Subgroups — #27180432

Order 12928.

764 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692301	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692280	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265755	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265338	C₂	1	2	yes	maximal, complemented
5	#20265326	C₂	1	2	yes	maximal, complemented
6	#20265321	C₂	1	2	yes	maximal, complemented
7	#20265309	C₂	1	2	yes	maximal, complemented
8	#20265178	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692177	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265143	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#135×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#15692301	—	2	4	no	—
13	#15692301	—	2	4	no	—
14	C₂×#13108075	C₂²	1	2	yes	—
15	C₂×#13108074	—	2	1	no	—
16	C₂×#13108074	C₂²	1	1	yes	characteristic
17	C₂²×D₄₀₄	—	2	1	no	—
18	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₁₀₁	—	2	2	no	—
20	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
21	#15692280	—	2	4	no	—
22	C₂×#13108061	C₂²	1	2	yes	complemented
23	C₂×#13108059	C₂²	1	2	yes	complemented
24	C₂×#13108058	C₂²	1	1	yes	characteristic, complemented
25	#15692234	—	2	1	no	—
26	#15692231	C₂²	1	2	yes	complemented
27	#15692229	—	2	1	no	—
28	#15692227	—	2	2	no	—
29	#15692226	C₂²	1	2	yes	complemented
30	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
31	#15692179	C₂²	1	2	yes	complemented
32	#15692177	C₂²	1	2	yes	complemented
33	#15692177	C₂²	1	2	yes	complemented
34	#15692176	C₂²	1	2	yes	complemented
35	#15692167	C₂²	1	2	yes	complemented
36	#15692167	C₂²	1	2	yes	complemented
37	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
39	#135×C₁₀₁	—	2	4	no	—
40	C₂×Q₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
41	C₂×SD₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
42	C₂×SD₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
43	C₂×D₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
44	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
45	C₂³×D₁₀₁	—	2	1	no	—
46	C₂³×D₁₀₁	—	2	2	no	—
47	#13108075	—	2	4	no	—
48	#13108075	—	2	4	no	—
49	#13108074	—	2	8	no	—
50	#13108074	—	2	2	no	—
51	#13108074	—	2	2	no	—
52	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
53	C₂×D₄₀₄	—	2	8	no	—
54	C₂×D₄₀₄	—	2	2	no	—
55	C₂×D₄₀₄	—	2	1	no	—
56	C₂×D₄₀₄	—	2	2	no	—
57	C₂×D₄₀₄	D₄	1	1	yes	characteristic
58	C₂×D₄₀₄	D₄	1	4	yes	complemented
59	C₂×D₄₀₄	C₂³	1	2	yes	—
60	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
61	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
62	C₂×C₂²⋊D₁₀₁	D₄	1	2	yes	—
63	D₄×D₁₀₁	—	2	16	no	—
64	D₄×D₁₀₁	—	2	4	no	—
65	D₄×D₁₀₁	—	2	4	no	—
66	C₂²×Dic₁₀₁	—	2	1	no	—
67	C₂×C₄×D₁₀₁	—	2	2	no	—
68	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
69	#13108061	—	2	4	no	—
70	#13108059	—	2	4	no	—
71	#13108058	—	2	2	no	—
72	C₂×#11717579	C₂³	1	2	yes	—
73	C₄⋊Dic₁₀₁	—	2	1	no	—
