Subgroups — #28407331

Order 14464.

506 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299667	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299665	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299639	C₂	1	2	yes	maximal, complemented
4	#20299637	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#90673	C₂	1	2	yes	maximal, complemented
6	D₈×#90673	C₂	1	2	yes	maximal, complemented
7	#20299609	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299588	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16813684	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299564	C₂	1	1	yes	maximal, characteristic, complemented
11	AGL(1,ℤ/8)×D₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268655	C₂²	1	1	yes	characteristic
13	Q₈×#90673	C₂²	1	2	yes	complemented
14	D₄×#90673	—	2	1	no	—
15	D₄×#90673	—	2	1	no	—
16	D₄×#90673	—	2	2	no	—
17	D₄×#90673	C₂²	1	2	yes	—
18	D₄×#90673	C₂²	1	2	yes	—
19	D₄×#90673	C₂²	1	2	yes	complemented
20	#16813701	—	2	1	no	—
21	#16813701	C₂²	1	1	yes	characteristic
22	#16813701	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#90673	—	2	1	no	—
24	C₂×C₄⋊#90673	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#90673	—	2	2	no	—
26	#16813691	C₂²	1	2	yes	complemented
27	#16813690	C₂²	1	2	yes	complemented
28	#16813685	C₂²	1	1	yes	characteristic, complemented
29	#16813685	C₂²	1	1	yes	characteristic, complemented
30	#16813684	C₂²	1	2	yes	—
31	#16813684	C₂²	1	2	yes	—
32	#16813684	C₂²	1	1	yes	characteristic, complemented
33	#16813684	C₂²	1	1	yes	characteristic, complemented
34	#16813676	C₂²	1	2	yes	complemented
35	C₈×#90673	C₂²	1	2	yes	complemented
36	#41×D₁₁₃	C₂²	1	1	yes	characteristic
37	C₂×D₄×D₁₁₃	C₂²	1	1	yes	characteristic
38	#16813650	C₄	1	1	yes	characteristic, complemented
39	#16813649	C₄	1	2	yes	complemented
40	#16813648	C₄	1	1	yes	characteristic, complemented
41	#16813647	C₄	1	1	yes	characteristic, complemented
42	#16813646	C₄	1	2	yes	complemented
43	SD₁₆×D₁₁₃	C₂²	1	2	yes	—
44	D₈×D₁₁₃	C₂²	1	2	yes	—
45	M₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₁₁₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#90673	—	2	1	no	—
48	C₂²×#90673	—	2	2	no	—
49	C₂²×#90673	—	2	2	no	—
50	C₂²×#90673	—	2	2	no	—
51	#14268655	—	2	1	no	—
52	#14268655	C₂³	1	1	yes	characteristic
53	#14268655	C₂³	1	1	yes	characteristic
54	C₂×#11744753	—	2	1	no	—
55	C₄⋊#90673	—	2	2	no	—
56	C₄⋊#90673	—	2	1	no	—
57	C₄⋊#90673	D₄	1	1	yes	characteristic
58	C₄⋊#90673	D₄	1	1	yes	characteristic
59	C₄⋊#90673	C₂³	1	2	yes	—
60	C₄⋊#90673	C₂³	1	1	yes	characteristic
61	C₄⋊#90673	C₂³	1	1	yes	characteristic
62	C₂²⋊#90673	—	2	1	no	—
63	C₂²⋊#90673	—	2	2	no	—
64	C₂²⋊#90673	—	2	2	no	—
