Subgroups — #137×#89137 #28379788

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#89137	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#89137	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296066	C₂	1	2	yes	maximal, complemented
4	#20296065	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#89137	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#89137	C₂	1	2	yes	maximal, complemented
7	#20296037	C₂	1	1	yes	maximal, characteristic, complemented
8	#20296019	C₂	1	1	yes	maximal, characteristic, complemented
9	#20296017	C₂	1	1	yes	maximal, characteristic
10	M₁₆×#89137	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₀₉	C₂	1	1	yes	maximal, characteristic
12	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
13	Q₈×#89137	—	2	1	no	—
14	Q₈×#89137	—	2	1	no	—
15	Q₈×#89137	C₂²	1	2	yes	—
16	Q₈×#89137	C₂²	1	1	yes	characteristic
17	Q₈×#89137	C₂²	1	2	yes	—
18	Q₈×#89137	C₂²	1	1	yes	characteristic
19	D₄×#89137	—	2	1	no	—
20	D₄×#89137	—	2	1	no	—
21	D₄×#89137	C₂²	1	1	yes	characteristic
22	D₄×#89137	C₂²	1	1	yes	characteristic
23	#16812778	—	2	1	no	—
24	#16812778	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#89137	—	2	1	no	—
26	C₂×C₄⋊#89137	C₂²	1	1	yes	characteristic
27	C₂×C₄×#89137	—	2	1	no	—
28	C₂×C₄×#89137	—	2	1	no	—
29	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
30	#16812768	C₂²	1	2	yes	—
31	#16812767	C₂²	1	2	yes	—
32	#16812762	C₂²	1	2	yes	—
33	#16812762	C₂²	1	2	yes	—
34	#16812762	C₂²	1	2	yes	—
35	#16812761	C₂²	1	2	yes	—
36	#16812753	C₂²	1	2	yes	—
37	C₈×#89137	C₂²	1	2	yes	—
38	#41×D₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812732	C₄	1	1	yes	characteristic, complemented
41	#16812731	C₄	1	2	yes	complemented
42	#16812730	C₄	1	1	yes	characteristic, complemented
43	#16812729	C₄	1	1	yes	characteristic, complemented
44	#16812728	C₄	1	2	yes	complemented
45	Q₁₆×D₁₀₉	C₂²	1	2	yes	—
46	SD₁₆×D₁₀₉	C₂²	1	2	yes	—
47	M₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
48	#137×C₁₀₉	C₄	1	1	yes	characteristic, complemented
49	C₂²×#89137	—	2	1	no	—
50	C₂²×#89137	—	2	1	no	—
51	C₂²×#89137	—	2	1	no	—
52	C₂²×#89137	D₄	1	1	yes	characteristic
53	C₂²×#89137	D₄	1	1	yes	characteristic
54	#13182039	—	2	1	no	—
55	#13182038	—	2	1	no	—
56	D₁₀₉⋊C₈	C₂³	1	2	yes	—
57	C₄⋊#89137	—	2	2	no	—
58	C₄⋊#89137	—	2	2	no	—
59	C₄⋊#89137	—	2	1	no	—
60	C₄⋊#89137	—	2	1	no	—
61	C₄⋊#89137	—	2	1	no	—
62	C₄⋊#89137	C₂³	1	2	yes	—
63	C₄⋊#89137	C₂³	1	1	yes	characteristic
64	C₄⋊#89137	C₂³	1	1	yes	characteristic
65	C₂²⋊#89137	—	2	1	no	—
66	C₂²⋊#89137	—	2	1	no	—
67	C₂²⋊#89137	—	2	1	no	—
68	C₄×#89137	—	2	1	no	—
69	C₄×#89137	—	2	1	no	—
70	C₄×#89137	—	2	2	no	—
71	C₄×#89137	—	2	2	no	—
72	C₄×#89137	—	2	1	no	—
73	C₄×#89137	D₄	1	1	yes	characteristic
74	C₄×#89137	D₄	1	1	yes	characteristic
75	C₄×#89137	C₂³	1	1	yes	characteristic
76	C₄×#89137	C₂³	1	1	yes	characteristic
77	#13182030	C₂×C₄	1	1	yes	characteristic
78	#13182029	—	2	1	no	—
79	#13182029	C₂×C₄	1	1	yes	characteristic
80	#13182028	—	2	1	no	—
81	#13182028	C₂×C₄	1	1	yes	characteristic
82	C₂×Dic₂₁₈	—	2	1	no	—
83	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic, complemented
84	Q₈×D₁₀₉	—	2	1	no	—
85	Q₈×D₁₀₉	—	2	1	no	—
