Subgroups — #27184460

Order 12928.

1556 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₀₁	C₂	1	2	yes	maximal, complemented
2	#20265758	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265755	C₂	1	2	yes	maximal, complemented
4	#20265750	C₂	1	4	yes	maximal, complemented
5	#20265743	C₂	1	4	yes	maximal, complemented
6	#20265735	C₂	1	4	yes	maximal, complemented
7	#20265722	C₂	1	4	yes	maximal, complemented
8	#20265690	C₂	1	4	yes	maximal, complemented
9	#20265687	C₂	1	4	yes	maximal, complemented
10	C₂×#15692213	C₂	1	1	yes	maximal, characteristic, complemented
11	#537×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108075	C₂²	1	2	yes	—
13	C₂×#13108074	C₂²	1	4	yes	—
14	C₂×#13108074	C₂²	1	2	yes	complemented
15	C₂×#13108073	C₂²	1	2	yes	—
16	C₂×#13108073	C₂²	1	4	yes	complemented
17	#41×D₁₀₁	—	2	16	no	—
18	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	complemented
19	C₂×D₄×D₁₀₁	C₂²	1	4	yes	—
20	C₂×D₄×D₁₀₁	C₂²	1	2	yes	complemented
21	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	complemented
23	#15692254	C₂²	1	4	yes	complemented
24	#15692251	C₂²	1	4	yes	complemented
25	#15692241	C₂²	1	8	yes	complemented
26	#15692240	C₂²	1	8	yes	complemented
27	#15692237	C₂²	1	8	yes	complemented
28	#15692235	C₂²	1	4	yes	complemented
29	#15692234	C₂²	1	4	yes	complemented
30	#15692231	C₂²	1	4	yes	complemented
31	#15692230	C₂²	1	4	yes	complemented
32	#15692229	C₂²	1	4	yes	complemented
33	#15692229	C₂²	1	4	yes	complemented
34	#15692227	C₂²	1	8	yes	complemented
35	#15692226	C₂²	1	4	yes	complemented
36	#15692225	C₂²	1	8	yes	complemented
37	#15692223	C₂²	1	8	yes	complemented
38	D₄×Dic₁₀₁	C₂²	1	4	yes	complemented
39	#15692217	C₂²	1	4	yes	complemented
40	#15692215	C₂²	1	8	yes	complemented
41	#15692213	C₂²	1	4	yes	complemented
42	#15692213	C₂²	1	4	yes	complemented
43	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
44	C₂×#13108051	C₂²	1	2	yes	—
45	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
46	C₂×#41×C₁₀₁	C₂²	1	2	yes	complemented
47	C₂²⋊Q₈×C₁₀₁	C₂²	1	4	yes	complemented
48	#121×C₁₀₁	C₂²	1	4	yes	complemented
49	#121×C₁₀₁	C₂²	1	4	yes	complemented
50	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
51	C₂³×D₁₀₁	C₂³	1	4	yes	—
52	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
53	#13108075	—	2	8	no	—
54	#13108074	—	2	8	no	—
55	#13108074	—	2	16	no	—
56	#13108073	—	2	16	no	—
57	#13108073	—	2	8	no	—
58	C₂×Dic₂₀₂	C₂³	1	2	yes	—
59	C₂×Dic₂₀₂	C₂³	1	4	yes	—
60	Q₈×D₁₀₁	—	2	8	no	—
61	C₂×D₄₀₄	C₂³	1	4	yes	—
62	C₂×D₄₀₄	C₂³	1	2	yes	—
63	C₂×D₄₀₄	C₂³	1	4	yes	—
64	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
65	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
66	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
67	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
68	D₄×D₁₀₁	—	2	16	no	—
69	D₄×D₁₀₁	—	2	8	no	—
70	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
71	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
72	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₀₁	—	2	16	no	—
74	C₂×C₄×D₁₀₁	D₄	1	8	yes	complemented
75	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
76	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
77	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
78	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