65	C₂²⋊#90673	—	2	2	no	—
66	C₂²⋊#90673	D₄	1	2	yes	—
67	C₄×#90673	—	2	2	no	—
68	C₄×#90673	—	2	1	no	—
69	C₄×#90673	C₂³	1	2	yes	—
70	C₂³×D₁₁₃	—	2	1	no	—
71	#14268648	C₂×C₄	1	1	yes	characteristic, complemented
72	#14268647	—	2	1	no	—
73	#14268647	C₂×C₄	1	1	yes	characteristic, complemented
74	#14268646	—	2	1	no	—
75	#14268646	C₂×C₄	1	1	yes	characteristic, complemented
76	Q₈×D₁₁₃	C₂³	1	1	yes	characteristic
77	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
79	D₄×D₁₁₃	—	2	1	no	—
80	D₄×D₁₁₃	—	2	1	no	—
81	D₄×D₁₁₃	—	2	1	no	—
82	D₄×D₁₁₃	C₂×C₄	1	2	yes	complemented
83	D₄×D₁₁₃	C₂³	1	1	yes	characteristic
84	D₄×D₁₁₃	C₂³	1	2	yes	—
85	C₂×C₄×D₁₁₃	—	2	1	no	—
86	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
87	#14268636	C₂×C₄	1	2	yes	complemented
88	#14268635	C₂×C₄	1	2	yes	complemented
89	#14268634	C₂×C₄	1	2	yes	complemented
90	D₉₀₄	C₂×C₄	1	2	yes	complemented
91	#14268632	C₂×C₄	1	2	yes	complemented
92	#14268632	C₂×C₄	1	2	yes	complemented
93	#14268631	C₂×C₄	1	1	yes	characteristic, complemented
94	#14268630	C₂×C₄	1	2	yes	complemented
95	C₈×D₁₁₃	C₂³	1	2	yes	—
96	#41×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
97	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
98	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
99	D₈×C₁₁₃	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
101	C₂³×C₁₁₃	—	2	1	no	—
102	C₂²×D₁₁₃	—	2	1	no	—
103	C₂²×D₁₁₃	—	2	1	no	—
104	C₂²×D₁₁₃	—	2	2	no	—
105	C₂²×D₁₁₃	—	2	2	no	—
106	C₂²×D₁₁₃	—	8	1	no	—
107	C₂²×D₁₁₃	—	8	1	no	—
108	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×#90673	—	2	1	no	—
110	C₂×#90673	—	2	1	no	—
111	C₂×#90673	—	2	2	no	—
112	C₂×#90673	—	8	2	no	—
113	C₂×#90673	—	8	2	no	—
114	C₂×#90673	—	8	2	no	—
115	C₂×#90673	—	8	1	no	—
116	C₂×#90673	C₂×D₄	1	2	yes	—
117	C₂×#90673	C₂×D₄	1	2	yes	—
118	Q₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
119	D₄×C₁₁₃	—	2	1	no	—
120	D₄×C₁₁₃	—	2	1	no	—
121	D₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
122	D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₁₁₃	—	2	1	no	—
124	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²⋊D₁₁₃	—	2	1	no	—
126	C₂²⋊D₁₁₃	—	2	1	no	—
127	C₂²⋊D₁₁₃	—	2	2	no	—
128	C₂²⋊D₁₁₃	—	2	2	no	—
129	C₂²⋊D₁₁₃	—	2	1	no	—
130	C₂×Dic₁₁₃	—	2	1	no	—
131	C₂×Dic₁₁₃	#41	1	1	yes	characteristic
132	D₄₅₂	—	2	1	no	—
133	D₄₅₂	—	2	1	no	—
134	D₄₅₂	C₂²×C₄	1	1	yes	characteristic
135	D₄₅₂	C₂²×C₄	1	2	yes	—
136	C₄×D₁₁₃	—	2	1	no	—
137	C₄×D₁₁₃	—	2	1	no	—
138	C₄×D₁₁₃	—	2	1	no	—
139	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
140	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
141	C₄×D₁₁₃	#41	1	1	yes	characteristic
142	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
