Subgroups — #13509349

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735914	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729170	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729168	C₂	1	2	yes	maximal, complemented
5	#11729167	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6374	C₂	1	2	yes	maximal, complemented
8	#11726816	C₂	1	1	yes	maximal, characteristic, complemented
9	#11726813	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#11726799	C₂	1	1	yes	maximal, characteristic, complemented
12	#90170	—	2	2	no	—
13	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	#89888	C₂²	1	2	yes	—
17	#89887	C₂²	1	2	yes	—
18	#89886	—	2	2	no	—
19	C₂×Dic₅₆	C₂²	1	1	yes	characteristic
20	C₂×#6374	C₂²	1	2	yes	—
21	C₂×D₁₁₂	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₈	C₂²	1	2	yes	—
23	#89469	—	2	2	no	—
24	#89468	C₂²	1	2	yes	—
25	C₄×D₅₆	C₂²	1	2	yes	—
26	C₄×#1132	—	2	2	no	—
27	#89455	C₂²	1	2	yes	—
28	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
31	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
32	#89304	C₂²	1	2	yes	—
33	#89304	C₂²	1	2	yes	—
34	#89300	C₂²	1	2	yes	—
35	#89299	C₂²	1	1	yes	characteristic
36	#89299	C₂²	1	1	yes	characteristic
37	#89298	C₂²	1	2	yes	—
38	#89298	C₂²	1	2	yes	—
39	#33735	—	7	1	no	maximal, Sylow
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
43	#6493	—	2	2	no	—
44	#6492	—	4	2	no	—
45	C₄×C₂²⋊D₇	—	4	2	no	—
46	#6487	—	2	2	no	—
47	C₂×Dic₂₈	C₂³	1	2	yes	—
48	C₂×D₅₆	C₂³	1	2	yes	—
49	C₂×#1132	—	2	2	no	—
50	#6439	—	4	2	no	—
51	#6438	—	2	2	no	—
52	C₄×D₂₈	—	2	2	no	—
53	#6435	—	4	2	no	—
54	#6433	—	2	2	no	—
55	C₄×Dic₁₄	—	2	2	no	—
56	C₂×C₇×C₁₆	—	2	2	no	—
57	C₂×C₇×C₁₆	—	2	2	no	—
58	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	D₄	1	1	yes	characteristic
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	#6395	—	2	2	no	—
67	#6395	—	2	2	no	—
68	#6392	C₂³	1	2	yes	—
69	#6392	C₂³	1	2	yes	—
70	#6391	—	2	2	no	—
71	#6390	—	2	2	no	—
72	#6390	—	2	2	no	—
73	Dic₅₆	—	2	2	no	—
74	#6374	—	2	4	no	—
75	D₁₁₂	—	2	2	no	—
76	#3168	—	7	2	no	—
77	#2253	—	7	1	no	—
78	#2249	—	7	1	no	—
79	#2242	—	7	2	no	—
80	#2217	—	7	2	no	—
81	#2216	—	7	1	no	—
82	#2215	—	7	1	no	—
83	C₄×Q₃₂	—	7	1	no	—
84	C₄×SD₃₂	—	7	2	no	—
85	C₄×D₁₆	—	7	1	no	—
86	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic, complemented
87	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₇	—	4	2	no	—
91	C₂×D₂₈	—	2	2	no	—
92	C₂×C₄×D₇	—	4	2	no	—
93	C₂×Dic₁₄	—	2	2	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	1	no	—
96	C₂×C₇×C₈	—	2	1	no	—
97	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
101	C₄²×C₇	D₈	1	1	yes	characteristic
102	C₄²×C₇	D₈	1	1	yes	characteristic
103	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂²⋊Dic₇	—	4	2	no	—
106	(C₂×D₇)⋊C₄	—	4	2	no	—
107	(C₂×D₇)⋊C₄	—	4	2	no	—
108	(C₂×D₇)⋊C₄	—	4	2	no	—
109	(C₂×D₇)⋊C₄	—	4	2	no	—
110	C₄⋊Dic₇	—	2	2	no	—
111	C₄⋊Dic₇	—	2	2	no	—
112	Dic₇⋊C₄	—	4	2	no	—
113	Dic₇⋊C₄	—	4	2	no	—
114	Dic₇⋊C₄	—	4	2	no	—
115	Dic₇⋊C₄	—	4	2	no	—
116	C₄×Dic₇	—	4	2	no	—
