Subgroups — #13324457

Order 1792.

640 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	#11736735	C₂	1	2	yes	maximal, complemented
2	#11735915	C₂	1	1	yes	maximal, characteristic
3	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728519	C₂	1	2	yes	maximal, complemented
5	#11727486	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725837	C₂	1	2	yes	maximal, complemented
7	#11725831	C₂	1	2	yes	maximal, complemented
8	#11725806	C₂	1	2	yes	maximal, complemented
9	C₈×D₅₆	C₂	1	2	yes	maximal, complemented
10	#90168	C₂²	1	2	yes	—
11	C₇×#431	C₂²	1	1	yes	characteristic
12	C₇×#431	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	#89887	C₄	1	2	yes	—
15	#89887	C₄	1	2	yes	—
16	#89698	C₂²	1	2	yes	—
17	#89608	—	2	4	no	—
18	#89499	—	2	4	no	—
19	#89467	C₄	1	1	yes	characteristic
20	#89466	C₄	1	2	yes	—
21	C₄×D₅₆	C₂²	1	2	yes	—
22	C₄×D₅₆	C₂²	1	2	yes	—
23	#89460	C₂²	1	4	yes	—
24	C₈×D₂₈	C₂²	1	4	yes	—
25	#89454	C₄	1	1	yes	characteristic
26	C₇×#322	C₂²	1	1	yes	characteristic
27	C₇×#322	C₂²	1	1	yes	characteristic
28	C₇×C₈²	C₂²	1	1	yes	characteristic
29	C₇×C₈²	C₂²	1	1	yes	characteristic
30	#89254	C₂²	1	4	yes	—
31	#89254	C₂²	1	4	yes	—
32	#89253	C₂²	1	2	yes	—
33	#89252	C₂²	1	2	yes	—
34	#17970	—	7	1	no	maximal, Sylow
35	C₂×#1157	—	2	2	no	—
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	#6493	—	2	2	no	—
40	#6493	C₂×C₄	1	4	yes	—
41	C₂×D₅₆	C₂×C₄	1	2	yes	—
42	C₂×D₅₆	C₂×C₄	1	2	yes	—
43	C₂×#1131	—	2	4	no	—
44	C₂×C₈×D₇	—	2	4	no	—
45	#6438	—	2	2	no	—
46	#6437	C₂×C₄	1	2	yes	—
47	C₄×D₂₈	—	2	2	no	—
48	C₄×D₂₈	C₂³	1	4	yes	—
49	#6432	C₂×C₄	1	2	yes	—
50	C₇×C₄⋊C₈	—	2	1	no	—
51	C₇×C₄⋊C₈	D₄	1	2	yes	—
52	C₇×C₂²⋊C₈	—	2	1	no	—
53	C₇×C₂²⋊C₈	D₄	1	2	yes	—
54	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
63	#6395	C₂×C₄	1	4	yes	—
64	#6395	C₂×C₄	1	4	yes	—
65	(C₂×D₇)⋊C₈	—	2	4	no	—
66	(C₂×D₇)⋊C₈	—	2	4	no	—
67	#6392	C₂×C₄	1	2	yes	—
68	#6392	C₂×C₄	1	2	yes	—
69	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
70	#6377	—	2	2	no	—
71	#3169	—	7	1	no	—
72	#2981	—	7	2	no	—
73	#1594	—	7	2	no	—
74	#1589	—	7	2	no	—
75	#1580	—	7	2	no	—
76	C₈×D₈	—	7	2	no	—
77	#1568	—	7	1	no	—
78	#1496	—	7	2	no	—
79	#1450	D₇	1	1	yes	characteristic, complemented
80	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	#41	1	1	yes	characteristic
83	C₂×C₂²⋊D₇	—	2	4	no	—
84	#1157	—	8	4	no	—
85	C₂×D₂₈	—	2	2	no	—
86	C₂×D₂₈	—	2	2	no	—
87	C₂×D₂₈	C₂²×C₄	1	4	yes	—
88	C₂×C₄×D₇	—	2	4	no	—
89	C₂×Dic₁₄	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	1	no	—
93	C₂×C₇×C₈	—	2	1	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
100	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇	#41	1	1	yes	characteristic
103	C₄²×C₇	C₂⁴	1	1	yes	characteristic
104	(C₂×D₇)⋊C₄	—	2	4	no	—
105	(C₂×D₇)⋊C₄	—	2	4	no	—
