Subgroups — #11861133

Order 1152.

687 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589925	C₂	1	1	yes	maximal, characteristic
2	#10589921	C₂	1	1	yes	maximal, characteristic, complemented
3	#10589847	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588799	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#3667	C₂	1	1	yes	maximal, characteristic, complemented
8	#10588504	C₂	1	4	yes	maximal, complemented
9	#10588503	C₂	1	4	yes	maximal, complemented
10	#78980	—	3	1	no	maximal
11	C₂×C₉×#41	C₂²	1	1	yes	characteristic
12	C₂×#3697	C₂²	1	1	yes	characteristic
13	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	#63513	C₂²	1	4	yes	—
16	#63512	C₄	1	2	yes	complemented
17	#63511	C₄	1	2	yes	complemented
18	#63510	—	2	2	no	—
19	#63510	C₂²	1	4	yes	complemented
20	#63505	C₄	1	2	yes	complemented
21	#63495	C₄	1	2	yes	complemented
22	C₂×#3667	—	2	2	no	—
23	C₂×#3667	C₂²	1	1	yes	characteristic
24	C₂×#3667	C₂²	1	1	yes	characteristic
25	C₂×#3667	C₂²	1	1	yes	characteristic
26	#63469	C₂²	1	4	yes	complemented
27	M₁₆×D₉	—	2	2	no	—
28	#63464	—	2	2	no	—
29	C₉×#104	C₂²	1	2	yes	complemented
30	C₉×#104	C₂²	1	2	yes	complemented
31	#63381	C₂²	1	4	yes	complemented
32	#63364	C₂²	1	2	yes	complemented
33	#63364	C₂²	1	2	yes	complemented
34	#63362	C₂²	1	2	yes	complemented
35	C₄×C₉⋊C₈	C₂²	1	2	yes	complemented
36	#5822	—	3	1	no	—
37	#5819	—	3	1	no	—
38	#5745	—	3	1	no	—
39	C₂×C₃×#104	D₃	1	1	yes	characteristic
40	#5124	—	3	1	no	—
41	C₂×#805	—	3	1	no	—
42	C₄×#208	—	3	1	no	—
43	#4824	—	3	4	no	—
44	#4823	—	3	4	no	—
45	C₉×#41	—	2	1	no	—
46	C₉×#41	—	2	1	no	—
47	C₉×#41	C₂×C₄	1	2	yes	—
48	C₉×#41	C₂³	1	2	yes	—
49	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₂×#611	—	2	1	no	—
56	#3697	—	2	1	no	—
57	#3697	—	2	1	no	—
58	#3697	C₂×C₄	1	2	yes	—
59	#3697	C₂³	1	2	yes	—
60	C₂×D₃₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₉	—	2	1	no	—
62	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
63	C₉×M₁₆	—	2	1	no	—
64	C₉×M₁₆	C₂³	1	2	yes	—
65	C₂×C₈×C₉	—	2	1	no	—
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	C₂³	1	1	yes	characteristic
68	C₄²×C₉	C₂³	1	1	yes	characteristic
69	#3675	C₂×C₄	1	4	yes	complemented
70	#3674	C₂×C₄	1	4	yes	complemented
71	#3673	C₂×C₄	1	4	yes	complemented
72	#3672	C₂×C₄	1	4	yes	complemented
73	#3667	—	2	1	no	—
74	#3667	—	2	4	no	—
75	#3667	C₂×C₄	1	2	yes	—
76	#3667	C₂×C₄	1	2	yes	—
77	#3667	D₄	1	2	yes	—
78	#3667	D₄	1	2	yes	—
79	#3667	C₂³	1	2	yes	—
80	C₂×C₉⋊C₈	—	2	1	no	—
81	C₂×C₉⋊C₈	—	2	4	no	—
82	C₂×C₉⋊C₈	C₂³	1	2	yes	—
83	C₂×C₉⋊C₈	C₂³	1	2	yes	—
84	#3663	—	2	4	no	—
85	C₈×D₉	—	2	4	no	—
86	#2956	—	9	1	no	Sylow
87	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
88	C₂×#235	—	3	1	no	—
89	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
90	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
91	#951	—	3	2	no	—
92	#950	—	3	4	no	—
93	#949	—	3	2	no	—
94	#948	—	3	4	no	—
95	#948	—	6	2	no	—
96	#942	—	3	2	no	—
97	#932	—	3	2	no	—
98	C₂×#208	—	3	1	no	—
99	C₂×#208	—	3	1	no	—
100	C₂×#208	—	3	1	no	—
101	C₂×#208	—	6	2	no	—