86	Q₈×D₁₀₉	C₂×C₄	1	2	yes	complemented
87	Q₈×D₁₀₉	C₂³	1	2	yes	—
88	Q₈×D₁₀₉	C₂³	1	1	yes	characteristic
89	D₄×D₁₀₉	—	2	1	no	—
90	D₄×D₁₀₉	C₂³	1	1	yes	characteristic
91	C₂×C₄×D₁₀₉	—	2	1	no	—
92	C₂×C₄×D₁₀₉	—	2	1	no	—
93	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
94	Dic₄₃₆	C₂×C₄	1	2	yes	complemented
95	#13182019	C₂×C₄	1	2	yes	complemented
96	#13182019	C₂×C₄	1	2	yes	complemented
97	#13182018	C₂×C₄	1	2	yes	complemented
98	#13182017	C₂×C₄	1	2	yes	complemented
99	#13182016	C₂×C₄	1	2	yes	complemented
100	#13182013	C₂×C₄	1	1	yes	characteristic, complemented
101	#13182012	C₂×C₄	1	2	yes	complemented
102	C₈×D₁₀₉	C₂³	1	2	yes	—
103	#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
104	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
105	Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
106	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
107	M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
108	C₂²×D₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×#89137	—	2	2	no	—
111	C₂×#89137	—	2	1	no	—
112	C₂×#89137	—	2	1	no	—
113	C₂×#89137	—	2	2	no	—
114	C₂×#89137	—	4	1	no	—
115	C₂×#89137	—	4	2	no	—
116	C₂×#89137	—	4	2	no	—
117	C₂×#89137	—	4	2	no	—
118	C₂×#89137	C₂×D₄	1	1	yes	characteristic
119	C₂×#89137	C₂×D₄	1	1	yes	characteristic
120	C₂×#89137	C₂×D₄	1	1	yes	characteristic
121	C₂×#89137	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₀₉	—	2	1	no	—
123	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
124	Q₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
125	D₄×C₁₀₉	—	2	1	no	—
126	D₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	—	2	1	no	—
128	C₂×C₄×C₁₀₉	—	2	1	no	—
129	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
130	C₂²⋊D₁₀₉	—	2	1	no	—
131	C₂²⋊D₁₀₉	—	2	1	no	—
132	C₂²⋊D₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
136	D₄₃₆	—	2	1	no	—
137	D₄₃₆	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₁₀₉	—	2	2	no	—
139	C₄×D₁₀₉	—	2	1	no	—
140	C₄×D₁₀₉	—	2	2	no	—
141	C₄×D₁₀₉	—	2	1	no	—
142	C₄×D₁₀₉	—	2	1	no	—
143	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
144	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
145	C₄×D₁₀₉	#41	1	1	yes	characteristic
146	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
147	Dic₂₁₈	—	2	1	no	—
148	Dic₂₁₈	—	2	2	no	—
149	Dic₂₁₈	—	2	2	no	—
150	Dic₂₁₈	—	2	1	no	—
151	Dic₂₁₈	C₂²×C₄	1	2	yes	—
152	Dic₂₁₈	C₂²×C₄	1	1	yes	characteristic
153	#11724786	—	2	2	no	—
154	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
155	#11724784	C₂²×C₄	1	2	yes	—
156	C₂²×C₁₀₉	—	2	1	no	—
157	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
158	C₂×D₁₀₉	—	2	1	no	—
159	C₂×D₁₀₉	—	2	1	no	—
160	C₂×D₁₀₉	—	2	1	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
165	#89137	—	2	1	no	—
166	#89137	—	2	1	no	—
167	#89137	—	2	1	no	—
168	#89137	—	4	1	no	—
169	#89137	—	4	1	no	—
170	#89137	—	4	2	no	—
171	#89137	#137	1	2	yes	direct product factor
172	C₄×C₁₀₉	—	2	2	no	—
173	C₄×C₁₀₉	—	2	1	no	—
174	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
176	Dic₁₀₉	—	2	2	no	—
177	Dic₁₀₉	—	2	1	no	—
178	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
179	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