106	C₄⋊Dic₇	—	2	2	no	—
107	C₄⋊Dic₇	C₂²×C₄	1	4	yes	—
108	C₂×C₇⋊C₈	—	2	4	no	—
109	D₅₆	—	2	4	no	—
110	D₅₆	—	2	4	no	—
111	#1131	—	8	4	no	—
112	C₈×D₇	—	8	4	no	—
113	#532	—	7	2	no	—
114	#500	—	7	1	no	—
115	#493	—	7	1	no	—
116	#486	—	7	2	no	—
117	#466	—	7	2	no	—
118	#466	—	7	2	no	—
119	C₄×D₈	—	7	2	no	—
120	C₄×D₈	—	7	2	no	—
121	#436	—	7	4	no	—
122	#435	—	14	4	no	—
123	C₈×D₄	—	7	4	no	—
124	#432	—	7	2	no	—
125	#431	C₂×D₇	1	1	yes	characteristic
126	#431	C₂×D₇	1	1	yes	characteristic
127	#408	—	14	4	no	—
128	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
129	#335	—	7	2	no	—
130	#334	—	7	2	no	—
131	D₄⋊C₈	—	7	4	no	—
132	D₄⋊C₈	—	7	4	no	—
133	#322	C₂×D₇	1	1	yes	characteristic
134	#322	C₂×D₇	1	1	yes	characteristic
135	C₈²	C₂×D₇	1	1	yes	characteristic
136	C₈²	C₂×D₇	1	1	yes	characteristic
137	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
138	C₂²×D₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	1	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	4	1	no	—
143	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
149	C₂²⋊D₇	—	8	4	no	—
150	C₂×Dic₇	—	2	4	no	—
151	D₂₈	—	2	4	no	—
152	D₂₈	—	2	4	no	—
153	D₂₈	—	8	2	no	—
154	D₂₈	—	8	2	no	—
155	C₄×D₇	—	4	4	no	—
156	C₄×D₇	—	4	4	no	—
157	Dic₁₄	—	8	2	no	—
158	C₇×C₈	—	2	1	no	—
159	C₇×C₈	—	2	1	no	—
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	—	2	2	no	—
162	C₇×C₈	—	4	1	no	—
163	C₇×C₈	—	4	1	no	—
164	C₇×C₈	#131	1	2	yes	—
165	C₇×C₈	#131	1	2	yes	—
166	C₇⋊C₈	—	4	4	no	—
167	C₂×#41	—	14	2	no	—
168	C₂×D₈	—	7	2	no	—
169	C₂×D₈	—	7	2	no	—
170	C₂×M₁₆	—	14	4	no	—
171	C₂²×C₈	—	14	4	no	—
172	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	7	2	no	—
175	#127	—	7	2	no	—
176	#124	—	14	2	no	—
177	#121	—	7	4	no	—
178	#121	—	14	2	no	—
179	C₄×D₄	—	7	4	no	—
180	C₄×D₄	—	14	2	no	—
181	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
182	#107	—	7	2	no	—
183	#107	—	7	2	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	7	4	no	—
186	C₄⋊C₈	D₂₈	1	2	yes	—
187	#102	—	7	4	no	—
188	#102	—	7	4	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	14	2	no	—
191	C₂²⋊C₈	—	14	4	no	—
192	C₂²⋊C₈	—	14	4	no	—
193	C₂²⋊C₈	D₂₈	1	2	yes	—
194	#97	—	14	2	no	—
195	C₄×C₈	C₄×D₇	1	1	yes	characteristic
196	C₄×C₈	C₄×D₇	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
200	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
203	C₂²×C₇	—	2	1	no	—
204	C₂²×C₇	—	2	1	no	—
205	C₂²×C₇	—	2	1	no	—
206	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂×D₇	—	4	4	no	—
208	C₂×D₇	—	4	4	no	—
209	C₂×D₇	—	4	4	no	—
210	C₄×C₇	—	2	1	no	—
211	C₄×C₇	—	2	1	no	—
212	C₄×C₇	—	2	1	no	—
213	C₄×C₇	—	2	1	no	—
214	C₄×C₇	C₂×#131	1	1	yes	characteristic
215	C₄×C₇	C₂×#131	1	1	yes	characteristic
216	C₄×C₇	#576	1	2	yes	—
217	Dic₇	—	4	4	no	—
