Subgroups — #28379789

Order 13952.

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	#20296095	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296093	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296067	C₂	1	2	yes	maximal, complemented
4	#20296065	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#89137	C₂	1	2	yes	maximal, complemented
6	D₈×#89137	C₂	1	2	yes	maximal, complemented
7	#20296037	C₂	1	1	yes	maximal, characteristic, complemented
8	#20296016	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812761	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295992	C₂	1	1	yes	maximal, characteristic, complemented
11	AGL(1,ℤ/8)×D₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×#89137	C₂²	1	2	yes	complemented
14	D₄×#89137	—	2	1	no	—
15	D₄×#89137	—	2	1	no	—
16	D₄×#89137	—	2	2	no	—
17	D₄×#89137	C₂²	1	2	yes	—
18	D₄×#89137	C₂²	1	2	yes	—
19	D₄×#89137	C₂²	1	2	yes	complemented
20	#16812778	—	2	1	no	—
21	#16812778	C₂²	1	1	yes	characteristic
22	#16812778	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#89137	—	2	1	no	—
24	C₂×C₄⋊#89137	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#89137	—	2	2	no	—
26	#16812768	C₂²	1	2	yes	complemented
27	#16812767	C₂²	1	2	yes	complemented
28	#16812762	C₂²	1	1	yes	characteristic, complemented
29	#16812762	C₂²	1	1	yes	characteristic, complemented
30	#16812761	C₂²	1	2	yes	—
31	#16812761	C₂²	1	2	yes	—
32	#16812761	C₂²	1	1	yes	characteristic, complemented
33	#16812761	C₂²	1	1	yes	characteristic, complemented
34	#16812753	C₂²	1	2	yes	complemented
35	C₈×#89137	C₂²	1	2	yes	complemented
36	#41×D₁₀₉	C₂²	1	1	yes	characteristic
37	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812727	C₄	1	1	yes	characteristic, complemented
39	#16812726	C₄	1	2	yes	complemented
40	#16812725	C₄	1	1	yes	characteristic, complemented
41	#16812724	C₄	1	1	yes	characteristic, complemented
42	#16812723	C₄	1	2	yes	complemented
43	SD₁₆×D₁₀₉	C₂²	1	2	yes	—
44	D₈×D₁₀₉	C₂²	1	2	yes	—
45	M₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₁₀₉	C₄	1	1	yes	characteristic, complemented
47	C₂²×#89137	—	2	1	no	—
48	C₂²×#89137	—	2	2	no	—
49	C₂²×#89137	—	2	2	no	—
50	C₂²×#89137	—	2	2	no	—
51	D₁₀₉⋊C₈	—	2	1	no	—
52	D₁₀₉⋊C₈	C₂³	1	1	yes	characteristic
53	D₁₀₉⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×#11724786	—	2	1	no	—
55	C₄⋊#89137	—	2	2	no	—
56	C₄⋊#89137	—	2	1	no	—
57	C₄⋊#89137	D₄	1	1	yes	characteristic
58	C₄⋊#89137	D₄	1	1	yes	characteristic
59	C₄⋊#89137	C₂³	1	2	yes	—
60	C₄⋊#89137	C₂³	1	1	yes	characteristic
61	C₄⋊#89137	C₂³	1	1	yes	characteristic
62	C₂²⋊#89137	—	2	1	no	—