79	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
80	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
81	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
82	#13108053	C₂³	1	8	yes	—
83	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
84	#13108051	C₂³	1	8	yes	—
85	#13108051	C₂³	1	8	yes	—
86	#13108051	C₂³	1	8	yes	—
87	C₄×Dic₁₀₁	C₂³	1	4	yes	—
88	C₄×Dic₁₀₁	C₂³	1	4	yes	—
89	#41×C₁₀₁	—	2	8	no	—
90	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
91	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
92	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
93	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
94	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
95	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
96	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
98	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
99	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
100	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
101	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
102	C₂²×D₁₀₁	—	2	8	no	—
103	C₂²×D₁₀₁	—	4	1	no	—
104	C₂²×D₁₀₁	—	4	1	no	—
105	C₂²×D₁₀₁	—	4	4	no	—
106	C₂²×D₁₀₁	—	4	4	no	—
107	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
108	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
109	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
110	Q₈×C₁₀₁	—	2	4	no	—
111	D₄×C₁₀₁	—	2	8	no	—
112	D₄×C₁₀₁	—	2	4	no	—
113	D₄×C₁₀₁	—	4	4	no	—
114	C₂×C₄×C₁₀₁	—	2	8	no	—
115	C₂×C₄×C₁₀₁	—	4	2	no	—
116	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
117	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
118	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
119	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
120	C₂²⋊D₁₀₁	—	2	8	no	—
121	C₂²⋊D₁₀₁	—	2	8	no	—
122	C₂²⋊D₁₀₁	—	2	8	no	—
123	C₂²⋊D₁₀₁	—	4	8	no	—
124	C₂×Dic₁₀₁	—	2	8	no	—
125	C₂×Dic₁₀₁	—	4	2	no	—
126	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
127	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
128	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
129	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
130	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
131	D₄₀₄	—	2	8	no	—
132	D₄₀₄	—	2	4	no	—
133	D₄₀₄	—	4	4	no	—
134	C₄×D₁₀₁	—	2	8	no	—
135	C₄×D₁₀₁	—	2	8	no	—
136	C₄×D₁₀₁	—	2	8	no	—
137	C₄×D₁₀₁	C₂×D₄	1	8	yes	—
138	C₄×D₁₀₁	C₂×D₄	1	8	yes	—
139	Dic₂₀₂	—	2	4	no	—
140	Dic₂₀₂	—	2	8	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	—	2	1	no	—
143	C₂²×C₁₀₁	—	2	4	no	—
144	C₂²×C₁₀₁	—	2	4	no	—
145	C₂²×C₁₀₁	—	2	4	no	—
146	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
147	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₀₁	—	2	2	no	—
149	C₂×D₁₀₁	—	2	2	no	—
150	C₂×D₁₀₁	—	2	4	no	—
151	C₂×D₁₀₁	—	2	4	no	—
152	C₂×D₁₀₁	—	2	4	no	—
153	C₂×D₁₀₁	—	4	4	no	—
154	C₂×D₁₀₁	—	4	16	no	—
155	C₂×D₁₀₁	C₂²×D₄	1	4	yes	—
156	C₄×C₁₀₁	—	2	4	no	—
157	C₄×C₁₀₁	—	2	4	no	—
158	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
159	Dic₁₀₁	—	2	4	no	—
160	Dic₁₀₁	—	2	4	no	—
161	Dic₁₀₁	C₂²×D₄	1	2	yes	—
162	Dic₁₀₁	C₂²×D₄	1	2	yes	—
163	C₂×C₁₀₁	—	2	2	no	—
164	C₂×C₁₀₁	—	4	4	no	—
165	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
166	C₂×C₁₀₁	#585	1	1	yes	characteristic
167	C₂×C₁₀₁	#585	1	1	yes	characteristic
168	D₁₀₁	—	2	4	no	—
