Subgroups — #13324658

Order 1792.

544 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	#11736711	C₂	1	2	yes	maximal
2	#11735920	C₂	1	1	yes	maximal, characteristic
3	C₇×#1452	C₂	1	1	yes	maximal, characteristic
4	#11728525	C₂	1	2	yes	maximal
5	#11727486	C₂	1	1	yes	maximal, characteristic
6	#11725841	C₂	1	4	yes	maximal, complemented
7	#11725840	C₂	1	4	yes	maximal, complemented
8	#90161	C₂²	1	2	yes	—
9	C₇×#431	C₂²	1	1	yes	characteristic
10	C₇×#431	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#89888	C₄	1	2	yes	—
13	#89888	C₄	1	2	yes	—
14	#89698	C₂²	1	2	yes	—
15	#89600	—	2	4	no	—
16	#89492	—	2	4	no	—
17	#89471	C₄	1	1	yes	characteristic
18	C₄×Dic₂₈	C₂²	1	2	yes	—
19	C₄×Dic₂₈	C₂²	1	2	yes	—
20	#89454	C₄	1	1	yes	characteristic
21	#89453	C₄	1	2	yes	—
22	#89451	C₂²	1	4	yes	—
23	C₈×Dic₁₄	C₂²	1	4	yes	—
24	C₇×#322	C₂²	1	2	yes	—
25	C₇×#322	C₂²	1	2	yes	—
26	#89253	C₂²	1	2	yes	—
27	#89252	C₂²	1	2	yes	—
28	#89251	C₂²	1	8	yes	—
29	#17991	—	7	1	no	maximal, Sylow
30	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	#6489	—	2	2	no	—
34	#6487	—	2	2	no	—
35	#6487	C₂×C₄	1	4	yes	—
36	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
38	#6433	—	2	2	no	—
39	#6432	C₂×C₄	1	2	yes	—
40	#6432	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₄	—	2	2	no	—
42	C₄×Dic₁₄	C₂³	1	4	yes	—
43	C₇×C₄⋊C₈	—	2	1	no	—
44	C₇×C₄⋊C₈	D₄	1	2	yes	—
45	C₇×C₂²⋊C₈	—	2	1	no	—
46	C₇×C₂²⋊C₈	D₄	1	2	yes	—
47	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂³	1	2	yes	—
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₇×C₈	C₂³	1	2	yes	—
53	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
54	#6392	C₂×C₄	1	2	yes	—
55	#6392	C₂×C₄	1	2	yes	—
56	#6390	C₂×C₄	1	4	yes	—
57	#6390	C₂×C₄	1	4	yes	—
58	#6389	—	2	4	no	—
59	Dic₇⋊C₈	—	2	4	no	—
60	Dic₇⋊C₈	—	2	4	no	—
61	C₈×Dic₇	—	2	4	no	—
62	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
63	#6377	—	2	2	no	—
64	#3170	—	7	1	no	—
65	#2985	—	7	2	no	—
66	#1587	—	7	4	no	—
67	#1584	—	7	4	no	—
68	#1568	—	7	1	no	—
69	#1497	—	7	2	no	—
70	#1452	D₇	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	#41	1	1	yes	characteristic
74	C₂×Dic₁₄	—	2	2	no	—
75	C₂×Dic₁₄	C₂²×C₄	1	4	yes	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	4	1	no	—
78	C₂×C₇×C₈	—	4	1	no	—
79	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
84	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇	#41	1	1	yes	characteristic
87	C₄²×C₇	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₇	—	2	4	no	—
89	C₄⋊Dic₇	—	2	2	no	—
90	C₄⋊Dic₇	—	2	2	no	—
91	C₄⋊Dic₇	—	2	2	no	—
92	C₄⋊Dic₇	C₂²×C₄	1	4	yes	—
93	Dic₇⋊C₄	—	2	4	no	—
94	Dic₇⋊C₄	—	2	4	no	—
95	C₄×Dic₇	—	2	4	no	—
96	C₂×C₇⋊C₈	—	2	4	no	—
97	Dic₂₈	—	4	2	no	—
98	Dic₂₈	—	4	2	no	—
99	#533	—	7	2	no	—
100	#500	—	7	1	no	—
101	#494	—	7	1	no	—
102	#487	—	7	2	no	—
103	#467	—	7	2	no	—
104	#467	—	7	2	no	—
105	#446	—	7	4	no	—
106	#446	—	14	4	no	—
107	C₈×Q₈	—	7	4	no	—
108	C₄×Q₁₆	—	7	2	no	—
109	C₄×Q₁₆	—	7	2	no	—
110	#433	—	14	4	no	—
111	#432	—	7	2	no	—
112	#431	C₂×D₇	1	1	yes	characteristic
113	#431	C₂×D₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
115	#335	—	7	2	no	—
116	#334	—	7	2	no	—
117	Q₈⋊C₈	—	7	8	no	—
118	#322	C₂×D₇	1	2	yes	—
119	#322	C₂×D₇	1	2	yes	—
120	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	4	1	no	—
125	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₇	—	2	4	no	—
132	C₂×Dic₇	—	2	4	no	—
133	Dic₁₄	—	2	4	no	—
134	Dic₁₄	—	2	4	no	—
135	Dic₁₄	—	8	2	no	—
136	C₇×C₈	—	2	1	no	—
137	C₇×C₈	—	2	1	no	—
138	C₇×C₈	—	2	1	no	—