74	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
75	#13108053	—	2	2	no	—
76	C₂²⋊Dic₁₀₁	—	2	2	no	—
77	#13108051	—	2	2	no	—
78	#41×C₁₀₁	—	2	2	no	—
79	#41×C₁₀₁	—	2	2	no	—
80	#41×C₁₀₁	—	2	2	no	—
81	#41×C₁₀₁	D₄	1	4	yes	complemented
82	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₁₀₁	—	2	1	no	—
85	C₂×D₄×C₁₀₁	—	2	1	no	—
86	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
87	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₁	—	2	1	no	—
89	C₂²×C₄×C₁₀₁	—	2	1	no	—
90	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
91	Q₁₆×C₁₀₁	—	2	2	no	—
92	SD₁₆×C₁₀₁	—	2	2	no	—
93	SD₁₆×C₁₀₁	—	2	2	no	—
94	D₈×C₁₀₁	—	2	2	no	—
95	C₂×C₈×C₁₀₁	—	2	2	no	—
96	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
98	C₂³×C₁₀₁	—	2	1	no	—
99	C₂³×C₁₀₁	—	2	1	no	—
100	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×D₁₀₁	—	2	2	no	—
102	C₂²×D₁₀₁	—	2	8	no	—
103	C₂²×D₁₀₁	—	2	4	no	—
104	C₂²×D₁₀₁	—	8	2	no	—
105	C₂²×D₁₀₁	—	8	2	no	—
106	C₂²×D₁₀₁	—	8	1	no	—
107	C₂²×D₁₀₁	—	8	1	no	—
108	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
109	Q₈×C₁₀₁	—	2	1	no	—
110	Q₈×C₁₀₁	—	2	1	no	—
111	Q₈×C₁₀₁	—	2	1	no	—
112	Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
113	D₄×C₁₀₁	—	2	4	no	—
114	D₄×C₁₀₁	—	2	1	no	—
115	D₄×C₁₀₁	—	2	1	no	—
116	D₄×C₁₀₁	—	2	1	no	—
117	D₄×C₁₀₁	—	4	2	no	—
118	D₄×C₁₀₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₁	—	2	1	no	—
120	C₂×C₄×C₁₀₁	—	2	4	no	—
121	C₂×C₄×C₁₀₁	—	2	1	no	—
122	C₂×C₄×C₁₀₁	—	4	2	no	—
123	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₀₁	—	2	8	no	—
128	C₂²⋊D₁₀₁	—	2	4	no	—
129	C₂²⋊D₁₀₁	—	8	2	no	—
130	C₂×Dic₁₀₁	—	2	2	no	—
131	C₂×Dic₁₀₁	—	8	1	no	—
132	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
133	D₄₀₄	—	2	8	no	—
134	D₄₀₄	—	2	2	no	—
135	D₄₀₄	—	2	4	no	—
136	D₄₀₄	—	2	2	no	—
137	D₄₀₄	C₂×D₄	1	4	yes	—
138	C₄×D₁₀₁	—	2	8	no	—
139	C₄×D₁₀₁	—	2	2	no	—
140	C₄×D₁₀₁	—	2	2	no	—
141	Dic₂₀₂	—	2	2	no	—
142	C₈×C₁₀₁	—	2	1	no	—
143	C₈×C₁₀₁	—	2	1	no	—
144	#11717579	—	2	2	no	—
145	C₂²×C₁₀₁	—	2	1	no	—
146	C₂²×C₁₀₁	—	2	1	no	—
147	C₂²×C₁₀₁	—	2	2	no	—
148	C₂²×C₁₀₁	—	4	1	no	—
149	C₂²×C₁₀₁	—	4	1	no	—
150	C₂²×C₁₀₁	—	4	1	no	—
151	C₂²×C₁₀₁	—	4	1	no	—
152	C₂²×C₁₀₁	—	4	1	no	—
153	C₂²×C₁₀₁	#120	1	2	yes	—
154	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₀₁	—	2	2	no	—
156	C₂×D₁₀₁	—	2	4	no	—
157	C₂×D₁₀₁	—	2	2	no	—
158	C₂×D₁₀₁	—	2	2	no	—
159	C₂×D₁₀₁	—	4	2	no	—
160	C₂×D₁₀₁	—	4	2	no	—
161	C₂×D₁₀₁	—	8	4	no	—
162	C₂×D₁₀₁	—	8	8	no	—
163	C₄×C₁₀₁	—	2	2	no	—
164	C₄×C₁₀₁	—	4	1	no	—
165	C₄×C₁₀₁	#120	1	2	yes	—
166	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
168	Dic₁₀₁	—	2	2	no	—
169	Dic₁₀₁	—	4	2	no	—
170	C₂×C₁₀₁	—	2	2	no	—
171	C₂×C₁₀₁	—	4	2	no	—
172	C₂×C₁₀₁	—	8	1	no	—
173	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
174	C₂×C₁₀₁	#576	1	1	yes	characteristic
175	C₂×C₁₀₁	#577	1	1	yes	characteristic
176	D₁₀₁	—	4	2	no	—
177	D₁₀₁	—	4	4	no	—
178	#3012	—	101	1	no	maximal, Sylow
