Subgroups — #11868016

Order 1152.

678 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	#10589850	C₂	1	2	yes	maximal, complemented
2	C₂×#63467	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63464	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588814	C₂	1	2	yes	maximal, complemented
6	#10588578	C₂	1	4	yes	maximal, complemented
7	#10588577	C₂	1	2	yes	maximal, complemented
8	#10588576	C₂	1	2	yes	maximal, complemented
9	#79256	—	3	1	no	maximal
10	C₂×#3697	C₂²	1	1	yes	characteristic
11	C₂×#3697	C₂²	1	2	yes	—
12	C₂×C₉×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×#3667	C₂²	1	1	yes	characteristic
15	#63471	—	2	2	no	—
16	#63471	C₂²	1	4	yes	—
17	#63470	—	2	2	no	—
18	#63470	C₂²	1	4	yes	—
19	#63467	—	2	4	no	—
20	#63467	—	2	1	no	—
21	#63467	—	2	1	no	—
22	#63467	—	2	1	no	—
23	#63467	—	2	1	no	—
24	C₂×D₇₂	C₂²	1	1	yes	characteristic, complemented
25	C₂×#3664	C₂²	1	2	yes	complemented
26	#63464	—	2	1	no	—
27	#63464	—	2	1	no	—
28	#63464	C₂²	1	2	yes	—
29	#63464	C₂²	1	2	yes	—
30	C₂×#3663	—	2	1	no	—
31	C₂×C₈×D₉	—	2	1	no	—
32	C₂×Dic₃₆	C₂²	1	1	yes	characteristic, complemented
33	C₉×#108	C₂²	1	2	yes	complemented
34	C₉×#108	C₂²	1	2	yes	complemented
35	#63383	C₂²	1	4	yes	complemented
36	#63382	C₂²	1	4	yes	complemented
37	#5748	—	3	2	no	—
38	C₂×#904	—	3	1	no	—
39	C₂×#901	—	3	1	no	—
40	C₂×C₃×#108	D₃	1	1	yes	characteristic
41	#5132	—	3	2	no	—
42	#4898	—	3	4	no	—
43	#4897	—	3	2	no	—
44	#4896	—	3	2	no	—
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×#611	—	2	1	no	—
47	C₂×#611	—	2	2	no	—
48	#3697	—	2	1	no	—
49	#3697	—	2	1	no	—
50	#3697	—	2	2	no	—
51	#3697	—	2	2	no	—
52	#3697	—	2	2	no	—
53	#3697	—	2	2	no	—
54	#3697	D₄	1	2	yes	—
55	#3697	C₂³	1	2	yes	—
56	C₂×D₃₆	D₄	1	1	yes	characteristic
57	C₂×D₃₆	C₂³	1	2	yes	—
58	C₂×C₄×D₉	—	2	1	no	—
59	C₂×C₄×D₉	—	2	2	no	—
60	C₂×Dic₁₈	D₄	1	1	yes	characteristic
61	C₂×Dic₁₈	C₂³	1	2	yes	—
62	C₉×M₁₆	—	2	2	no	—
63	C₉×M₁₆	C₂³	1	4	yes	—
64	C₂×C₈×C₉	—	2	2	no	—
65	C₂×C₈×C₉	D₄	1	1	yes	characteristic
66	C₂×C₈×C₉	D₄	1	1	yes	characteristic
67	C₂×C₈×C₉	D₄	1	1	yes	characteristic
68	C₂×C₈×C₉	D₄	1	1	yes	characteristic
69	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
71	#3667	—	2	1	no	—
72	#3667	C₂³	1	2	yes	—
73	C₂×C₉⋊C₈	—	2	1	no	—
74	D₇₂	—	2	4	no	—
75	D₇₂	—	2	1	no	—
76	D₇₂	—	2	1	no	—
77	#3664	—	2	4	no	—
78	#3664	—	2	4	no	—
79	#3664	—	2	2	no	—
80	#3664	—	2	2	no	—
81	#3663	—	2	2	no	—
82	#3663	—	2	2	no	—
83	C₈×D₉	—	2	2	no	—
84	C₈×D₉	—	2	2	no	—
85	Dic₃₆	—	2	4	no	—
86	Dic₃₆	—	2	1	no	—
87	Dic₃₆	—	2	1	no	—
88	#3067	—	9	1	no	Sylow
89	C₂×#235	—	3	1	no	—
90	C₂×#235	—	3	2	no	—
91	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
92	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
93	C₂×#208	—	3	1	no	—
94	#909	—	3	4	no	—
95	#909	—	6	2	no	—
96	#908	—	3	4	no	—
97	#908	—	6	2	no	—
98	C₂×Dic₁₂	—	3	1	no	—
99	#904	—	6	4	no	—
100	#904	—	6	1	no	—
101	#904	—	6	1	no	—
102	#904	—	6	1	no	—
103	#904	—	6	1	no	—
104	C₂×D₂₄	—	3	1	no	—
105	C₂×#204	—	3	2	no	—