117	Dic₂₈	—	2	2	no	—
118	Dic₂₈	—	2	2	no	—
119	D₅₆	—	2	2	no	—
120	D₅₆	—	2	2	no	—
121	#1132	—	4	4	no	—
122	C₇×C₁₆	—	2	1	no	—
123	C₇×C₁₆	—	2	1	no	—
124	C₇×C₁₆	#41	1	2	yes	—
125	C₇×C₁₆	#41	1	2	yes	—
126	#529	—	14	2	no	—
127	C₂×Q₃₂	—	7	1	no	—
128	C₂×SD₃₂	—	7	2	no	—
129	C₂×D₁₆	—	7	1	no	—
130	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
131	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
132	#499	—	7	2	no	—
133	#495	—	7	2	no	—
134	#488	—	14	2	no	—
135	#467	—	7	2	no	—
136	#466	—	7	2	no	—
137	#465	—	14	2	no	—
138	C₄×Q₁₆	—	7	2	no	—
139	C₄×SD₁₆	—	14	2	no	—
140	C₄×D₈	—	7	2	no	—
141	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
142	#367	—	7	2	no	—
143	#366	—	7	1	no	—
144	#366	—	7	1	no	—
145	#358	—	7	2	no	—
146	#358	—	7	2	no	—
147	#357	—	7	2	no	—
148	#357	—	7	2	no	—
149	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
150	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
151	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
152	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
153	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
154	C₂²×D₇	—	4	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	1	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
165	C₂²⋊D₇	—	8	4	no	—
166	C₂×Dic₇	—	4	2	no	—
167	C₂×Dic₇	—	4	2	no	—
168	C₂×Dic₇	—	4	2	no	—
169	D₂₈	—	4	2	no	—
170	D₂₈	—	4	2	no	—
171	C₄×D₇	—	8	4	no	—
172	Dic₁₄	—	4	2	no	—
173	Dic₁₄	—	4	2	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	2	1	no	—
176	C₇×C₈	#135	1	2	yes	—
177	C₇×C₈	C₂×#41	1	1	yes	characteristic
178	C₇×C₈	C₂×#41	1	1	yes	characteristic
179	C₂×Q₁₆	—	7	2	no	—
180	C₂×SD₁₆	—	14	2	no	—
181	C₂×D₈	—	7	2	no	—
182	C₂²×C₈	D₂₈	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
184	#126	—	28	2	no	—
185	#125	—	14	2	no	—
186	#124	—	14	2	no	—
187	#123	—	28	2	no	—
188	C₂²⋊Q₈	—	14	2	no	—
189	#121	—	14	2	no	—
190	C₄×Q₈	—	14	2	no	—
191	C₄×D₄	—	14	2	no	—
192	C₄×D₄	—	28	2	no	—
193	#117	—	28	2	no	—
194	C₂×C₄²	D₂₈	1	1	yes	characteristic
195	Q₃₂	—	14	2	no	—
196	SD₃₂	—	14	4	no	—
197	D₁₆	—	14	2	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
202	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
203	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
204	#107	—	7	2	no	—
205	#107	—	7	2	no	—
206	#106	—	14	2	no	—
207	#103	—	14	2	no	—
208	#103	—	14	2	no	—
209	#102	—	14	2	no	—
210	#102	—	14	2	no	—
211	C₄×C₈	D₂₈	1	1	yes	characteristic
212	C₄×C₈	D₂₈	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
215	C₂²×C₇	—	2	1	no	—
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	#508	1	2	yes	—
218	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
219	C₂×D₇	—	8	2	no	—
220	C₂×D₇	—	8	2	no	—
221	C₂×D₇	—	8	2	no	—
222	C₄×C₇	—	2	1	no	—
223	C₄×C₇	—	2	1	no	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	#508	1	2	yes	—
227	C₄×C₇	#532	1	1	yes	characteristic
228	C₄×C₇	C₂×#135	1	1	yes	characteristic
229	Dic₇	—	8	2	no	—
230	Dic₇	—	8	2	no	—
231	Dic₇	—	8	2	no	—
232	C₂×Q₈	—	14	2	no	—
233	C₂×D₄	—	14	2	no	—
234	C₂×D₄	—	28	2	no	—