179	C₁₀₁	#3012	1	1	yes	characteristic, complemented, Sylow
180	C₂×H(5,𝔽₂)	—	101	1	no	—
181	C₂×AGL(1,ℤ/8)	—	101	1	no	—
182	C₂×#135	D₁₀₁	1	1	yes	characteristic, complemented
183	#537	—	101	1	no	—
184	#452	—	101	2	no	—
185	#450	—	101	2	no	—
186	#449	—	101	2	no	—
187	#447	—	101	2	no	—
188	#417	—	101	1	no	—
189	C₂×#102	—	101	1	no	—
190	#408	—	101	1	no	—
191	H(5,𝔽₂)	—	202	4	no	—
192	H(5,𝔽₂)	—	202	4	no	—
193	C₂×#41	—	101	1	no	—
194	C₂×#41	—	101	2	no	—
195	C₂×#41	—	202	1	no	—
196	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
197	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic, complemented
198	C₂²×D₄	—	101	2	no	—
199	C₂²×D₄	—	101	1	no	—
200	C₂²×D₄	—	202	1	no	—
201	C₂²×D₄	—	202	2	no	—
202	AGL(1,ℤ/8)	—	202	4	no	—
203	#135	—	2	4	no	—
204	C₂×Q₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
205	C₂×SD₁₆	—	101	2	no	—
206	C₂×SD₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
208	C₂×D₈	—	101	2	no	—
209	C₂×D₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
210	C₂×M₁₆	—	101	1	no	—
211	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
212	C₂²⋊Q₈	—	101	2	no	—
213	C₂²⋊Q₈	—	202	1	no	—
214	#121	—	101	2	no	—
215	#121	—	202	1	no	—
216	#121	—	202	2	no	—
217	C₂×C₄⋊C₄	—	101	1	no	—
218	#103	—	101	2	no	—
219	#102	—	101	2	no	—
220	#102	—	101	2	no	—
221	#102	—	101	2	no	—
222	C₂²⋊C₈	—	101	2	no	—
223	C₂²⋊C₈	—	101	2	no	—
224	C₂⁴	—	202	2	no	—
225	C₂⁴	—	202	1	no	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	202	8	no	—
230	#41	—	202	4	no	—
231	#41	—	202	4	no	—
232	#41	—	202	2	no	—
233	#41	—	202	2	no	—
234	#41	C₂²⋊D₁₀₁	1	4	yes	complemented
235	C₂×Q₈	—	101	1	no	—
236	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	101	2	no	—
241	C₂×D₄	—	101	4	no	—
242	C₂×D₄	—	101	1	no	—
243	C₂×D₄	—	101	2	no	—
244	C₂×D₄	—	202	2	no	—
245	C₂×D₄	—	202	8	no	—
246	C₂×D₄	—	202	2	no	—
247	C₂×D₄	—	202	4	no	—
248	C₂×D₄	—	202	16	no	—
249	C₂×D₄	—	202	2	no	—
250	C₂×D₄	—	202	4	no	—
251	C₂×D₄	—	202	1	no	—
252	C₂×D₄	—	202	2	no	—
253	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
254	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	101	2	no	—
258	C₂²×C₄	—	202	2	no	—
259	C₂²×C₄	—	202	1	no	—
260	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
261	Q₁₆	—	2	2	no	—
262	SD₁₆	—	2	2	no	—
263	SD₁₆	—	2	2	no	—
264	SD₁₆	—	202	4	no	—
265	D₈	—	2	2	no	—
266	D₈	—	202	4	no	—
267	M₁₆	—	202	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	101	2	no	—
270	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
272	C₄⋊C₄	—	101	2	no	—
273	C₄⋊C₄	—	202	2	no	—
274	C₄⋊C₄	—	202	1	no	—
275	C₂²⋊C₄	—	202	2	no	—
276	C₂²⋊C₄	—	202	2	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	808	1	no	—
280	C₂³	—	808	1	no	—
281	C₂³	—	808	2	no	—
282	C₂³	—	808	2	no	—
283	C₂³	—	101	2	no	—
284	C₂³	—	202	8	no	—
285	C₂³	—	202	4	no	—
286	C₂³	—	202	2	no	—
287	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	202	2	no	—