180	C₂×C₁₀₉	—	2	1	no	—
181	C₂×C₁₀₉	—	4	1	no	—
182	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
183	D₁₀₉	—	2	1	no	—
184	D₁₀₉	—	4	1	no	—
185	D₁₀₉	C₂×#137	1	1	yes	characteristic
186	D₁₀₉	#575	1	1	yes	characteristic
187	C₄×#137	—	109	1	no	maximal, Sylow
188	C₁₀₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
189	C₂×#137	—	109	1	no	—
190	C₄×#41	—	109	1	no	—
191	C₂×C₄×Q₈	—	109	1	no	—
192	#441	—	109	2	no	—
193	#440	—	109	2	no	—
194	C₄×Q₁₆	—	109	2	no	—
195	C₄×SD₁₆	—	109	2	no	—
196	#428	—	109	1	no	—
197	#419	—	109	1	no	—
198	#417	—	109	1	no	—
199	C₄×M₁₆	—	109	1	no	—
200	C₂×#41	—	109	1	no	—
201	C₂²×Q₈	—	109	1	no	—
202	#137	—	109	1	no	—
203	#137	—	109	1	no	—
204	#137	—	109	2	no	—
205	#137	—	109	2	no	—
206	#137	—	109	1	no	—
207	#137	#89137	1	1	yes	characteristic, direct product factor
208	C₂×Q₁₆	—	109	2	no	—
209	C₂×SD₁₆	—	109	2	no	—
210	C₂×M₁₆	—	109	1	no	—
211	C₂×M₁₆	—	109	1	no	—
212	C₄×Q₈	—	109	2	no	—
213	C₄×Q₈	—	109	2	no	—
214	C₄×Q₈	—	109	1	no	—
215	C₄×Q₈	—	109	1	no	—
216	C₄×Q₈	—	218	1	no	—
217	C₄×Q₈	—	218	1	no	—
218	C₄×D₄	—	109	1	no	—
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	218	1	no	—
221	C₄×D₄	—	218	1	no	—
222	#117	—	109	1	no	—
223	#117	—	218	1	no	—
224	C₂×C₄⋊C₄	—	109	1	no	—
225	C₂×C₄⋊C₄	—	218	1	no	—
226	C₂×C₄²	—	109	1	no	—
227	C₂×C₄²	—	218	1	no	—
228	C₂×C₄²	—	218	1	no	—
229	#107	—	109	2	no	—
230	#106	—	109	2	no	—
231	#103	—	109	2	no	—
232	#103	—	109	2	no	—
233	#103	—	109	2	no	—
234	#102	—	109	2	no	—
235	#97	—	109	2	no	—
236	C₄×C₈	—	109	2	no	—
237	#41	—	109	1	no	—
238	#41	—	109	1	no	—
239	#41	—	109	1	no	—
240	#41	—	218	1	no	—
241	#41	—	218	1	no	—
242	#41	C₂×#89137	1	1	yes	characteristic
243	C₂×Q₈	—	109	2	no	—
244	C₂×Q₈	—	109	2	no	—
245	C₂×Q₈	—	109	1	no	—
246	C₂×Q₈	—	109	1	no	—
247	C₂×Q₈	—	218	1	no	—
248	C₂×Q₈	—	218	1	no	—
249	C₂×Q₈	—	218	1	no	—
250	C₂×Q₈	C₂×#89137	1	1	yes	characteristic, complemented
251	C₂×D₄	—	109	1	no	—
252	C₂×D₄	—	218	1	no	—
253	C₂²×C₄	—	109	1	no	—
254	C₂²×C₄	—	109	1	no	—
255	C₂²×C₄	—	109	1	no	—
256	C₂²×C₄	—	218	1	no	—
257	C₂²×C₄	—	218	1	no	—
258	C₂²×C₄	—	218	1	no	—
259	C₂²×C₄	—	218	1	no	—
260	C₂²×C₄	—	218	1	no	—
261	Q₁₆	—	109	2	no	—
262	Q₁₆	—	109	2	no	—
263	Q₁₆	—	109	2	no	—
264	Q₁₆	C₂×#89137	1	2	yes	complemented
265	SD₁₆	—	109	2	no	—
266	SD₁₆	—	109	2	no	—
267	SD₁₆	—	109	2	no	—
268	SD₁₆	C₂×#89137	1	2	yes	complemented
269	M₁₆	—	109	1	no	—
270	M₁₆	—	109	2	no	—
271	M₁₆	—	218	1	no	—
272	M₁₆	—	218	1	no	—
273	M₁₆	C₂×#89137	1	1	yes	characteristic, complemented
274	C₂×C₈	—	109	2	no	—
275	C₂×C₈	—	109	2	no	—
276	C₄⋊C₄	—	109	2	no	—
277	C₄⋊C₄	—	109	1	no	—
278	C₄⋊C₄	—	109	1	no	—
279	C₄⋊C₄	—	218	1	no	—
280	C₄⋊C₄	—	218	2	no	—
281	C₄⋊C₄	—	218	2	no	—
282	C₄⋊C₄	—	218	1	no	—
283	C₄⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₂²⋊C₄	—	218	1	no	—
286	C₂²⋊C₄	—	218	1	no	—
287	C₄²	—	109	1	no	—
288	C₄²	—	109	1	no	—
289	C₄²	—	109	1	no	—
290	C₄²	—	109	1	no	—
291	C₄²	—	218	1	no	—
292	C₄²	—	218	1	no	—
293	C₄²	—	218	1	no	—
294	C₄²	—	218	2	no	—
295	C₄²	—	218	2	no	—
296	C₂³	—	109	1	no	—
297	C₂³	—	218	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	109	2	no	—
300	Q₈	—	109	1	no	—
301	Q₈	—	218	2	no	—