235	C₂²×C₄	—	28	2	no	—
236	C₂²×C₄	D₅₆	1	1	yes	characteristic
237	C₂²×C₄	D₅₆	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
239	Q₁₆	—	14	2	no	—
240	Q₁₆	—	14	2	no	—
241	SD₁₆	—	28	4	no	—
242	D₈	—	14	2	no	—
243	D₈	—	14	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
248	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
249	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
251	C₄⋊C₄	—	14	2	no	—
252	C₄⋊C₄	—	14	2	no	—
253	C₄⋊C₄	—	28	2	no	—
254	C₄⋊C₄	—	28	2	no	—
255	C₄⋊C₄	—	28	2	no	—
256	C₄⋊C₄	—	28	2	no	—
257	C₂²⋊C₄	—	28	2	no	—
258	C₂²⋊C₄	—	28	2	no	—
259	C₂²⋊C₄	—	28	2	no	—
260	C₂²⋊C₄	—	28	2	no	—
261	C₂²⋊C₄	—	28	2	no	—
262	C₄²	—	28	2	no	—
263	C₄²	D₅₆	1	1	yes	characteristic
264	C₄²	D₅₆	1	1	yes	characteristic
265	C₄²	C₂×D₂₈	1	1	yes	characteristic
266	C₄²	C₂×D₂₈	1	1	yes	characteristic
267	C₁₆	—	2	1	no	—
268	C₁₆	—	2	1	no	—
269	C₁₆	#1157	1	2	yes	—
270	C₁₆	#1157	1	2	yes	—
271	C₂×C₇	—	2	2	no	—
272	C₂×C₇	#3169	1	1	yes	characteristic
273	C₂×C₇	C₂×#508	1	1	yes	characteristic
274	C₂×C₇	C₂×#508	1	1	yes	characteristic
275	D₇	—	16	2	no	—
276	C₂³	—	28	2	no	—
277	C₂³	C₂×D₅₆	1	1	yes	characteristic
278	Q₈	—	28	2	no	—
279	Q₈	—	28	2	no	—
280	D₄	—	28	2	no	—
281	D₄	—	28	2	no	—
282	D₄	—	56	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	28	2	no	—
288	C₂×C₄	—	28	2	no	—
289	C₂×C₄	—	28	2	no	—
290	C₂×C₄	—	56	4	no	—
291	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
292	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
293	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
294	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
295	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	#6467	1	2	yes	—
300	C₈	C₂×#1157	1	1	yes	characteristic
301	C₈	C₂×#1157	1	1	yes	characteristic
302	C₇	#33735	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	56	2	no	—
306	C₂²	—	56	2	no	—
307	C₂²	—	56	2	no	—
308	C₂²	#89676	1	2	yes	—
309	C₂²	C₂²×D₅₆	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	56	2	no	—
315	C₄	—	56	2	no	—
316	C₄	—	56	2	no	—
317	C₄	#89676	1	2	yes	—
318	C₄	#90168	1	1	yes	characteristic
319	C₄	C₂×#6467	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	112	2	no	—
322	C₂	#11735915	1	1	yes	characteristic
323	C₂	C₂×#89676	1	1	yes	characteristic
324	C₂	C₂×#89676	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	#11729167	C₂
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#33735
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #11735914	C₂	semidirect
2. C₂×C₄×C₇×C₁₆	C₂	semidirect
3. #11729170	C₂	semidirect
4. #11729168	C₂	semidirect
6. C₄×Dic₅₆	C₂	semidirect
7. C₄×#6374	C₂	semidirect
8. #11726816	C₂	semidirect
9. #11726813	C₂	semidirect
10. C₄×D₁₁₂	C₂	semidirect
11. #11726799	C₂	semidirect
86. C₂×C₄×C₁₆	D₇	semidirect
302. C₇	#33735	semidirect

Maximal subgroups

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

subgroup	index
#11735914	2
C₂×C₄×C₇×C₁₆	2
#11729170	2
#11729168	2
#11729167	2
C₄×Dic₅₆	2
C₄×#6374	2
#11726816	2
#11726813	2
C₄×D₁₁₂	2
#11726799	2
#33735	7