102	#907	—	3	4	no	—
103	D₃×M₁₆	—	6	2	no	—
104	#901	—	6	2	no	—
105	C₃×#104	C₂×D₃	1	2	yes	—
106	C₃×#104	C₂×D₃	1	2	yes	—
107	#822	—	3	4	no	—
108	#805	—	3	2	no	—
109	#805	—	3	2	no	—
110	#803	—	3	2	no	—
111	C₄×C₃⋊C₈	—	3	2	no	—
112	C₂³×C₉	—	2	1	no	—
113	C₂³×C₉	#41	1	1	yes	characteristic
114	C₂²×D₉	—	2	1	no	—
115	Q₈×C₉	—	2	1	no	—
116	Q₈×C₉	C₂²×C₄	1	2	yes	—
117	D₄×C₉	—	2	2	no	—
118	D₄×C₉	—	2	2	no	—
119	D₄×C₉	—	2	1	no	—
120	D₄×C₉	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	2	no	—
124	C₂×C₄×C₉	—	2	2	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	4	2	no	—
127	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₉	#41	1	1	yes	characteristic
132	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
133	#611	—	2	2	no	—
134	#611	—	2	2	no	—
135	C₂×Dic₉	—	2	1	no	—
136	D₃₆	—	2	1	no	—
137	D₃₆	C₂²×C₄	1	2	yes	—
138	C₄×D₉	—	2	2	no	—
139	C₄×D₉	—	2	2	no	—
140	Dic₁₈	—	2	1	no	—
141	Dic₁₈	C₂²×C₄	1	2	yes	—
142	C₈×C₉	—	2	2	no	—
143	C₉⋊C₈	—	2	2	no	—
144	C₉⋊C₈	C₂²×C₄	1	4	yes	—
145	C₉⋊C₈	C₂×D₄	1	4	yes	—
146	#575	—	9	1	no	—
147	#568	—	9	1	no	—
148	#568	—	9	1	no	—
149	#444	—	9	4	no	—
150	#443	—	9	4	no	—
151	#430	—	9	1	no	—
152	C₂×#104	—	9	1	no	—
153	C₂×#104	D₉	1	1	yes	characteristic, complemented
154	C₄×M₁₆	—	9	1	no	—
155	C₃×#41	—	2	1	no	—
156	C₃×#41	—	2	1	no	—
157	C₃×#41	C₄×D₃	1	2	yes	—
158	C₃×#41	C₂²×D₃	1	2	yes	—
159	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
160	C₂×C₃×D₄	—	2	1	no	—
161	C₂×C₃×D₄	C₄×D₃	1	1	yes	characteristic
162	C₂²×C₃×C₄	—	2	1	no	—
163	C₂²×C₃×C₄	—	2	1	no	—
164	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
165	C₂×#67	—	6	1	no	—
166	#235	—	3	2	no	—
167	#235	—	3	2	no	—
168	#235	—	6	1	no	—
169	#235	—	6	1	no	—
170	C₂×D₁₂	—	3	1	no	—
171	C₂×C₄×D₃	—	6	1	no	—
172	C₂×Dic₆	—	3	1	no	—
173	C₃×M₁₆	—	2	1	no	—
174	C₃×M₁₆	C₂²×D₃	1	2	yes	—
175	C₂×C₃×C₈	—	2	1	no	—
176	C₃×C₄²	—	2	1	no	—
177	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
179	#216	—	3	4	no	—
180	#215	—	3	4	no	—
181	#214	—	3	4	no	—
182	#213	—	3	4	no	—
183	#208	—	3	2	no	—
184	#208	—	3	2	no	—
185	#208	—	3	2	no	—
186	#208	—	3	2	no	—
187	#208	—	3	2	no	—
188	#208	—	6	4	no	—
189	#208	—	6	1	no	—
190	C₂×C₃⋊C₈	—	3	2	no	—
191	C₂×C₃⋊C₈	—	3	2	no	—
192	C₂×C₃⋊C₈	—	6	4	no	—
193	C₂×C₃⋊C₈	—	6	1	no	—
194	ΓL(1,𝔽₂₅)	—	6	4	no	—
195	D₃×C₈	—	6	4	no	—
196	C₂²×C₉	—	2	2	no	—
197	C₂²×C₉	—	4	1	no	—
198	C₂²×C₉	—	8	1	no	—
199	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
201	C₂²×C₉	C₂×#41	1	1	yes	characteristic
202	C₂×D₉	—	2	2	no	—
203	C₂×D₉	—	8	1	no	—
204	C₄×C₉	—	2	2	no	—
205	C₄×C₉	—	2	2	no	—
206	C₄×C₉	—	4	1	no	—
207	C₄×C₉	C₄×D₄	1	1	yes	characteristic
208	C₄×C₉	C₄×D₄	1	1	yes	characteristic
209	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
210	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
211	Dic₉	—	2	2	no	—
212	C₂×#41	—	9	1	no	—
213	C₂×#41	C₂×D₉	1	1	yes	characteristic
214	#137	—	9	2	no	—
215	#137	—	9	2	no	—
216	AGL(1,ℤ/8)	—	9	2	no	—
217	AGL(1,ℤ/8)	—	9	2	no	—
218	#135	—	9	4	no	—
219	#131	—	9	4	no	—
220	#131	—	9	4	no	—
221	#131	—	18	2	no	—