63	C₂²⋊#89137	—	2	2	no	—
64	C₂²⋊#89137	—	2	2	no	—
65	C₂²⋊#89137	—	2	2	no	—
66	C₂²⋊#89137	D₄	1	2	yes	—
67	C₄×#89137	—	2	2	no	—
68	C₄×#89137	—	2	1	no	—
69	C₄×#89137	C₂³	1	2	yes	—
70	C₂³×D₁₀₉	—	2	1	no	—
71	#13182030	C₂×C₄	1	1	yes	characteristic, complemented
72	#13182029	—	2	1	no	—
73	#13182029	C₂×C₄	1	1	yes	characteristic, complemented
74	#13182028	—	2	1	no	—
75	#13182028	C₂×C₄	1	1	yes	characteristic, complemented
76	Q₈×D₁₀₉	C₂³	1	1	yes	characteristic
77	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
79	D₄×D₁₀₉	—	2	1	no	—
80	D₄×D₁₀₉	—	2	1	no	—
81	D₄×D₁₀₉	—	2	1	no	—
82	D₄×D₁₀₉	C₂×C₄	1	2	yes	complemented
83	D₄×D₁₀₉	C₂³	1	1	yes	characteristic
84	D₄×D₁₀₉	C₂³	1	2	yes	—
85	C₂×C₄×D₁₀₉	—	2	1	no	—
86	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
87	#13182018	C₂×C₄	1	2	yes	complemented
88	#13182017	C₂×C₄	1	2	yes	complemented
89	#13182016	C₂×C₄	1	2	yes	complemented
90	D₈₇₂	C₂×C₄	1	2	yes	complemented
91	#13182014	C₂×C₄	1	2	yes	complemented
92	#13182014	C₂×C₄	1	2	yes	complemented
93	#13182013	C₂×C₄	1	1	yes	characteristic, complemented
94	#13182012	C₂×C₄	1	2	yes	complemented
95	C₈×D₁₀₉	C₂³	1	2	yes	—
96	#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
97	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
98	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
99	D₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
101	C₂³×C₁₀₉	—	2	1	no	—
102	C₂²×D₁₀₉	—	2	1	no	—
103	C₂²×D₁₀₉	—	2	1	no	—
104	C₂²×D₁₀₉	—	2	2	no	—
105	C₂²×D₁₀₉	—	2	2	no	—
106	C₂²×D₁₀₉	—	8	1	no	—
107	C₂²×D₁₀₉	—	8	1	no	—
108	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×#89137	—	2	1	no	—
110	C₂×#89137	—	2	1	no	—
111	C₂×#89137	—	2	2	no	—
112	C₂×#89137	—	8	2	no	—
113	C₂×#89137	—	8	2	no	—
114	C₂×#89137	—	8	2	no	—
115	C₂×#89137	—	8	1	no	—
116	C₂×#89137	C₂×D₄	1	2	yes	—
117	C₂×#89137	C₂×D₄	1	2	yes	—
118	Q₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
122	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
125	C₂²⋊D₁₀₉	—	2	1	no	—
126	C₂²⋊D₁₀₉	—	2	1	no	—
127	C₂²⋊D₁₀₉	—	2	2	no	—
128	C₂²⋊D₁₀₉	—	2	2	no	—
129	C₂²⋊D₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	—	2	1	no	—
134	D₄₃₆	C₂²×C₄	1	1	yes	characteristic
135	D₄₃₆	C₂²×C₄	1	2	yes	—
136	C₄×D₁₀₉	—	2	1	no	—
137	C₄×D₁₀₉	—	2	1	no	—
138	C₄×D₁₀₉	—	2	1	no	—
139	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
140	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
141	C₄×D₁₀₉	#41	1	1	yes	characteristic
142	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	C₂²×C₄	1	1	yes	characteristic
145	#11724786	—	2	1	no	—
146	#11724786	—	2	1	no	—
147	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
