dic₇	C₂×Q₈	1	2	yes	—
95	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
97	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
101	C₂×#125	—	7	4	no	—
102	#499	—	14	8	no	—
103	#499	—	14	8	no	—
104	C₂×#107	—	7	2	no	—
105	C₂×#107	—	7	4	no	—
106	C₂×#107	C₂×D₇	1	2	yes	—
107	C₂×#106	—	7	2	no	—
108	C₂×#106	—	7	4	no	—
109	C₂×#106	C₂×D₇	1	2	yes	—
110	C₂×C₄×C₈	—	7	1	no	—
111	C₂×C₄×C₈	—	7	1	no	—
112	C₂×C₄×C₈	—	7	1	no	—
113	C₂×C₄×C₈	—	7	1	no	—
114	C₂×C₄×C₈	—	7	1	no	—
115	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
116	#389	—	14	8	no	—
117	#384	—	7	4	no	—
118	#382	—	7	2	no	—
119	#382	C₂×D₇	1	2	yes	—
120	C₄³	—	7	1	no	—
121	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	4	4	no	—
124	C₂×C₄×C₇	—	4	8	no	—
125	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
126	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×Dic₇	—	2	4	no	—
133	C₂×Dic₇	—	4	4	no	—
134	C₂×Dic₇	—	4	8	no	—
135	C₂×Dic₇	#127	1	4	yes	—
136	C₂×Dic₇	C₄⋊Q₈	1	4	yes	—
137	C₂×Dic₇	#135	1	8	yes	—
138	C₇×C₈	C₄⋊Q₈	1	4	yes	—
139	C₇×C₈	C₄⋊Q₈	1	4	yes	—
140	C₇⋊C₈	C₄⋊Q₈	1	4	yes	—
141	C₇⋊C₈	C₄⋊Q₈	1	4	yes	—
142	C₂²×C₈	—	7	1	no	—
143	C₂²×C₈	—	7	1	no	—
144	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
146	#125	—	14	8	no	—
147	#125	—	14	8	no	—
148	C₂×C₄⋊C₄	—	7	4	no	—
149	C₂×C₄⋊C₄	—	14	4	no	—
150	C₂×C₄⋊C₄	—	14	4	no	—
151	C₂×C₄⋊C₄	—	14	8	no	—
152	C₂×C₄⋊C₄	C₂²×D₇	1	4	yes	—
153	C₂×C₄²	—	7	1	no	—
154	C₂×C₄²	—	7	1	no	—
155	C₂×C₄²	—	7	4	no	—
156	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
157	#107	—	2	4	no	—
158	#107	—	14	8	no	—
159	#107	—	14	4	no	—
160	#106	—	2	4	no	—
161	#106	—	14	8	no	—
162	#106	—	14	4	no	—
163	C₄×C₈	—	7	4	no	—
164	C₄×C₈	—	7	4	no	—
165	C₄×C₈	—	7	4	no	—
166	C₄×C₈	—	7	4	no	—
167	C₄×C₈	—	7	4	no	—
168	C₄×C₈	Dic₁₄	1	4	yes	—
169	#95	—	2	4	no	—
170	#95	—	14	4	no	—
171	C₂²×C₇	C₂×#127	1	1	yes	characteristic
172	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₂²×C₇	C₂×#135	1	4	yes	—
175	C₄×C₇	—	2	4	no	—
176	C₄×C₇	—	8	4	no	—
177	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
178	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
180	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	Dic₇	—	2	4	no	—
182	Dic₇	—	8	4	no	—
183	Dic₇	#499	1	8	yes	complemented
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	7	2	no	—
186	C₂²×C₄	—	7	2	no	—
187	C₂²×C₄	—	14	4	no	—
188	C₂²×C₄	D₄×D₇	1	2	yes	—
189	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
190	C₂×C₈	—	7	2	no	—
191	C₂×C₈	—	7	2	no	—
192	C₂×C₈	—	7	2	no	—
193	C₂×C₈	—	7	2	no	—
194	C₂×C₈	—	7	2	no	—
195	C₂×C₈	—	7	2	no	—
196	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
197	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
198	C₂×C₈	D₄×D₇	1	2	yes	—
199	C₂×C₈	D₄×D₇	1	2	yes	—
200	C₂×C₈	Q₈×D₇	1	2	yes	—
201	C₂×C₈	Q₈×D₇	1	2	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	14	4	no	—
205	C₄⋊C₄	—	14	4	no	—
206	C₄⋊C₄	—	28	16	no	—
207	C₄⋊C₄	—	28	8	no	—
208	C₄⋊C₄	—	28	8	no	—
209	C₄²	—	7	2	no	—
210	C₄²	—	7	2	no	—
211	C₄²	—	7	2	no	—
212	C₄²	—	7	2	no	—
213	C₄²	—	14	8	no	—
214	C₄²	C₂×Dic₁₄	1	2	yes	—
215	C₄²	Q₈×D₇	1	2	yes	—
216	C₂×C₇	#2868	1	1	yes	characteristic
217	C₂×C₇	#3147	1	2	yes	—
218	C₂×C₇	C₂×#499	1	2	yes	—
219	C₂×C₇	#3160	1	2	yes	—
220	C₂³	C₂×D₄×D₇	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	7	4	no	—
225	C₂×C₄	—	7	8	no	—
226	C₂×C₄	—	7	4	no	—
227	C₂×C₄	—	14	4	no	—
228	C₂×C₄	—	28	4	no	—
229	C₂×C₄	—	28	8	no	—
230	C₂×C₄	#6451	1	4	yes	—
231	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
234	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
237	C₈	—	7	4	no	—
238	C₈	—	7	4	no	—
239	C₈	#6451	1	4	yes	—
240	C₈	#6451	1	4	yes	—
241	C₇	#33311	1	1	yes	characteristic, complemented, Sylow
242	C₂²	C₂×#6451	1	1	yes	characteristic
243	C₂²	D₇×#127	1	1	yes	characteristic
244	C₂²	D₇×C₄⋊Q₈	1	1	yes	characteristic
245	C₂²	D₇×#135	1	4	yes	—
246	C₄	—	2	4	no	—
247	C₄	—	7	8	no	—
248	C₄	—	8	4	no	—
249	C₄	—	14	4	no	—
250	C₄	—	56	4	no	—
251	C₄	C₂×#6451	1	1	yes	characteristic
252	C₄	C₂×#6451	1	1	yes	characteristic
253	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
254	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
255	C₂	#11731801	1	1	yes	characteristic
256	C₂	#11736615	1	2	yes	—
257	C₂	#11739265	1	2	yes	—
258	C₂	D₇×#499	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×#1846	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	#1846	D₇
hypocenter	C₇	#33311
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
183. Dic₇	#499	semidirect
241. 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×#89650	2
C₂×#89630	2
#11731665	2
C₇×#1846	2
#11728360	2
#11728308	2
#11727520	2
C₄×C₈×Dic₇	2
#11727303	2
#33311	7