106	#901	—	3	2	no	—
107	#901	—	3	2	no	—
108	#901	—	6	1	no	—
109	#901	—	6	1	no	—
110	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
111	C₂×D₃×C₈	—	6	1	no	—
112	C₃×#108	C₂×D₃	1	2	yes	—
113	C₃×#108	C₂×D₃	1	2	yes	—
114	#824	—	3	4	no	—
115	#823	—	3	4	no	—
116	C₂³×C₉	#41	1	1	yes	characteristic
117	C₂²×D₉	—	2	1	no	—
118	C₂²×D₉	—	2	2	no	—
119	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₉	#41	1	1	yes	characteristic
124	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
125	#611	—	2	2	no	—
126	#611	—	2	2	no	—
127	#611	—	4	2	no	—
128	#611	—	4	2	no	—
129	C₂×Dic₉	—	2	1	no	—
130	C₂×Dic₉	—	2	2	no	—
131	D₃₆	—	2	1	no	—
132	D₃₆	—	2	2	no	—
133	D₃₆	—	2	2	no	—
134	D₃₆	C₂×D₄	1	2	yes	—
135	C₄×D₉	—	2	2	no	—
136	C₄×D₉	—	2	2	no	—
137	C₄×D₉	—	4	2	no	—
138	C₄×D₉	—	4	2	no	—
139	Dic₁₈	—	2	1	no	—
140	Dic₁₈	—	2	2	no	—
141	Dic₁₈	—	2	2	no	—
142	Dic₁₈	C₂×D₄	1	2	yes	—
143	C₈×C₉	—	2	4	no	—
144	C₈×C₉	C₂×D₄	1	1	yes	characteristic
145	C₈×C₉	C₂×D₄	1	1	yes	characteristic
146	C₈×C₉	C₂×D₄	1	1	yes	characteristic
147	C₈×C₉	C₂×D₄	1	1	yes	characteristic
148	C₉⋊C₈	—	2	2	no	—
149	#575	—	9	2	no	—
150	C₂×#135	—	9	1	no	—
151	C₂×#131	—	9	1	no	—
152	#473	—	9	2	no	—
153	#472	—	9	2	no	—
154	#471	—	9	4	no	—
155	C₂×#108	D₉	1	1	yes	characteristic, complemented
156	#413	—	9	2	no	—
157	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
158	C₂×#67	—	6	1	no	—
159	C₂×#67	—	6	2	no	—
160	#235	—	3	2	no	—
161	#235	—	3	2	no	—
162	#235	—	6	1	no	—
163	#235	—	6	1	no	—
164	#235	—	6	2	no	—
165	#235	—	6	2	no	—
166	#235	—	6	2	no	—
167	#235	—	6	2	no	—
168	C₂×D₁₂	—	3	1	no	—
169	C₂×D₁₂	—	3	2	no	—
170	C₂×C₄×D₃	—	6	1	no	—
171	C₂×C₄×D₃	—	6	2	no	—
172	C₂×Dic₆	—	3	1	no	—
173	C₂×Dic₆	—	3	2	no	—
174	C₃×M₁₆	—	2	2	no	—
175	C₃×M₁₆	C₂²×D₃	1	4	yes	—
176	C₂×C₃×C₈	—	2	2	no	—
177	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
178	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
179	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
180	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
181	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
182	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
183	#208	—	3	2	no	—
184	#208	—	6	1	no	—
185	C₂×C₃⋊C₈	—	6	1	no	—
186	Dic₁₂	—	6	4	no	—
187	Dic₁₂	—	6	1	no	—
188	Dic₁₂	—	6	1	no	—
189	D₂₄	—	6	4	no	—
190	D₂₄	—	6	1	no	—
191	D₂₄	—	6	1	no	—
192	#204	—	6	4	no	—
193	#204	—	6	4	no	—
194	#204	—	6	2	no	—
195	#204	—	6	2	no	—
196	ΓL(1,𝔽₂₅)	—	6	2	no	—
197	ΓL(1,𝔽₂₅)	—	6	2	no	—
198	D₃×C₈	—	6	2	no	—
199	D₃×C₈	—	6	2	no	—
200	C₂²×C₉	—	4	1	no	—
201	C₂²×C₉	#121	1	1	yes	characteristic
202	C₂²×C₉	#121	1	1	yes	characteristic
203	C₂²×C₉	C₂×#41	1	1	yes	characteristic
204	C₂×D₉	—	2	2	no	—
205	C₂×D₉	—	4	2	no	—
206	C₂×D₉	—	4	2	no	—
207	C₂×D₉	—	8	2	no	—
208	C₄×C₉	#121	1	1	yes	characteristic
209	C₄×C₉	#121	1	1	yes	characteristic
210	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
211	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
212	Dic₉	—	2	2	no	—
213	Dic₉	—	4	2	no	—
214	C₂×#41	—	9	1	no	—