222	#131	—	18	2	no	—
223	C₂×M₁₆	—	9	1	no	—
224	C₂×M₁₆	—	9	1	no	—
225	C₂×M₁₆	—	9	1	no	—
226	C₂×M₁₆	—	18	2	no	—
227	C₂×M₁₆	—	18	2	no	—
228	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
229	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
230	#108	—	9	4	no	—
231	#104	—	9	2	no	—
232	#104	—	9	2	no	—
233	#104	C₂×D₉	1	2	yes	complemented
234	#104	C₂×D₉	1	2	yes	complemented
235	#97	—	9	2	no	—
236	C₄×C₈	—	9	2	no	—
237	C₂³×C₃	—	2	1	no	—
238	C₂³×C₃	#237	1	1	yes	characteristic
239	C₂²×D₃	—	6	1	no	—
240	C₃×Q₈	—	2	1	no	—
241	C₃×Q₈	C₂×C₄×D₃	1	2	yes	—
242	C₃×D₄	—	2	2	no	—
243	C₃×D₄	—	2	2	no	—
244	C₃×D₄	—	2	1	no	—
245	C₃×D₄	C₂×C₄×D₃	1	2	yes	—
246	C₂×C₃×C₄	—	2	1	no	—
247	C₂×C₃×C₄	—	2	1	no	—
248	C₂×C₃×C₄	—	2	2	no	—
249	C₂×C₃×C₄	—	2	2	no	—
250	C₂×C₃×C₄	—	2	1	no	—
251	C₂×C₃×C₄	—	4	2	no	—
252	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
253	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
254	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
255	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
256	C₂×C₃×C₄	#237	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
258	#67	—	6	2	no	—
259	#67	—	6	2	no	—
260	C₂×Dic₃	—	6	1	no	—
261	D₁₂	—	3	2	no	—
262	D₁₂	—	6	1	no	—
263	C₄×D₃	—	6	2	no	—
264	C₄×D₃	—	6	2	no	—
265	Dic₆	—	3	2	no	—
266	Dic₆	—	6	1	no	—
267	C₃×C₈	—	2	2	no	—
268	C₃⋊C₈	—	3	4	no	—
269	C₃⋊C₈	—	3	4	no	—
270	C₃⋊C₈	—	6	2	no	—
271	C₂×C₉	—	2	1	no	—
272	C₂×C₉	—	2	1	no	—
273	C₂×C₉	—	2	1	no	—
274	C₂×C₉	—	4	2	no	—
275	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
276	D₉	—	4	2	no	—
277	#41	—	2	1	no	—
278	#41	—	2	1	no	—
279	#41	—	9	2	no	—
280	#41	—	9	2	no	—
281	#41	—	18	1	no	—
282	#41	—	18	1	no	—
283	#41	C₄×D₉	1	2	yes	—
284	#41	C₂²×D₉	1	2	yes	—
285	C₂×Q₈	—	9	1	no	—
286	C₂×Q₈	C₄×D₉	1	1	yes	characteristic
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	9	1	no	—
289	C₂×D₄	—	18	1	no	—
290	C₂×D₄	C₄×D₉	1	1	yes	characteristic
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	18	1	no	—
294	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
295	Q₁₆	—	9	4	no	—
296	SD₁₆	—	9	4	no	—
297	SD₁₆	—	9	4	no	—
298	D₈	—	9	4	no	—
299	M₁₆	—	2	1	no	—
300	M₁₆	—	9	2	no	—
301	M₁₆	—	9	2	no	—
302	M₁₆	—	9	2	no	—
303	M₁₆	—	9	2	no	—
304	M₁₆	—	9	2	no	—
305	M₁₆	—	18	4	no	—
306	M₁₆	—	18	4	no	—
307	M₁₆	—	18	1	no	—
308	M₁₆	C₂²×D₉	1	2	yes	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	9	2	no	—
311	C₂×C₈	—	9	2	no	—
312	C₂×C₈	—	18	4	no	—
313	C₂×C₈	—	18	4	no	—
314	C₂×C₈	—	18	1	no	—
315	C₄²	—	2	1	no	—
316	C₄²	C₂²×D₉	1	1	yes	characteristic
317	C₄²	C₂²×D₉	1	1	yes	characteristic
318	C₂²×C₃	—	2	2	no	—
319	C₂²×C₃	—	4	1	no	—
320	C₂²×C₃	—	8	1	no	—
321	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
322	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
323	C₂²×C₃	C₂×#237	1	1	yes	characteristic
324	C₂×D₃	—	6	2	no	—
325	C₂×D₃	—	24	1	no	—
326	C₃×C₄	—	2	2	no	—
327	C₃×C₄	—	2	2	no	—
328	C₃×C₄	—	4	1	no	—
329	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
330	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