143	Dic₂₂₆	—	2	1	no	—
144	Dic₂₂₆	C₂²×C₄	1	1	yes	characteristic
145	#11744753	—	2	1	no	—
146	#11744753	—	2	1	no	—
147	C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
148	#11744751	C₂²×C₄	1	2	yes	—
149	C₂²×C₁₁₃	—	2	1	no	—
150	C₂²×C₁₁₃	—	2	2	no	—
151	C₂²×C₁₁₃	—	8	1	no	—
152	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₁₃	—	2	1	no	—
154	C₂×D₁₁₃	—	2	1	no	—
155	C₂×D₁₁₃	—	2	1	no	—
156	C₂×D₁₁₃	—	2	2	no	—
157	C₂×D₁₁₃	—	4	1	no	—
158	C₂×D₁₁₃	—	4	2	no	—
159	C₂×D₁₁₃	—	4	2	no	—
160	C₂×D₁₁₃	—	4	1	no	—
161	C₂×D₁₁₃	—	8	2	no	—
162	C₂×D₁₁₃	—	8	1	no	—
163	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₁₃	C₂²×D₄	1	1	yes	characteristic
165	#90673	—	2	2	no	—
166	#90673	—	2	2	no	—
167	#90673	—	4	1	no	—
168	#90673	—	4	2	no	—
169	#90673	—	4	1	no	—
170	C₄×C₁₁₃	—	2	1	no	—
171	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
172	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
173	Dic₁₁₃	—	2	1	no	—
174	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
175	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₁₁₃	—	2	1	no	—
177	C₂×C₁₁₃	—	4	2	no	—
178	C₂×C₁₁₃	—	4	1	no	—
179	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	D₁₁₃	—	2	1	no	—
181	D₁₁₃	—	4	1	no	—
182	D₁₁₃	—	4	2	no	—
183	D₁₁₃	#576	1	1	yes	characteristic
184	D₁₁₃	#577	1	1	yes	characteristic
185	#2946	—	113	1	no	maximal, Sylow
186	C₁₁₃	#2946	1	1	yes	characteristic, complemented, Sylow
187	C₂×AGL(1,ℤ/8)	—	113	1	no	—
188	#520	—	113	1	no	—
189	#518	—	113	1	no	—
190	#442	—	113	2	no	—
191	#440	—	113	2	no	—
192	C₄×SD₁₆	—	113	2	no	—
193	C₄×D₈	—	113	2	no	—
194	#428	—	113	1	no	—
195	#416	—	113	1	no	—
196	C₂×#102	—	113	1	no	—
197	#405	—	113	1	no	—
198	C₂×#41	—	113	1	no	—
199	C₂²×D₄	—	113	1	no	—
200	AGL(1,ℤ/8)	—	113	1	no	—
201	AGL(1,ℤ/8)	—	113	1	no	—
202	AGL(1,ℤ/8)	—	113	2	no	—
203	AGL(1,ℤ/8)	—	113	2	no	—
204	AGL(1,ℤ/8)	—	113	1	no	—
205	AGL(1,ℤ/8)	#90673	1	1	yes	characteristic, complemented
206	C₂×SD₁₆	—	113	2	no	—
207	C₂×D₈	—	113	2	no	—
208	C₂×M₁₆	—	113	1	no	—
209	C₂²×C₈	—	113	1	no	—
210	C₄×Q₈	—	113	2	no	—
211	C₄×D₄	—	113	2	no	—
212	C₄×D₄	—	113	2	no	—
213	C₄×D₄	—	113	2	no	—
214	C₄×D₄	—	226	2	no	—
215	C₄×D₄	—	226	1	no	—
216	C₄×D₄	—	226	1	no	—
217	#117	—	113	1	no	—
218	#117	—	113	1	no	—
219	#117	—	226	1	no	—
220	C₂×C₄⋊C₄	—	113	1	no	—
221	C₂×C₄⋊C₄	—	226	1	no	—
222	C₂×C₂²⋊C₄	—	226	2	no	—
223	#107	—	113	2	no	—
224	#106	—	113	2	no	—
225	#103	—	113	1	no	—
226	#103	—	113	1	no	—
227	#102	—	113	2	no	—
228	#102	—	113	2	no	—
229	#102	—	113	1	no	—
230	#102	—	113	1	no	—
231	#97	—	113	2	no	—
232	C₄×C₈	—	113	2	no	—
233	C₂⁴	—	226	1	no	—
234	#41	—	113	1	no	—
235	#41	—	113	1	no	—
236	#41	—	113	1	no	—
237	#41	—	226	1	no	—
238	#41	—	226	1	no	—