139	C₇×C₈	—	2	1	no	—
140	C₇×C₈	—	2	4	no	—
141	C₇×C₈	—	4	2	no	—
142	C₇⋊C₈	—	4	4	no	—
143	C₂×Q₁₆	—	7	2	no	—
144	C₂×Q₁₆	—	7	2	no	—
145	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
146	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	7	2	no	—
148	C₄⋊Q₈	—	7	2	no	—
149	#125	—	14	2	no	—
150	C₂²⋊Q₈	—	7	4	no	—
151	C₂²⋊Q₈	—	14	2	no	—
152	C₄×Q₈	—	7	4	no	—
153	C₄×Q₈	—	14	2	no	—
154	#117	—	14	2	no	—
155	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
156	#107	—	7	2	no	—
157	#107	—	7	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	7	4	no	—
160	C₄⋊C₈	—	14	4	no	—
161	C₄⋊C₈	—	14	4	no	—
162	C₄⋊C₈	D₂₈	1	2	yes	—
163	#103	—	7	4	no	—
164	#103	—	7	4	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₂²⋊C₈	—	14	2	no	—
167	C₂²⋊C₈	D₂₈	1	2	yes	—
168	#97	—	14	4	no	—
169	#97	—	14	2	no	—
170	C₄×C₈	—	14	4	no	—
171	C₄×C₈	C₄×D₇	1	1	yes	characteristic
172	C₄×C₈	C₄×D₇	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₇	1	2	yes	—
174	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₇	1	2	yes	—
177	C₂²×C₇	—	2	1	no	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	1	no	—
185	C₄×C₇	#568	1	1	yes	characteristic
186	C₄×C₇	#568	1	1	yes	characteristic
187	C₄×C₇	#578	1	2	yes	—
188	Dic₇	—	4	4	no	—
189	Dic₇	—	4	4	no	—
190	C₂×Q₈	—	7	4	no	—
191	C₂×Q₈	—	14	2	no	—
192	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
194	C₂²×C₄	#1157	1	1	yes	characteristic
195	Q₁₆	—	28	2	no	—
196	Q₁₆	—	28	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	14	4	no	—
201	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
202	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
204	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
205	C₂×C₈	C₂×D₂₈	1	4	yes	—
206	C₄⋊C₄	—	7	4	no	—
207	C₄⋊C₄	—	14	4	no	—
208	C₄⋊C₄	—	14	2	no	—
209	C₄⋊C₄	—	14	2	no	—
210	C₄⋊C₄	—	14	2	no	—
211	C₄⋊C₄	—	14	4	no	—
212	C₂²⋊C₄	—	14	4	no	—
213	C₄²	—	14	4	no	—
214	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
215	C₄²	C₂×D₂₈	1	1	yes	characteristic
216	C₄²	#1157	1	1	yes	characteristic
217	C₄²	C₂³×D₇	1	1	yes	characteristic
218	C₂×C₇	—	4	1	no	—
219	C₂×C₇	#2932	1	1	yes	characteristic
220	C₂×C₇	#2950	1	1	yes	characteristic
221	C₂×C₇	#2957	1	1	yes	characteristic
222	C₂³	C₄×D₂₈	1	1	yes	characteristic
223	Q₈	—	14	4	no	—
224	Q₈	—	14	4	no	—
225	Q₈	—	56	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	14	4	no	—
231	C₂×C₄	—	14	4	no	—
232	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
233	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
234	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
237	C₂×C₄	C₂×#1157	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	4	no	—
243	C₈	—	4	2	no	—
244	C₈	—	28	4	no	—
245	C₇	#17991	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	28	4	no	—
255	C₄	—	28	4	no	—
256	C₄	#90436	1	1	yes	characteristic
257	C₄	#90436	1	1	yes	characteristic
258	C₄	#90444	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#11735936	1	1	yes	characteristic
261	C₂	#11736018	1	1	yes	characteristic
262	C₂	#11736025	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1452	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1452	D₇
hypocenter	C₇	#17991
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
6. #11725841	C₂	semidirect
7. #11725840	C₂	semidirect
245. C₇	#17991	semidirect

Maximal subgroups

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

subgroup	index
#11736711	2
#11735920	2
C₇×#1452	2
#11728525	2
#11727486	2
#11725841	2
#11725840	2
#17991	7