169	D₁₀₁	—	4	4	no	—
170	#3449	—	101	1	no	maximal, Sylow
171	C₁₀₁	#3449	1	1	yes	characteristic, complemented, Sylow
172	C₂²×#41	—	101	2	no	—
173	#537	—	101	2	no	—
174	#537	—	101	1	no	—
175	#537	D₁₀₁	1	1	yes	characteristic, complemented
176	#536	—	101	4	no	—
177	#535	—	101	4	no	—
178	#534	—	101	4	no	—
179	#532	—	101	4	no	—
180	#525	—	101	4	no	—
181	#525	—	101	4	no	—
182	C₂×#117	—	101	1	no	—
183	C₂×#41	—	101	2	no	—
184	C₂×#41	—	101	2	no	—
185	C₂×#41	—	101	4	no	—
186	C₂×#41	—	101	4	no	—
187	C₂×#41	—	101	2	no	—
188	C₂×#41	—	202	16	no	—
189	C₂×#41	C₂×D₁₀₁	1	2	yes	complemented
190	C₂²×Q₈	—	101	2	no	—
191	C₂²×D₄	—	101	4	no	—
192	C₂²×D₄	—	101	2	no	—
193	C₂³×C₄	—	101	1	no	—
194	C₂³×C₄	—	101	4	no	—
195	C₄⋊Q₈	—	101	4	no	—
196	#127	—	101	4	no	—
197	#124	—	101	8	no	—
198	#123	—	101	8	no	—
199	#123	—	101	8	no	—
200	C₂²⋊Q₈	—	101	4	no	—
201	C₂²⋊Q₈	—	101	4	no	—
202	C₂²⋊Q₈	—	101	4	no	—
203	C₂²⋊Q₈	C₂×D₁₀₁	1	4	yes	complemented
204	#121	—	101	4	no	—
205	#121	—	101	8	no	—
206	#121	—	101	4	no	—
207	#121	—	101	4	no	—
208	#121	—	101	4	no	—
209	#121	C₂×D₁₀₁	1	4	yes	complemented
210	#121	C₂×D₁₀₁	1	4	yes	complemented
211	#120	—	101	8	no	—
212	#120	—	101	8	no	—
213	C₄×D₄	—	101	4	no	—
214	C₄×D₄	—	101	4	no	—
215	C₄×D₄	—	101	8	no	—
216	#117	—	101	4	no	—
217	#117	—	101	4	no	—
218	C₂×C₄⋊C₄	—	101	1	no	—
219	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
220	C₂×C₂²⋊C₄	—	101	2	no	—
221	C₂×C₄²	—	101	2	no	—
222	C₂⁴	—	101	1	no	—
223	C₂⁴	—	101	4	no	—
224	#41	—	2	8	no	—
225	#41	—	202	16	no	—
226	#41	—	202	16	no	—
227	#41	—	202	8	no	—
228	#41	—	202	8	no	—
229	#41	—	202	8	no	—
230	C₂×Q₈	—	101	2	no	—
231	C₂×Q₈	—	101	4	no	—
232	C₂×Q₈	—	202	8	no	—
233	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
234	C₂×D₄	—	101	8	no	—
235	C₂×D₄	—	101	4	no	—
236	C₂×D₄	—	101	4	no	—
237	C₂×D₄	—	101	4	no	—
238	C₂×D₄	—	101	4	no	—
239	C₂×D₄	—	101	4	no	—
240	C₂×D₄	—	101	2	no	—
241	C₂×D₄	—	202	8	no	—
242	C₂×D₄	—	202	16	no	—
243	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
244	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
245	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
246	C₂²×C₄	—	101	1	no	—
247	C₂²×C₄	—	101	8	no	—
248	C₂²×C₄	—	101	2	no	—
249	C₂²×C₄	—	101	2	no	—
250	C₂²×C₄	—	101	2	no	—
251	C₂²×C₄	—	101	4	no	—
252	C₂²×C₄	—	101	4	no	—
253	C₂²×C₄	—	101	4	no	—
254	C₂²×C₄	—	101	4	no	—
255	C₂²×C₄	—	202	16	no	—
256	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
257	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
258	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
259	C₄⋊C₄	—	101	8	no	—
260	C₄⋊C₄	—	101	4	no	—
261	C₄⋊C₄	—	101	4	no	—
262	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
263	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
264	C₂²⋊C₄	—	101	8	no	—
265	C₂²⋊C₄	—	101	8	no	—
266	C₂²⋊C₄	—	101	8	no	—
267	C₂²⋊C₄	—	101	8	no	—
268	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
269	C₄²	—	101	4	no	—
270	C₄²	—	101	4	no	—
271	C₂³	—	101	4	no	—
272	C₂³	—	101	2	no	—
273	C₂³	—	101	4	no	—
274	C₂³	—	404	1	no	—
275	C₂³	—	404	1	no	—
276	C₂³	—	404	4	no	—
277	C₂³	—	404	4	no	—
278	C₂³	—	202	8	no	—
279	C₂³	C₂³×D₁₀₁	1	4	yes	—
280	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	202	8	no	—
283	Q₈	—	202	4	no	—
284	D₄	—	2	8	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	404	4	no	—