239	#41	C₂×#90673	1	1	yes	characteristic, complemented
240	C₂×Q₈	—	113	1	no	—
241	C₂×D₄	—	113	1	no	—
242	C₂×D₄	—	113	2	no	—
243	C₂×D₄	—	113	2	no	—
244	C₂×D₄	—	113	1	no	—
245	C₂×D₄	—	226	1	no	—
246	C₂×D₄	—	226	1	no	—
247	C₂×D₄	—	226	1	no	—
248	C₂×D₄	—	226	1	no	—
249	C₂×D₄	C₂×#90673	1	1	yes	characteristic, complemented
250	C₂²×C₄	—	113	1	no	—
251	C₂²×C₄	—	226	1	no	—
252	C₂²×C₄	—	226	2	no	—
253	C₂²×C₄	—	226	2	no	—
254	C₂²×C₄	—	226	2	no	—
255	C₂²×C₄	—	226	1	no	—
256	SD₁₆	—	113	2	no	—
257	SD₁₆	—	113	2	no	—
258	SD₁₆	—	113	2	no	—
259	SD₁₆	C₂×#90673	1	2	yes	complemented
260	D₈	—	113	2	no	—
261	D₈	—	113	2	no	—
262	D₈	—	113	2	no	—
263	D₈	C₂×#90673	1	2	yes	complemented
264	M₁₆	—	113	1	no	—
265	M₁₆	—	113	2	no	—
266	M₁₆	C₂×#90673	1	1	yes	characteristic, complemented
267	C₂×C₈	—	113	2	no	—
268	C₂×C₈	—	113	1	no	—
269	C₂×C₈	—	113	1	no	—
270	C₂×C₈	—	226	1	no	—
271	C₂×C₈	—	226	1	no	—
272	C₄⋊C₄	—	113	1	no	—
273	C₄⋊C₄	—	113	1	no	—
274	C₄⋊C₄	—	113	2	no	—
275	C₄⋊C₄	—	113	1	no	—
276	C₄⋊C₄	—	113	1	no	—
277	C₄⋊C₄	—	226	1	no	—
278	C₄⋊C₄	—	226	2	no	—
279	C₂²⋊C₄	—	113	2	no	—
280	C₂²⋊C₄	—	226	2	no	—
281	C₂²⋊C₄	—	226	2	no	—
282	C₂²⋊C₄	—	226	2	no	—
283	C₂²⋊C₄	—	226	1	no	—
284	C₄²	—	113	2	no	—
285	C₄²	—	226	1	no	—
286	C₄²	—	226	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	113	1	no	—
289	C₂³	—	904	1	no	—
290	C₂³	—	904	1	no	—
291	C₂³	—	226	2	no	—
292	C₂³	—	226	1	no	—
293	C₂³	—	226	2	no	—
294	C₂³	—	226	1	no	—
295	Q₈	—	113	1	no	—
296	Q₈	—	226	1	no	—
297	Q₈	C₂²×#90673	1	1	yes	characteristic
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	113	1	no	—
301	D₄	—	113	2	no	—
302	D₄	—	226	1	no	—
303	D₄	—	226	2	no	—
304	D₄	—	226	2	no	—
305	D₄	—	226	1	no	—
306	D₄	—	226	1	no	—
307	D₄	—	226	1	no	—
308	D₄	—	226	1	no	—
309	D₄	C₂²×#90673	1	1	yes	characteristic
310	D₄	C₂²×#90673	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	113	1	no	—
313	C₂×C₄	—	113	1	no	—
314	C₂×C₄	—	113	1	no	—
315	C₂×C₄	—	113	1	no	—
316	C₂×C₄	—	113	1	no	—
317	C₂×C₄	—	113	2	no	—
318	C₂×C₄	—	113	2	no	—
319	C₂×C₄	—	904	1	no	—
320	C₂×C₄	—	904	2	no	—
321	C₂×C₄	—	904	2	no	—
322	C₂×C₄	—	904	2	no	—
323	C₂×C₄	—	226	1	no	—
324	C₂×C₄	—	226	1	no	—
325	C₂×C₄	—	226	1	no	—
326	C₂×C₄	—	226	1	no	—
327	C₂×C₄	—	226	1	no	—
328	C₂×C₄	—	226	2	no	—
329	C₂×C₄	—	226	1	no	—
330	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
331	C₈	—	113	2	no	—
332	C₈	—	226	1	no	—
333	C₈	—	226	1	no	—
334	C₈	C₂²×#90673	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	113	1	no	—
339	C₂²	—	113	1	no	—
340	C₂²	—	904	1	no	—
341	C₂²	—	904	2	no	—
342	C₂²	—	452	1	no	—
343	C₂²	—	452	2	no	—
344	C₂²	—	452	2	no	—
345	C₂²	—	452	1	no	—
346	C₂²	—	226	1	no	—
347	C₂²	—	226	1	no	—
348	C₂²	—	226	1	no	—
