Subgroups — #13268645

Order 1792.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#89873	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725617	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×#6489	C₄	1	1	yes	characteristic
7	C₂²×#1136	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	—	2	2	no	—
9	C₂×C₇×M₃₂	—	2	2	no	—
10	C₂×C₇×M₃₂	C₄	1	2	yes	complemented
11	C₂×C₇×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₇×C₁₆	C₄	1	2	yes	complemented
13	#89873	—	2	2	no	—
14	#89873	—	2	2	no	—
15	C₂×#6389	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇	C₄	1	2	yes	—
17	C₂×#6386	—	2	2	no	—
18	C₂×#6386	—	2	2	no	—
19	C₂×#6386	C₄	1	2	yes	complemented
20	C₂×#6386	C₂²	1	2	yes	—
21	C₂²×C₇⋊C₁₆	C₄	1	2	yes	complemented
22	#12547	—	7	1	no	maximal, Sylow
23	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	2	yes	—
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	2	yes	—
32	C₂²×C₇×C₈	Q₈	1	2	yes	—
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₇	—	2	1	no	—
35	#6489	—	2	2	no	—
36	#6489	C₈	1	4	yes	—
37	C₂×C₄⋊Dic₇	—	2	1	no	—
38	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
39	C₂×#1136	—	2	2	no	—
40	C₂×#1136	—	2	1	no	—
41	C₂×#1136	—	2	1	no	—
42	C₂×#1136	C₈	1	4	yes	—
43	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
44	C₇×M₃₂	—	2	4	no	—
45	C₇×M₃₂	—	2	2	no	—
46	C₇×M₃₂	—	2	2	no	—
47	C₂×C₇×C₁₆	—	2	4	no	—
48	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
49	#6389	—	2	2	no	—
50	#6389	—	2	2	no	—
51	C₈×Dic₇	—	2	4	no	—
52	#6386	—	2	4	no	—
53	#6386	—	2	2	no	—
54	#6386	—	2	2	no	—
55	C₂×C₇⋊C₁₆	—	2	4	no	—
56	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
57	C₂²×M₃₂	—	7	1	no	—
58	C₂²×M₃₂	D₇	1	1	yes	characteristic
59	C₂×#405	—	7	1	no	—
60	#1375	—	7	4	no	—
61	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	2	no	—
73	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
74	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
81	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
82	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₂²⋊Dic₇	—	2	4	no	—
92	C₄⋊Dic₇	—	2	4	no	—
93	C₄×Dic₇	—	2	2	no	—
94	C₄×Dic₇	C₂×C₈	1	4	yes	—
95	#1136	—	2	4	no	—
96	#1136	—	2	4	no	—
97	C₂×C₇⋊C₈	—	2	2	no	—
98	C₂×C₇⋊C₈	—	2	2	no	—
99	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
100	C₇×C₁₆	—	2	4	no	—
101	C₇⋊C₁₆	—	2	4	no	—
102	C₂²×M₁₆	—	7	1	no	—
103	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
104	C₂×#117	—	7	1	no	—
105	C₂×M₃₂	—	2	2	no	—
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	—	7	2	no	—
108	C₂×M₃₂	—	7	2	no	—
109	C₂×M₃₂	—	14	2	no	—
110	C₂×M₃₂	—	14	2	no	—
111	C₂×M₃₂	Dic₇	1	2	yes	complemented
112	C₂×M₃₂	C₂×D₇	1	2	yes	—
113	C₂²×C₁₆	—	7	2	no	—
114	C₂²×C₁₆	Dic₇	1	2	yes	complemented
115	#405	—	14	2	no	—
116	#405	—	14	2	no	—
117	C₂×#97	—	7	2	no	—
118	C₂×C₄×C₈	—	7	2	no	—
119	C₂³×C₇	—	4	2	no	—
120	C₂³×C₇	#97	1	1	yes	characteristic
121	C₂³×C₇	#97	1	1	yes	characteristic
122	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
123	C₂³×C₇	C₄⋊C₈	1	2	yes	—
124	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇	#95	1	2	yes	—
126	C₂×C₄×C₇	#95	1	2	yes	—
127	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
128	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
129	C₂×C₄×C₇	#97	1	1	yes	characteristic
130	C₂×C₄×C₇	#97	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
132	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
133	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
135	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
136	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
137	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
143	C₂×Dic₇	—	2	4	no	—
144	C₂×Dic₇	—	4	4	no	—
145	C₇×C₈	#95	1	2	yes	—
146	C₇×C₈	#95	1	2	yes	—
147	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₇⋊C₈	—	2	4	no	—
152	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
153	C₂×M₁₆	—	7	4	no	—
154	C₂×M₁₆	—	14	1	no	—
155	C₂×M₁₆	—	14	1	no	—
156	C₂×M₁₆	—	14	2	no	—
157	C₂²×C₈	—	7	2	no	—
158	C₂²×C₈	Dic₁₄	1	2	yes	—
159	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
160	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
162	C₂²×C₈	D₂₈	1	2	yes	—
163	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
167	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
168	#117	—	7	4	no	—
169	#117	—	14	2	no	—
170	C₂×C₄⋊C₄	—	14	1	no	—
171	C₂×C₂²⋊C₄	—	14	1	no	—
172	C₂×C₄²	—	7	2	no	—
173	M₃₂	—	2	4	no	—
174	M₃₂	—	2	2	no	—
175	M₃₂	—	2	2	no	—
176	M₃₂	—	14	4	no	—
177	M₃₂	—	14	2	no	—
178	M₃₂	—	14	2	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	7	4	no	—
181	C₂×C₁₆	—	14	4	no	—
182	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
