Subgroups — #13243271

Order 1792.

674 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×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89697	C₂	1	1	yes	maximal, characteristic
3	C₂×#89696	C₂	1	1	yes	maximal, characteristic
4	#11725509	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂²×#1136	—	2	1	no	—
8	C₂×C₇×C₄⋊C₈	—	2	1	no	—
9	C₂×C₇×C₄⋊C₈	—	2	2	no	—
10	C₂×C₇×C₄⋊C₈	C₄	1	4	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₇⋊C₈)	C₄	1	2	yes	—
13	#89697	—	2	2	no	—
14	#89697	—	2	2	no	—
15	#89696	—	2	2	no	—
16	#89696	C₄	1	4	yes	—
17	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₇⋊C₈)	C₄	1	2	yes	—
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
21	#7877	—	7	1	no	maximal, Sylow
22	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
25	C₂²×C₇×C₈	—	2	4	no	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	—	4	2	no	—
28	C₂²×C₇×C₈	—	4	2	no	—
29	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	2	yes	—
35	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₂×#1136	—	2	4	no	—
38	C₂×#1136	—	4	2	no	—
39	C₂×#1136	—	4	2	no	—
40	C₂²×C₇⋊C₈	—	2	2	no	—
41	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₇×C₄⋊C₈	—	2	4	no	—
43	C₇×C₄⋊C₈	—	2	2	no	—
44	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
46	C₄⋊(C₇⋊C₈)	—	2	4	no	—
47	C₄⋊(C₇⋊C₈)	—	2	4	no	—
48	C₄⋊(C₇⋊C₈)	—	2	2	no	—
49	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
50	C₄×C₇⋊C₈	—	2	2	no	—
51	C₄×C₇⋊C₈	—	2	2	no	—
52	C₂×#431	—	7	1	no	—
53	C₂×#423	—	7	1	no	—
54	C₂²×C₄⋊C₈	D₇	1	1	yes	characteristic
55	#1277	—	7	4	no	—
56	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	D₈	1	2	yes	—
69	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₇×C₈	—	2	4	no	—
76	C₂×C₇×C₈	—	4	8	no	—
77	C₂×C₇×C₈	—	4	2	no	—
78	C₂×C₇×C₈	—	4	2	no	—
79	C₄²×C₇	C₄²	1	2	yes	—
80	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
89	#1136	—	4	8	no	—
90	C₂×C₇⋊C₈	—	2	4	no	—
91	C₂×C₇⋊C₈	—	2	4	no	—
92	C₂×C₇⋊C₈	—	2	8	no	—
93	C₂×C₇⋊C₈	—	4	2	no	—
94	C₂×C₇⋊C₈	—	4	2	no	—
95	C₂²×M₁₆	—	14	1	no	—
96	C₂³×C₈	—	2	1	no	—
97	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
98	#431	—	14	2	no	—
99	#431	—	14	2	no	—
100	#423	—	7	4	no	—
101	#423	—	14	2	no	—
102	C₂×C₄⋊C₈	—	2	1	no	—
103	C₂×C₄⋊C₈	—	2	2	no	—
104	C₂×C₄⋊C₈	—	7	2	no	—
105	C₂×C₄⋊C₈	—	7	2	no	—
106	C₂×C₄⋊C₈	—	14	1	no	—
107	C₂×C₄⋊C₈	Dic₇	1	4	yes	—
108	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
109	C₂×C₂²⋊C₈	—	7	2	no	—
110	C₂×C₄×C₈	—	7	2	no	—
111	C₂³×C₇	—	2	2	no	—
112	C₂³×C₇	—	2	2	no	—
113	C₂³×C₇	#95	1	2	yes	—
114	C₂³×C₇	#106	1	2	yes	—
115	C₂³×C₇	#107	1	2	yes	—
116	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	4	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	4	2	no	—
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	#95	1	2	yes	—
125	C₂×C₄×C₇	#95	1	2	yes	—
126	C₂×C₄×C₇	#95	1	2	yes	—
127	C₂×C₄×C₇	#102	1	4	yes	—
128	C₂×C₄×C₇	#102	1	4	yes	—
129	C₂×C₄×C₇	#103	1	4	yes	—
130	C₂×C₄×C₇	#103	1	4	yes	—
131	C₂×C₄×C₇	#106	1	2	yes	—
132	C₂×C₄×C₇	#107	1	2	yes	—
133	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	#131	1	4	yes	—
140	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
144	C₇×C₈	—	4	4	no	—
145	C₇⋊C₈	—	4	4	no	—
146	C₇⋊C₈	—	4	4	no	—
147	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
148	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
149	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
150	C₂×M₁₆	—	14	4	no	—
151	C₂×M₁₆	—	28	2	no	—
152	C₂×M₁₆	—	28	2	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	2	no	—
156	C₂²×C₈	—	4	2	no	—
157	C₂²×C₈	—	7	4	no	—
158	C₂²×C₈	—	14	2	no	—
159	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₇	1	2	yes	—
162	C₂×C₄²	D₂₈	1	2	yes	—
163	C₂×C₄²	C₂×Dic₇	1	2	yes	—
164	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
165	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	7	4	no	—
170	C₄⋊C₈	—	14	4	no	—
171	C₄⋊C₈	—	14	4	no	—
172	C₄⋊C₈	—	14	2	no	—