331	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
332	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
333	Dic₃	—	6	2	no	—
334	C₉	#2956	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	2	1	no	—
336	C₂³	—	18	1	no	—
337	C₂³	#3699	1	1	yes	characteristic
338	Q₈	—	2	1	no	—
339	Q₈	—	9	2	no	—
340	Q₈	—	18	1	no	—
341	Q₈	C₂×C₄×D₉	1	2	yes	—
342	D₄	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	2	1	no	—
345	D₄	—	9	2	no	—
346	D₄	—	18	2	no	—
347	D₄	—	18	2	no	—
348	D₄	—	18	1	no	—
349	D₄	C₂×C₄×D₉	1	2	yes	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	18	1	no	—
357	C₂×C₄	—	18	2	no	—
358	C₂×C₄	—	18	2	no	—
359	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
360	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
361	C₂×C₄	D₄×D₉	1	1	yes	characteristic
362	C₂×C₄	D₄×D₉	1	1	yes	characteristic
363	C₂×C₄	#3699	1	1	yes	characteristic
364	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	9	4	no	—
367	C₈	—	9	4	no	—
368	C₈	—	18	2	no	—
369	C₂×C₃	—	2	1	no	—
370	C₂×C₃	—	2	1	no	—
371	C₂×C₃	—	2	1	no	—
372	C₂×C₃	—	4	2	no	—
373	C₂×C₃	C₂×#67⋊C₄	1	1	yes	characteristic
374	D₃	—	12	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	18	2	no	—
379	C₂²	—	72	1	no	—
380	C₂²	#611⋊C₄	1	1	yes	characteristic
381	C₂²	#611⋊C₄	1	1	yes	characteristic
382	C₂²	C₂×#3699	1	1	yes	characteristic
383	C₄	—	2	2	no	—
384	C₄	—	2	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	18	2	no	—
387	C₄	#611⋊C₄	1	1	yes	characteristic
388	C₄	#611⋊C₄	1	1	yes	characteristic
389	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
390	C₄	C₂×D₄×D₉	1	1	yes	characteristic
391	C₃	#78980	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	4	2	no	—
396	C₂	—	36	2	no	—
397	C₂	C₂×#611⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₉
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#104	C₂
socle	C₂×C₃	C₂×#67⋊C₄
prime radical	#10589925	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#104	D₉
hypocenter	C₉	#2956
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #10589921	C₂	semidirect
3. #10589847	C₂	semidirect
4. C₂×C₉×#104	C₂	semidirect
5. #10588799	C₂	semidirect
6. C₂×#63364	C₂	semidirect
7. C₄×#3667	C₂	semidirect
8. #10588504	C₂	semidirect
9. #10588503	C₂	semidirect
16. #63512	C₄	semidirect
17. #63511	C₄	semidirect
19. #63510	C₂²	semidirect
20. #63505	C₄	semidirect
21. #63495	C₄	semidirect
26. #63469	C₂²	semidirect
29. C₉×#104	C₂²	semidirect
30. C₉×#104	C₂²	semidirect
31. #63381	C₂²	semidirect
32. #63364	C₂²	semidirect
33. #63364	C₂²	semidirect
34. #63362	C₂²	semidirect
35. C₄×C₉⋊C₈	C₂²	semidirect
69. #3675	C₂×C₄	semidirect
70. #3674	C₂×C₄	semidirect
71. #3673	C₂×C₄	semidirect
72. #3672	C₂×C₄	semidirect
153. C₂×#104	D₉	semidirect
233. #104	C₂×D₉	semidirect
234. #104	C₂×D₉	semidirect
334. C₉	#2956	semidirect

Maximal subgroups

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

subgroup	index
#10589925	2
#10589921	2
#10589847	2
C₂×C₉×#104	2
#10588799	2
C₂×#63364	2
C₄×#3667	2
#10588504	2
#10588503	2
#78980	3