183	#97	—	14	2	no	—
184	#97	—	14	2	no	—
185	C₄×C₈	—	14	4	no	—
186	C₂²×C₇	—	2	2	no	—
187	C₂²×C₇	—	2	2	no	—
188	C₂²×C₇	—	2	2	no	—
189	C₂²×C₇	—	4	2	no	—
190	C₂²×C₇	—	4	2	no	—
191	C₂²×C₇	#336	1	2	yes	—
192	C₂²×C₇	#336	1	2	yes	—
193	C₂²×C₇	C₂×#97	1	1	yes	characteristic
194	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
196	C₄×C₇	#336	1	2	yes	—
197	C₄×C₇	#336	1	2	yes	—
198	C₄×C₇	C₂×#95	1	1	yes	characteristic
199	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
200	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
201	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
202	Dic₇	—	4	4	no	—
203	C₂⁴	#1131	1	1	yes	characteristic
204	C₂²×C₄	—	14	2	no	—
205	C₂²×C₄	C₈×D₇	1	2	yes	—
206	C₂²×C₄	C₈×D₇	1	2	yes	—
207	C₂²×C₄	#1131	1	1	yes	characteristic
208	C₂²×C₄	#1131	1	1	yes	characteristic
209	C₂²×C₄	#1131	1	1	yes	characteristic
210	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
211	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
212	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
213	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
215	M₁₆	—	14	4	no	—
216	M₁₆	—	14	4	no	—
217	C₂×C₈	—	7	4	no	—
218	C₂×C₈	—	14	2	no	—
219	C₂×C₈	—	14	2	no	—
220	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
221	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
222	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
223	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
224	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
225	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
226	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
227	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
228	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
229	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
230	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
231	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
232	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
234	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
238	C₄⋊C₄	—	14	4	no	—
239	C₂²⋊C₄	—	14	4	no	—
240	C₄²	—	7	4	no	—
241	C₄²	—	14	2	no	—
242	C₁₆	—	2	4	no	—
243	C₁₆	—	14	4	no	—
244	C₂×C₇	—	2	1	no	—
245	C₂×C₇	—	2	2	no	—
246	C₂×C₇	—	2	2	no	—
247	C₂×C₇	—	2	1	no	—
248	C₂×C₇	C₂×#336	1	1	yes	characteristic
249	C₂×C₇	#2110	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	Dic₇⋊C₈	1	2	yes	—
252	C₂³	#6389	1	1	yes	characteristic
253	C₂³	#6389	1	1	yes	characteristic
254	C₂³	(C₂×D₇)⋊C₈	1	2	yes	—
255	C₂³	C₂×#1131	1	1	yes	characteristic
256	C₂×C₄	—	14	4	no	—
257	C₂×C₄	—	28	4	no	—
258	C₂×C₄	C₈×Dic₇	1	2	yes	—
259	C₂×C₄	C₈×Dic₇	1	2	yes	—
260	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
261	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
262	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
263	C₂×C₄	#6389	1	1	yes	characteristic
264	C₂×C₄	#6389	1	1	yes	characteristic
265	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
266	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
267	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
268	C₂×C₄	C₇⋊#95	1	2	yes	—
269	C₂×C₄	C₇⋊#95	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
272	C₂×C₄	C₂×#1131	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
274	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
276	C₈	—	14	4	no	—
277	C₈	C₇⋊#95	1	2	yes	—
278	C₈	C₇⋊#95	1	2	yes	—
279	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
280	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
281	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
282	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
283	C₇	#12547	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#89345	1	2	yes	—
290	C₂²	#89345	1	2	yes	—
291	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
292	C₂²	C₂×#6389	1	1	yes	characteristic
293	C₂²	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
294	C₄	—	28	4	no	—
295	C₄	#89345	1	2	yes	—
296	C₄	#89345	1	2	yes	—
297	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
298	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
299	C₄	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
300	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	#11729179	1	2	yes	—
306	C₂	C₂×#89345	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	#12547
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #11725617	C₂	semidirect
10. C₂×C₇×M₃₂	C₄	semidirect
12. C₂²×C₇×C₁₆	C₄	semidirect
19. C₂×#6386	C₄	semidirect
21. C₂²×C₇⋊C₁₆	C₄	semidirect
111. C₂×M₃₂	Dic₇	semidirect
114. C₂²×C₁₆	Dic₇	semidirect
283. C₇	#12547	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×M₃₂	2
C₂×#89873	2
C₂²×#6386	2
#11725617	2
#12547	7