173	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
174	C₂²⋊C₈	—	14	4	no	—
175	C₄×C₈	—	14	2	no	—
176	C₄×C₈	—	14	2	no	—
177	C₂²×C₇	—	2	4	no	—
178	C₂²×C₇	—	2	4	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	#340	1	4	yes	—
182	C₂²×C₇	C₂×#95	1	1	yes	characteristic
183	C₂²×C₇	#424	1	2	yes	—
184	C₂²×C₇	C₂×#106	1	1	yes	characteristic
185	C₂²×C₇	C₂×#107	1	1	yes	characteristic
186	C₂²×C₇	#430	1	2	yes	—
187	C₄×C₇	—	2	2	no	—
188	C₄×C₇	—	2	2	no	—
189	C₄×C₇	—	2	1	no	—
190	C₄×C₇	—	2	1	no	—
191	C₄×C₇	#340	1	4	yes	—
192	C₄×C₇	#405	1	2	yes	—
193	C₄×C₇	#408	1	2	yes	—
194	C₄×C₇	C₂×#102	1	1	yes	characteristic
195	C₄×C₇	C₂×#102	1	1	yes	characteristic
196	C₄×C₇	C₂×#103	1	1	yes	characteristic
197	C₄×C₇	C₂×#103	1	1	yes	characteristic
198	C₂⁴	Dic₇⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄×Dic₇	1	2	yes	—
202	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
205	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
207	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
208	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
209	C₂²×C₄	#1141	1	2	yes	—
210	C₂²×C₄	#1142	1	2	yes	—
211	C₂²×C₄	#1143	1	2	yes	—
212	C₂²×C₄	#1144	1	2	yes	—
213	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
214	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
217	M₁₆	—	28	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	8	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	14	8	no	—
223	C₂×C₈	—	14	4	no	—
224	C₂×C₈	—	14	4	no	—
225	C₂×C₈	—	28	2	no	—
226	C₂×C₈	—	28	2	no	—
227	C₄²	C₄×Dic₇	1	2	yes	—
228	C₄²	Dic₇⋊C₄	1	2	yes	—
229	C₄²	C₄⋊Dic₇	1	2	yes	—
230	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
231	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
232	C₄²	C₂²⋊Dic₇	1	2	yes	—
233	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
234	C₄²	C₂×D₂₈	1	1	yes	characteristic
235	C₄²	C₂²×Dic₇	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
237	C₂×C₇	—	2	4	no	—
238	C₂×C₇	#1570	1	2	yes	—
239	C₂×C₇	#1571	1	2	yes	—
240	C₂×C₇	#1729	1	1	yes	characteristic
241	C₂×C₇	C₂×#340	1	1	yes	characteristic
242	C₂×C₇	#1738	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	#6381	1	2	yes	—
246	C₂³	#6382	1	2	yes	—
247	C₂³	C₇⋊#95	1	2	yes	—
248	C₂³	C₂×Dic₇⋊C₄	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	#6381	1	2	yes	—
257	C₂×C₄	#6382	1	2	yes	—
258	C₂×C₄	#6383	1	4	yes	—
259	C₂×C₄	#6384	1	4	yes	—
260	C₂×C₄	C₇⋊#95	1	2	yes	—
261	C₂×C₄	C₇⋊#95	1	2	yes	—
262	C₂×C₄	C₇⋊#95	1	2	yes	—
263	C₂×C₄	#6406	1	4	yes	—
264	C₂×C₄	#6409	1	4	yes	—
265	C₂×C₄	#6464	1	4	yes	—
266	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
269	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×#1141	1	1	yes	characteristic
272	C₂×C₄	C₂×#1142	1	1	yes	characteristic
273	C₂×C₄	C₂×#1143	1	1	yes	characteristic
274	C₂×C₄	C₂×#1144	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	28	4	no	—
278	C₈	—	28	4	no	—
279	C₇	#7877	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#89324	1	4	yes	—
285	C₂²	C₂×#6381	1	1	yes	characteristic
286	C₂²	C₂×#6382	1	1	yes	characteristic
287	C₂²	#89874	1	2	yes	—
288	C₂²	#89896	1	2	yes	—
289	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	#89324	1	4	yes	—
295	C₄	C₂×#6383	1	1	yes	characteristic
296	C₄	C₂×#6384	1	1	yes	characteristic
297	C₄	#89873	1	2	yes	—
298	C₄	#89882	1	2	yes	—
299	C₄	C₂×#6406	1	1	yes	characteristic
300	C₄	C₂×#6409	1	1	yes	characteristic
301	C₂	—	2	4	no	—
302	C₂	#11728175	1	2	yes	—
303	C₂	#11728176	1	2	yes	—
304	C₂	C₂×#89324	1	1	yes	characteristic
305	C₂	#11729719	1	1	yes	characteristic
306	C₂	#11729748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂²×C₇×C₄⋊C₈	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂²×C₄⋊C₈	D₇
hypocenter	C₇	#7877
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #11725509	C₂	semidirect
279. C₇	#7877	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₄⋊C₈	2
C₂×#89697	2
C₂×#89696	2
#11725509	2
#7877	7