215	C₂×#41	—	9	2	no	—
216	#137	—	9	4	no	—
217	#137	—	18	2	no	—
218	AGL(1,ℤ/8)	—	9	4	no	—
219	AGL(1,ℤ/8)	—	18	2	no	—
220	#135	—	18	4	no	—
221	#135	—	18	1	no	—
222	#135	—	18	1	no	—
223	#135	—	18	1	no	—
224	#135	—	18	1	no	—
225	C₂×Q₁₆	—	9	1	no	—
226	C₂×SD₁₆	—	9	2	no	—
227	C₂×D₈	—	9	1	no	—
228	#131	—	9	2	no	—
229	#131	—	9	2	no	—
230	#131	—	18	1	no	—
231	#131	—	18	1	no	—
232	C₂×M₁₆	—	9	1	no	—
233	C₂×M₁₆	—	18	1	no	—
234	C₂×M₁₆	C₂×D₉	1	2	yes	—
235	C₂²×C₈	—	18	1	no	—
236	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
237	#108	C₂×D₉	1	2	yes	complemented
238	#108	C₂×D₉	1	2	yes	complemented
239	#101	—	9	4	no	—
240	#100	—	9	4	no	—
241	C₂³×C₃	#239	1	1	yes	characteristic
242	C₂²×D₃	—	6	1	no	—
243	C₂²×D₃	—	6	2	no	—
244	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
246	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
247	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
248	C₂×C₃×C₄	#239	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
250	#67	—	6	2	no	—
251	#67	—	6	2	no	—
252	#67	—	12	2	no	—
253	#67	—	12	2	no	—
254	C₂×Dic₃	—	6	1	no	—
255	C₂×Dic₃	—	6	2	no	—
256	D₁₂	—	3	2	no	—
257	D₁₂	—	6	1	no	—
258	D₁₂	—	6	2	no	—
259	D₁₂	—	6	2	no	—
260	C₄×D₃	—	6	2	no	—
261	C₄×D₃	—	6	2	no	—
262	C₄×D₃	—	12	2	no	—
263	C₄×D₃	—	12	2	no	—
264	Dic₆	—	3	2	no	—
265	Dic₆	—	6	1	no	—
266	Dic₆	—	6	2	no	—
267	Dic₆	—	6	2	no	—
268	C₃×C₈	—	2	4	no	—
269	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
270	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
271	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
272	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
273	C₃⋊C₈	—	6	2	no	—
274	C₂×C₉	—	2	1	no	—
275	C₂×C₉	—	2	1	no	—
276	C₂×C₉	—	2	1	no	—
277	C₂×C₉	C₂×#121	1	1	yes	characteristic
278	D₉	—	4	2	no	—
279	D₉	—	8	2	no	—
280	#41	—	9	2	no	—
281	#41	—	9	2	no	—
282	#41	—	18	1	no	—
283	#41	—	18	1	no	—
284	#41	—	18	2	no	—
285	#41	—	18	2	no	—
286	#41	—	18	2	no	—
287	#41	—	18	2	no	—
288	C₂×Q₈	—	9	1	no	—
289	C₂×Q₈	—	9	2	no	—
290	C₂×D₄	—	9	1	no	—
291	C₂×D₄	—	9	2	no	—
292	C₂×D₄	—	18	1	no	—
293	C₂×D₄	—	18	2	no	—
294	C₂²×C₄	—	18	1	no	—
295	C₂²×C₄	—	18	2	no	—
296	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
297	Q₁₆	—	18	4	no	—
298	Q₁₆	—	18	1	no	—
299	Q₁₆	—	18	1	no	—
300	SD₁₆	—	18	4	no	—
301	SD₁₆	—	18	4	no	—
302	SD₁₆	—	18	2	no	—
303	SD₁₆	—	18	2	no	—
304	D₈	—	18	4	no	—
305	D₈	—	18	1	no	—
306	D₈	—	18	1	no	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	9	2	no	—
309	M₁₆	—	18	1	no	—
310	M₁₆	—	18	2	no	—
311	M₁₆	—	18	2	no	—
312	M₁₆	C₂²×D₉	1	4	yes	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	18	1	no	—
315	C₂×C₈	—	18	2	no	—
316	C₂×C₈	—	18	2	no	—
317	C₂×C₈	D₃₆	1	1	yes	characteristic
318	C₂×C₈	D₃₆	1	1	yes	characteristic
319	C₂×C₈	D₃₆	1	1	yes	characteristic
320	C₂×C₈	D₃₆	1	1	yes	characteristic
321	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
322	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
323	C₂²×C₃	—	4	1	no	—
324	C₂²×C₃	#895	1	1	yes	characteristic