148	#11724784	C₂²×C₄	1	2	yes	—
149	C₂²×C₁₀₉	—	2	1	no	—
150	C₂²×C₁₀₉	—	2	2	no	—
151	C₂²×C₁₀₉	—	8	1	no	—
152	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₀₉	—	2	1	no	—
154	C₂×D₁₀₉	—	2	1	no	—
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	2	2	no	—
157	C₂×D₁₀₉	—	4	1	no	—
158	C₂×D₁₀₉	—	4	2	no	—
159	C₂×D₁₀₉	—	4	2	no	—
160	C₂×D₁₀₉	—	4	1	no	—
161	C₂×D₁₀₉	—	8	2	no	—
162	C₂×D₁₀₉	—	8	1	no	—
163	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
165	#89137	—	2	2	no	—
166	#89137	—	2	2	no	—
167	#89137	—	4	1	no	—
168	#89137	—	4	2	no	—
169	#89137	—	4	1	no	—
170	C₄×C₁₀₉	—	2	1	no	—
171	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
172	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
173	Dic₁₀₉	—	2	1	no	—
174	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
175	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
176	C₂×C₁₀₉	—	2	1	no	—
177	C₂×C₁₀₉	—	4	2	no	—
178	C₂×C₁₀₉	—	4	1	no	—
179	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
180	D₁₀₉	—	2	1	no	—
181	D₁₀₉	—	4	1	no	—
182	D₁₀₉	—	4	2	no	—
183	D₁₀₉	#576	1	1	yes	characteristic
184	D₁₀₉	#577	1	1	yes	characteristic
185	#2946	—	109	1	no	maximal, Sylow
186	C₁₀₉	#2946	1	1	yes	characteristic, complemented, Sylow
187	C₂×AGL(1,ℤ/8)	—	109	1	no	—
188	#520	—	109	1	no	—
189	#518	—	109	1	no	—
190	#442	—	109	2	no	—
191	#440	—	109	2	no	—
192	C₄×SD₁₆	—	109	2	no	—
193	C₄×D₈	—	109	2	no	—
194	#428	—	109	1	no	—
195	#416	—	109	1	no	—
196	C₂×#102	—	109	1	no	—
197	#405	—	109	1	no	—
198	C₂×#41	—	109	1	no	—
199	C₂²×D₄	—	109	1	no	—
200	AGL(1,ℤ/8)	—	109	1	no	—
201	AGL(1,ℤ/8)	—	109	1	no	—
202	AGL(1,ℤ/8)	—	109	2	no	—
203	AGL(1,ℤ/8)	—	109	2	no	—
204	AGL(1,ℤ/8)	—	109	1	no	—
205	AGL(1,ℤ/8)	#89137	1	1	yes	characteristic, complemented
206	C₂×SD₁₆	—	109	2	no	—
207	C₂×D₈	—	109	2	no	—
208	C₂×M₁₆	—	109	1	no	—
209	C₂²×C₈	—	109	1	no	—
210	C₄×Q₈	—	109	2	no	—
211	C₄×D₄	—	109	2	no	—
212	C₄×D₄	—	109	2	no	—
213	C₄×D₄	—	109	2	no	—
214	C₄×D₄	—	218	2	no	—
215	C₄×D₄	—	218	1	no	—
216	C₄×D₄	—	218	1	no	—
217	#117	—	109	1	no	—
218	#117	—	109	1	no	—
219	#117	—	218	1	no	—
220	C₂×C₄⋊C₄	—	109	1	no	—
221	C₂×C₄⋊C₄	—	218	1	no	—
222	C₂×C₂²⋊C₄	—	218	2	no	—
223	#107	—	109	2	no	—
224	#106	—	109	2	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#102	—	109	2	no	—
228	#102	—	109	2	no	—
229	#102	—	109	1	no	—
230	#102	—	109	1	no	—
231	#97	—	109	2	no	—
232	C₄×C₈	—	109	2	no	—
233	C₂⁴	—	218	1	no	—
234	#41	—	109	1	no	—
235	#41	—	109	1	no	—
236	#41	—	109	1	no	—
237	#41	—	218	1	no	—
238	#41	—	218	1	no	—
239	#41	C₂×#89137	1	1	yes	characteristic, complemented
240	C₂×Q₈	—	109	1	no	—