349	C₂²	—	226	2	no	—
350	C₂²	D₄×#90673	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	113	1	no	—
353	C₄	—	113	1	no	—
354	C₄	—	452	2	no	—
355	C₄	—	452	1	no	—
356	C₄	—	452	1	no	—
357	C₄	—	226	2	no	—
358	C₄	—	226	2	no	—
359	C₄	—	226	1	no	—
360	C₄	D₄×#90673	1	1	yes	characteristic
361	C₄	C₂³×#90673	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	4	2	no	—
364	C₂	—	4	1	no	—
365	C₂	—	113	1	no	—
366	C₂	—	113	1	no	—
367	C₂	—	452	2	no	—
368	C₂	—	452	1	no	—
369	C₂	—	226	1	no	—
370	C₂	C₂×D₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#90673
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₄	C₂³×#90673
Fitting	AGL(1,ℤ/8)×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄×D₄
prime radical	AGL(1,ℤ/8)×D₁₁₃	C₂
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	AGL(1,ℤ/8)	#90673
hypocenter	C₁₁₃	#2946
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #20299667	C₂	semidirect
2. #20299665	C₂	semidirect
3. #20299639	C₂	semidirect
4. #20299637	C₂	semidirect
5. SD₁₆×#90673	C₂	semidirect
6. D₈×#90673	C₂	semidirect
7. #20299609	C₂	semidirect
8. #20299588	C₂	semidirect
9. C₂×#16813684	C₂	semidirect
10. #20299564	C₂	semidirect
13. Q₈×#90673	C₂²	semidirect
19. D₄×#90673	C₂²	semidirect
26. #16813691	C₂²	semidirect
27. #16813690	C₂²	semidirect
28. #16813685	C₂²	semidirect
29. #16813685	C₂²	semidirect
32. #16813684	C₂²	semidirect
33. #16813684	C₂²	semidirect
34. #16813676	C₂²	semidirect
35. C₈×#90673	C₂²	semidirect
38. #16813650	C₄	semidirect
39. #16813649	C₄	semidirect
40. #16813648	C₄	semidirect
41. #16813647	C₄	semidirect
42. #16813646	C₄	semidirect
46. AGL(1,ℤ/8)×C₁₁₃	C₄	semidirect
71. #14268648	C₂×C₄	semidirect
73. #14268647	C₂×C₄	semidirect
75. #14268646	C₂×C₄	semidirect
77. C₂×D₄₅₂	C₂×C₄	semidirect
82. D₄×D₁₁₃	C₂×C₄	semidirect
87. #14268636	C₂×C₄	semidirect
88. #14268635	C₂×C₄	semidirect
89. #14268634	C₂×C₄	semidirect
90. D₉₀₄	C₂×C₄	semidirect
91. #14268632	C₂×C₄	semidirect
92. #14268632	C₂×C₄	semidirect
93. #14268631	C₂×C₄	semidirect
94. #14268630	C₂×C₄	semidirect
96. #41×C₁₁₃	C₂×C₄	semidirect
97. C₂×D₄×C₁₁₃	C₂×C₄	semidirect
98. SD₁₆×C₁₁₃	C₂×C₄	semidirect
99. D₈×C₁₁₃	C₂×C₄	semidirect
100. M₁₆×C₁₁₃	C₂×C₄	semidirect
186. C₁₁₃	#2946	semidirect
205. AGL(1,ℤ/8)	#90673	semidirect
239. #41	C₂×#90673	semidirect
249. C₂×D₄	C₂×#90673	semidirect
259. SD₁₆	C₂×#90673	semidirect
263. D₈	C₂×#90673	semidirect
266. M₁₆	C₂×#90673	semidirect

Maximal subgroups

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

subgroup	index
#20299667	2
#20299665	2
#20299639	2
#20299637	2
SD₁₆×#90673	2
D₈×#90673	2
#20299609	2
#20299588	2
C₂×#16813684	2
#20299564	2
AGL(1,ℤ/8)×D₁₁₃	2
#2946	113