288	D₄	—	404	8	no	—
289	D₄	—	202	8	no	—
290	D₄	—	202	8	no	—
291	D₄	—	202	8	no	—
292	D₄	—	202	8	no	—
293	D₄	—	202	4	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	101	4	no	—
297	C₂×C₄	—	101	1	no	—
298	C₂×C₄	—	101	1	no	—
299	C₂×C₄	—	101	8	no	—
300	C₂×C₄	—	101	8	no	—
301	C₂×C₄	—	101	4	no	—
302	C₂×C₄	—	101	4	no	—
303	C₂×C₄	—	404	2	no	—
304	C₂×C₄	—	202	8	no	—
305	C₂×C₄	—	202	8	no	—
306	C₂×C₄	—	202	8	no	—
307	C₂×C₄	—	202	8	no	—
308	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
309	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
310	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
311	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	101	4	no	—
318	C₂²	—	404	4	no	—
319	C₂²	—	404	16	no	—
320	C₂²	—	202	4	no	—
321	C₂²	—	202	4	no	—
322	C₂²	—	202	4	no	—
323	C₂²	—	202	2	no	—
324	C₂²	—	202	2	no	—
325	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
326	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
327	C₄	—	2	4	no	—
328	C₄	—	2	4	no	—
329	C₄	—	101	2	no	—
330	C₄	—	101	2	no	—
331	C₄	—	202	4	no	—
332	C₄	—	202	4	no	—
333	C₄	C₂×D₄×D₁₀₁	1	4	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	4	4	no	—
336	C₂	—	404	4	no	—
337	C₂	—	202	4	no	—
338	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
339	C₂	#20266012	1	1	yes	characteristic
340	C₂	#20266014	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	#537×C₁₀₁	C₂
hypercenter	#537	D₁₀₁
hypocenter	C₁₀₁	#3449
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #537

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₁₀₁	C₂	semidirect
2. #20265758	C₂	semidirect
3. #20265755	C₂	semidirect
4. #20265750	C₂	semidirect
5. #20265743	C₂	semidirect
6. #20265735	C₂	semidirect
7. #20265722	C₂	semidirect
8. #20265690	C₂	semidirect
9. #20265687	C₂	semidirect
10. C₂×#15692213	C₂	semidirect
11. #537×C₁₀₁	C₂	semidirect
14. C₂×#13108074	C₂²	semidirect
16. C₂×#13108073	C₂²	semidirect
18. C₂×Q₈×D₁₀₁	C₂²	semidirect
20. C₂×D₄×D₁₀₁	C₂²	semidirect
22. C₂²×C₄×D₁₀₁	C₂²	semidirect
23. #15692254	C₂²	semidirect
24. #15692251	C₂²	semidirect
25. #15692241	C₂²	semidirect
26. #15692240	C₂²	semidirect
27. #15692237	C₂²	semidirect
28. #15692235	C₂²	semidirect
29. #15692234	C₂²	semidirect
30. #15692231	C₂²	semidirect
31. #15692230	C₂²	semidirect
32. #15692229	C₂²	semidirect
33. #15692229	C₂²	semidirect
34. #15692227	C₂²	semidirect
35. #15692226	C₂²	semidirect
36. #15692225	C₂²	semidirect
37. #15692223	C₂²	semidirect
38. D₄×Dic₁₀₁	C₂²	semidirect
39. #15692217	C₂²	semidirect
40. #15692215	C₂²	semidirect
41. #15692213	C₂²	semidirect
42. #15692213	C₂²	semidirect
43. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
45. C₂×C₄×Dic₁₀₁	C₂²	semidirect
46. C₂×#41×C₁₀₁	C₂²	semidirect
47. C₂²⋊Q₈×C₁₀₁	C₂²	semidirect
48. #121×C₁₀₁	C₂²	semidirect
49. #121×C₁₀₁	C₂²	semidirect
50. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
74. C₂×C₄×D₁₀₁	D₄	semidirect
171. C₁₀₁	#3449	semidirect
175. #537	D₁₀₁	semidirect
189. C₂×#41	C₂×D₁₀₁	semidirect
203. C₂²⋊Q₈	C₂×D₁₀₁	semidirect
209. #121	C₂×D₁₀₁	semidirect
210. #121	C₂×D₁₀₁	semidirect
219. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₀₁	2
#20265758	2
#20265755	2
#20265750	2
#20265743	2
#20265735	2
#20265722	2
#20265690	2
#20265687	2
C₂×#15692213	2
#537×C₁₀₁	2
#3449	101