241	C₂×D₄	—	109	2	no	—
242	C₂×D₄	—	109	1	no	—
243	C₂×D₄	—	109	2	no	—
244	C₂×D₄	—	109	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	—	218	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	C₂×#89137	1	1	yes	characteristic, complemented
250	C₂²×C₄	—	109	1	no	—
251	C₂²×C₄	—	218	1	no	—
252	C₂²×C₄	—	218	2	no	—
253	C₂²×C₄	—	218	2	no	—
254	C₂²×C₄	—	218	2	no	—
255	C₂²×C₄	—	218	1	no	—
256	SD₁₆	—	109	2	no	—
257	SD₁₆	—	109	2	no	—
258	SD₁₆	—	109	2	no	—
259	SD₁₆	C₂×#89137	1	2	yes	complemented
260	D₈	—	109	2	no	—
261	D₈	—	109	2	no	—
262	D₈	—	109	2	no	—
263	D₈	C₂×#89137	1	2	yes	complemented
264	M₁₆	—	109	1	no	—
265	M₁₆	—	109	2	no	—
266	M₁₆	C₂×#89137	1	1	yes	characteristic, complemented
267	C₂×C₈	—	109	2	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	—	109	1	no	—
270	C₂×C₈	—	218	1	no	—
271	C₂×C₈	—	218	1	no	—
272	C₄⋊C₄	—	109	1	no	—
273	C₄⋊C₄	—	109	1	no	—
274	C₄⋊C₄	—	109	2	no	—
275	C₄⋊C₄	—	109	1	no	—
276	C₄⋊C₄	—	109	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₄⋊C₄	—	218	2	no	—
279	C₂²⋊C₄	—	109	2	no	—
280	C₂²⋊C₄	—	218	2	no	—
281	C₂²⋊C₄	—	218	2	no	—
282	C₂²⋊C₄	—	218	2	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₄²	—	109	2	no	—
285	C₄²	—	218	1	no	—
286	C₄²	—	218	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	872	1	no	—
289	C₂³	—	872	1	no	—
290	C₂³	—	109	1	no	—
291	C₂³	—	218	2	no	—
292	C₂³	—	218	1	no	—
293	C₂³	—	218	2	no	—
294	C₂³	—	218	1	no	—
295	Q₈	—	109	1	no	—
296	Q₈	—	218	1	no	—
297	Q₈	C₂²×#89137	1	1	yes	characteristic
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	109	1	no	—
301	D₄	—	109	2	no	—
302	D₄	—	218	1	no	—
303	D₄	—	218	2	no	—
304	D₄	—	218	2	no	—
305	D₄	—	218	1	no	—
306	D₄	—	218	1	no	—
307	D₄	—	218	1	no	—
308	D₄	—	218	1	no	—
309	D₄	C₂²×#89137	1	1	yes	characteristic
310	D₄	C₂²×#89137	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	872	1	no	—
313	C₂×C₄	—	872	2	no	—
314	C₂×C₄	—	872	2	no	—
315	C₂×C₄	—	872	2	no	—
316	C₂×C₄	—	109	1	no	—
317	C₂×C₄	—	109	1	no	—
318	C₂×C₄	—	109	1	no	—
319	C₂×C₄	—	109	1	no	—
320	C₂×C₄	—	109	1	no	—
321	C₂×C₄	—	109	2	no	—
322	C₂×C₄	—	109	2	no	—
323	C₂×C₄	—	218	1	no	—
324	C₂×C₄	—	218	1	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	2	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
331	C₈	—	109	2	no	—
332	C₈	—	218	1	no	—
333	C₈	—	218	1	no	—
334	C₈	C₂²×#89137	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	872	1	no	—
339	C₂²	—	872	2	no	—
340	C₂²	—	109	1	no	—
341	C₂²	—	109	1	no	—
342	C₂²	—	436	1	no	—
343	C₂²	—	436	2	no	—
344	C₂²	—	436	2	no	—
345	C₂²	—	436	1	no	—
346	C₂²	—	218	1	no	—
347	C₂²	—	218	1	no	—
348	C₂²	—	218	1	no	—