218	#41	—	56	4	no	—
219	C₂×Q₈	—	14	2	no	—
220	C₂×D₄	—	7	4	no	—
221	C₂×D₄	—	14	4	no	—
222	C₂×D₄	—	14	2	no	—
223	C₂×D₄	—	14	2	no	—
224	C₂²×C₄	—	14	4	no	—
225	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
227	C₂²×C₄	#1157	1	1	yes	characteristic
228	D₈	—	14	4	no	—
229	D₈	—	14	4	no	—
230	M₁₆	—	56	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	14	4	no	—
236	C₂×C₈	—	56	4	no	—
237	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
241	C₂×C₈	C₂×D₂₈	1	2	yes	—
242	C₂×C₈	C₂×D₂₈	1	2	yes	—
243	C₄⋊C₄	—	7	4	no	—
244	C₄⋊C₄	—	14	2	no	—
245	C₂²⋊C₄	—	14	4	no	—
246	C₂²⋊C₄	—	14	4	no	—
247	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
248	C₄²	C₂×D₂₈	1	1	yes	characteristic
249	C₄²	#1157	1	1	yes	characteristic
250	C₄²	C₂³×D₇	1	1	yes	characteristic
251	C₂×C₇	—	4	1	no	—
252	C₂×C₇	#2931	1	1	yes	characteristic
253	C₂×C₇	#2948	1	1	yes	characteristic
254	C₂×C₇	#2957	1	1	yes	characteristic
255	D₇	—	8	4	no	—
256	C₂³	—	14	4	no	—
257	C₂³	C₄×D₂₈	1	1	yes	characteristic
258	Q₈	—	56	2	no	—
259	D₄	—	14	4	no	—
260	D₄	—	14	4	no	—
261	D₄	—	56	2	no	—
262	D₄	—	56	2	no	—
263	D₄	—	56	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	14	4	no	—
269	C₂×C₄	—	28	4	no	—
270	C₂×C₄	—	28	4	no	—
271	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
272	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
273	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
276	C₂×C₄	C₂×#1157	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	4	1	no	—
282	C₈	—	4	1	no	—
283	C₈	—	28	4	no	—
284	C₈	#6464	1	2	yes	—
285	C₈	#6464	1	2	yes	—
286	C₇	#17970	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	28	4	no	—
291	C₂²	—	28	4	no	—
292	C₂²	—	28	4	no	—
293	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	28	4	no	—
299	C₄	C₂×#6464	1	1	yes	characteristic
300	C₄	C₂×#6464	1	1	yes	characteristic
301	C₄	#90443	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	56	4	no	—
304	C₂	#11735903	1	1	yes	characteristic
305	C₂	#11736006	1	1	yes	characteristic
306	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×#1450	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#11735915	C₂
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1450	D₇
hypocenter	C₇	#17970
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #11736735	C₂	semidirect
3. C₇×#1450	C₂	semidirect
4. #11728519	C₂	semidirect
5. #11727486	C₂	semidirect
6. #11725837	C₂	semidirect
7. #11725831	C₂	semidirect
8. #11725806	C₂	semidirect
9. C₈×D₅₆	C₂	semidirect
79. #1450	D₇	semidirect
286. C₇	#17970	semidirect

Maximal subgroups

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

subgroup	index
#11736735	2
#11735915	2
C₇×#1450	2
#11728519	2
#11727486	2
#11725837	2
#11725831	2
#11725806	2
C₈×D₅₆	2
#17970	7