292	Q₈	C₂×C₂²⋊D₁₀₁	1	2	yes	—
293	D₄	—	2	4	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	808	2	no	—
299	D₄	—	101	4	no	—
300	D₄	—	202	8	no	—
301	D₄	—	202	8	no	—
302	D₄	—	202	4	no	—
303	D₄	—	202	2	no	—
304	D₄	—	202	4	no	—
305	D₄	—	202	2	no	—
306	D₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	808	1	no	—
312	C₂×C₄	—	101	2	no	—
313	C₂×C₄	—	202	8	no	—
314	C₂×C₄	—	202	2	no	—
315	C₂×C₄	—	202	2	no	—
316	C₂×C₄	—	202	2	no	—
317	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
318	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
320	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	202	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	808	4	no	—
333	C₂²	—	808	8	no	—
334	C₂²	—	404	2	no	—
335	C₂²	—	404	2	no	—
336	C₂²	—	202	2	no	—
337	C₂²	—	202	4	no	—
338	C₂²	—	202	2	no	—
339	C₂²	—	202	2	no	—
340	C₂²	#15692223	1	2	yes	—
341	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	404	2	no	—
345	C₄	—	202	2	no	—
346	C₄	#15692223	1	2	yes	—
347	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
348	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	2	no	—
351	C₂	—	8	1	no	—
352	C₂	—	404	4	no	—
353	C₂	—	404	2	no	—
354	C₂	C₂×#15692223	1	1	yes	characteristic
355	C₂	#20265979	1	1	yes	characteristic
356	C₂	#20265984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×#135×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂×#135	D₁₀₁
hypocenter	C₁₀₁	#3012
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15692301	C₂	semidirect
2. C₂×#15692280	C₂	semidirect
3. #20265755	C₂	semidirect
4. #20265338	C₂	semidirect
5. #20265326	C₂	semidirect
6. #20265321	C₂	semidirect
7. #20265309	C₂	semidirect
8. #20265178	C₂	semidirect
9. C₂×#15692177	C₂	semidirect
10. #20265143	C₂	semidirect
11. C₂×#135×C₁₀₁	C₂	semidirect
22. C₂×#13108061	C₂²	semidirect
23. C₂×#13108059	C₂²	semidirect
24. C₂×#13108058	C₂²	semidirect
26. #15692231	C₂²	semidirect
29. #15692226	C₂²	semidirect
30. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
31. #15692179	C₂²	semidirect
32. #15692177	C₂²	semidirect
33. #15692177	C₂²	semidirect
34. #15692176	C₂²	semidirect
35. #15692167	C₂²	semidirect
36. #15692167	C₂²	semidirect
38. C₂×#41×C₁₀₁	C₂²	semidirect
40. C₂×Q₁₆×C₁₀₁	C₂²	semidirect
41. C₂×SD₁₆×C₁₀₁	C₂²	semidirect
42. C₂×SD₁₆×C₁₀₁	C₂²	semidirect
43. C₂×D₈×C₁₀₁	C₂²	semidirect
44. C₂²×C₈×C₁₀₁	C₂²	semidirect
58. C₂×D₄₀₄	D₄	semidirect
81. #41×C₁₀₁	D₄	semidirect
179. C₁₀₁	#3012	semidirect
182. C₂×#135	D₁₀₁	semidirect
197. C₂×#41	C₂×D₁₀₁	semidirect
204. C₂×Q₁₆	C₂×D₁₀₁	semidirect
206. C₂×SD₁₆	C₂×D₁₀₁	semidirect
207. C₂×SD₁₆	C₂×D₁₀₁	semidirect
209. C₂×D₈	C₂×D₁₀₁	semidirect
211. C₂²×C₈	C₂×D₁₀₁	semidirect
234. #41	C₂²⋊D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692301	2
C₂×#15692280	2
#20265755	2
#20265338	2
#20265326	2
#20265321	2
#20265309	2
#20265178	2
C₂×#15692177	2
#20265143	2
C₂×#135×C₁₀₁	2
#3012	101