349	C₂²	—	218	2	no	—
350	C₂²	D₄×#89137	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	109	1	no	—
353	C₄	—	109	1	no	—
354	C₄	—	436	2	no	—
355	C₄	—	436	1	no	—
356	C₄	—	436	1	no	—
357	C₄	—	218	2	no	—
358	C₄	—	218	2	no	—
359	C₄	—	218	1	no	—
360	C₄	D₄×#89137	1	1	yes	characteristic
361	C₄	C₂³×#89137	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	4	2	no	—
364	C₂	—	4	1	no	—
365	C₂	—	109	1	no	—
366	C₂	—	109	1	no	—
367	C₂	—	436	2	no	—
368	C₂	—	436	1	no	—
369	C₂	—	218	1	no	—
370	C₂	C₂×D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	AGL(1,ℤ/8)×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	AGL(1,ℤ/8)×D₁₀₉	C₂
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	AGL(1,ℤ/8)	#89137
hypocenter	C₁₀₉	#2946
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
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. #20296095	C₂	semidirect
2. #20296093	C₂	semidirect
3. #20296067	C₂	semidirect
4. #20296065	C₂	semidirect
5. SD₁₆×#89137	C₂	semidirect
6. D₈×#89137	C₂	semidirect
7. #20296037	C₂	semidirect
8. #20296016	C₂	semidirect
9. C₂×#16812761	C₂	semidirect
10. #20295992	C₂	semidirect
13. Q₈×#89137	C₂²	semidirect
19. D₄×#89137	C₂²	semidirect
26. #16812768	C₂²	semidirect
27. #16812767	C₂²	semidirect
28. #16812762	C₂²	semidirect
29. #16812762	C₂²	semidirect
32. #16812761	C₂²	semidirect
33. #16812761	C₂²	semidirect
34. #16812753	C₂²	semidirect
35. C₈×#89137	C₂²	semidirect
38. #16812727	C₄	semidirect
39. #16812726	C₄	semidirect
40. #16812725	C₄	semidirect
41. #16812724	C₄	semidirect
42. #16812723	C₄	semidirect
46. AGL(1,ℤ/8)×C₁₀₉	C₄	semidirect
71. #13182030	C₂×C₄	semidirect
73. #13182029	C₂×C₄	semidirect
75. #13182028	C₂×C₄	semidirect
77. C₂×D₄₃₆	C₂×C₄	semidirect
82. D₄×D₁₀₉	C₂×C₄	semidirect
87. #13182018	C₂×C₄	semidirect
88. #13182017	C₂×C₄	semidirect
89. #13182016	C₂×C₄	semidirect
90. D₈₇₂	C₂×C₄	semidirect
91. #13182014	C₂×C₄	semidirect
92. #13182014	C₂×C₄	semidirect
93. #13182013	C₂×C₄	semidirect
94. #13182012	C₂×C₄	semidirect
96. #41×C₁₀₉	C₂×C₄	semidirect
97. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
98. SD₁₆×C₁₀₉	C₂×C₄	semidirect
99. D₈×C₁₀₉	C₂×C₄	semidirect
100. M₁₆×C₁₀₉	C₂×C₄	semidirect
186. C₁₀₉	#2946	semidirect
205. AGL(1,ℤ/8)	#89137	semidirect
239. #41	C₂×#89137	semidirect
249. C₂×D₄	C₂×#89137	semidirect
259. SD₁₆	C₂×#89137	semidirect
263. D₈	C₂×#89137	semidirect
266. M₁₆	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296095	2
#20296093	2
#20296067	2
#20296065	2
SD₁₆×#89137	2
D₈×#89137	2
#20296037	2
#20296016	2
C₂×#16812761	2
#20295992	2
AGL(1,ℤ/8)×D₁₀₉	2
#2946	109