302	Q₈	—	218	2	no	—
303	Q₈	—	218	1	no	—
304	Q₈	—	218	1	no	—
305	Q₈	C₂²×#89137	1	2	yes	—
306	Q₈	C₂²×#89137	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	109	1	no	—
309	D₄	—	218	1	no	—
310	D₄	—	218	1	no	—
311	D₄	—	218	1	no	—
312	D₄	—	218	1	no	—
313	D₄	C₂²×#89137	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	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	1	no	—
322	C₂×C₄	—	109	1	no	—
323	C₂×C₄	—	109	1	no	—
324	C₂×C₄	—	109	1	no	—
325	C₂×C₄	—	436	2	no	—
326	C₂×C₄	—	436	2	no	—
327	C₂×C₄	—	436	2	no	—
328	C₂×C₄	—	436	1	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	—	218	1	no	—
331	C₂×C₄	—	218	2	no	—
332	C₂×C₄	—	218	2	no	—
333	C₂×C₄	—	218	1	no	—
334	C₂×C₄	—	218	2	no	—
335	C₂×C₄	—	218	1	no	—
336	C₂×C₄	—	218	1	no	—
337	C₂×C₄	—	218	2	no	—
338	C₂×C₄	—	218	1	no	—
339	C₂×C₄	—	218	1	no	—
340	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
341	C₈	—	109	2	no	—
342	C₈	—	218	2	no	—
343	C₈	C₂²×#89137	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	109	1	no	—
346	C₂²	—	109	1	no	—
347	C₂²	—	436	1	no	—
348	C₂²	—	436	1	no	—
349	C₂²	—	218	1	no	—
350	C₂²	—	218	1	no	—
351	C₂²	—	218	1	no	—
352	C₂²	D₄×#89137	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	109	2	no	—
356	C₄	—	109	1	no	—
357	C₄	—	109	1	no	—
358	C₄	—	436	1	no	—
359	C₄	—	436	2	no	—
360	C₄	—	436	1	no	—
361	C₄	—	218	1	no	—
362	C₄	—	218	1	no	—
363	C₄	—	218	1	no	—
364	C₄	—	218	1	no	—
365	C₄	—	218	2	no	—
366	C₄	D₄×#89137	1	1	yes	characteristic
367	C₄	C₂³×#89137	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	109	1	no	—
371	C₂	—	109	1	no	—
372	C₂	—	436	1	no	—
373	C₂	—	218	1	no	—
374	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	#137×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	#41×D₁₀₉	C₂²
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	#137	#89137
hypocenter	C₁₀₉	C₄×#137
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×#89137	C₂	semidirect
3. #20296066	C₂	semidirect
4. #20296065	C₂	semidirect
5. Q₁₆×#89137	C₂	semidirect
6. SD₁₆×#89137	C₂	semidirect
7. #20296037	C₂	semidirect
8. #20296019	C₂	semidirect
10. M₁₆×#89137	C₂	semidirect
40. #16812732	C₄	semidirect
41. #16812731	C₄	semidirect
42. #16812730	C₄	semidirect
43. #16812729	C₄	semidirect
44. #16812728	C₄	semidirect
48. #137×C₁₀₉	C₄	semidirect
83. C₂×Dic₂₁₈	C₂×C₄	semidirect
86. Q₈×D₁₀₉	C₂×C₄	semidirect
94. Dic₄₃₆	C₂×C₄	semidirect
95. #13182019	C₂×C₄	semidirect
96. #13182019	C₂×C₄	semidirect
97. #13182018	C₂×C₄	semidirect
98. #13182017	C₂×C₄	semidirect
99. #13182016	C₂×C₄	semidirect
100. #13182013	C₂×C₄	semidirect
101. #13182012	C₂×C₄	semidirect
104. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
105. Q₁₆×C₁₀₉	C₂×C₄	semidirect
106. SD₁₆×C₁₀₉	C₂×C₄	semidirect
107. M₁₆×C₁₀₉	C₂×C₄	semidirect
171. #89137	#137	direct (N × H)
188. C₁₀₉	C₄×#137	semidirect
207. #137	#89137	direct (N × H)
250. C₂×Q₈	C₂×#89137	semidirect
264. Q₁₆	C₂×#89137	semidirect
268. SD₁₆	C₂×#89137	semidirect
273. 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
#41×#89137	2
C₂×Q₈×#89137	2
#20296066	2
#20296065	2
Q₁₆×#89137	2
SD₁₆×#89137	2
#20296037	2
#20296019	2
#20296017	2
M₁₆×#89137	2
#137×D₁₀₉	2
C₄×#137	109