325	C₂²×C₃	#895	1	1	yes	characteristic
326	C₂²×C₃	C₂×#239	1	1	yes	characteristic
327	C₂×D₃	—	6	2	no	—
328	C₂×D₃	—	12	2	no	—
329	C₂×D₃	—	12	2	no	—
330	C₂×D₃	—	24	2	no	—
331	C₃×C₄	#895	1	1	yes	characteristic
332	C₃×C₄	#895	1	1	yes	characteristic
333	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
334	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
335	Dic₃	—	6	2	no	—
336	Dic₃	—	12	2	no	—
337	C₉	#3067	1	1	yes	characteristic, complemented, Sylow
338	C₂³	—	18	1	no	—
339	C₂³	—	18	2	no	—
340	C₂³	#3701	1	1	yes	characteristic
341	Q₈	—	9	2	no	—
342	Q₈	—	18	1	no	—
343	Q₈	—	18	2	no	—
344	Q₈	—	18	2	no	—
345	D₄	—	9	2	no	—
346	D₄	—	18	2	no	—
347	D₄	—	18	2	no	—
348	D₄	—	18	2	no	—
349	D₄	—	18	2	no	—
350	D₄	—	18	1	no	—
351	D₄	—	36	2	no	—
352	D₄	—	36	2	no	—
353	C₂×C₄	—	18	1	no	—
354	C₂×C₄	—	18	2	no	—
355	C₂×C₄	—	18	2	no	—
356	C₂×C₄	—	18	2	no	—
357	C₂×C₄	—	36	2	no	—
358	C₂×C₄	—	36	2	no	—
359	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
360	C₂×C₄	C₂×D₃₆	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	#3701	1	1	yes	characteristic
364	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
365	C₈	—	2	4	no	—
366	C₈	—	18	2	no	—
367	C₈	C₂×D₃₆	1	1	yes	characteristic
368	C₈	C₂×D₃₆	1	1	yes	characteristic
369	C₈	C₂×D₃₆	1	1	yes	characteristic
370	C₈	C₂×D₃₆	1	1	yes	characteristic
371	C₂×C₃	—	2	1	no	—
372	C₂×C₃	—	2	1	no	—
373	C₂×C₃	—	2	1	no	—
374	C₂×C₃	C₂×#895	1	1	yes	characteristic
375	D₃	—	12	2	no	—
376	D₃	—	24	2	no	—
377	C₂²	—	4	1	no	—
378	C₂²	—	18	2	no	—
379	C₂²	—	36	2	no	—
380	C₂²	—	36	2	no	—
381	C₂²	—	72	2	no	—
382	C₂²	#63457	1	1	yes	characteristic
383	C₂²	#63457	1	1	yes	characteristic
384	C₂²	C₂×#3701	1	1	yes	characteristic
385	C₄	—	18	2	no	—
386	C₄	—	36	2	no	—
387	C₄	#63457	1	1	yes	characteristic
388	C₄	#63457	1	1	yes	characteristic
389	C₄	C₂²×D₃₆	1	1	yes	characteristic
390	C₄	C₂×D₄×D₉	1	1	yes	characteristic
391	C₃	#79256	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	36	2	no	—
396	C₂	—	72	2	no	—
397	C₂	C₂×#63457	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63457
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#108	C₂
socle	C₂×C₃	C₂×#895
hypercenter	C₂×#108	D₉
hypocenter	C₉	#3067
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
1. #10589850	C₂	semidirect
2. C₂×#63467	C₂	semidirect
3. C₂×#63464	C₂	semidirect
4. C₂×C₉×#108	C₂	semidirect
5. #10588814	C₂	semidirect
6. #10588578	C₂	semidirect
7. #10588577	C₂	semidirect
8. #10588576	C₂	semidirect
24. C₂×D₇₂	C₂²	semidirect
25. C₂×#3664	C₂²	semidirect
32. C₂×Dic₃₆	C₂²	semidirect
33. C₉×#108	C₂²	semidirect
34. C₉×#108	C₂²	semidirect
35. #63383	C₂²	semidirect
36. #63382	C₂²	semidirect
155. C₂×#108	D₉	semidirect
237. #108	C₂×D₉	semidirect
238. #108	C₂×D₉	semidirect
337. C₉	#3067	semidirect

Maximal subgroups

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

subgroup	index
#10589850	2
C₂×#63467	2
C₂×#63464	2
C₂×C₉×#108	2
#10588814	2
#10588578	2
#10588577	2
#10588576	2
#79256	3