Subgroups — #26106329

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20241270	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂×#13093983	C₄	1	2	yes	—
7	C₄×#10625329	—	2	2	no	—
8	C₄×#10625329	C₄	1	4	yes	complemented
9	C₂×#13093935	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
11	#15688993	C₄	1	2	yes	—
12	#15688989	C₂²	1	2	yes	—
13	#15688983	—	2	1	no	—
14	#15688983	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	—	2	1	no	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	C₂²×M₁₆×C₄₇	—	2	1	no	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₄₇	—	2	2	no	—
20	C₄×M₁₆×C₄₇	C₄	1	4	yes	complemented
21	C₂×#97×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	#13093983	—	4	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	#13093935	—	2	2	no	—
33	#13093935	C₂×C₄	1	4	yes	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₄₇	—	2	8	no	—
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#97×C₄₇	—	2	2	no	—
52	#97×C₄₇	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	16	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	C₄²	1	2	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
85	C₂²⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	C₄⋊Dic₄₇	—	4	4	no	—
88	C₄⋊Dic₄₇	—	4	4	no	—
89	C₄×Dic₄₇	—	4	4	no	—
90	#10625329	—	2	16	no	—
91	C₂×C₄₇⋊C₈	—	2	8	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	#95	1	2	yes	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	#95	1	2	yes	—
110	C₂×C₄×C₄₇	#95	1	2	yes	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	#100	1	2	yes	—
113	C₂×C₄×C₄₇	#101	1	2	yes	—
114	C₂×C₄×C₄₇	#104	1	4	yes	—
115	C₂×C₄×C₄₇	#104	1	4	yes	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₈×C₄₇	—	2	8	no	—
126	C₄₇⋊C₈	—	2	8	no	—
127	#8028	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	4	2	no	—
133	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×#104	1	2	yes	—
137	C₂²×C₄₇	C₂×#104	1	2	yes	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	#339	1	4	yes	—
145	C₄×C₄₇	#339	1	4	yes	—
146	C₄×C₄₇	#343	1	4	yes	—
147	Dic₄₇	—	8	4	no	—
148	C₂×C₄×M₁₆	—	47	1	no	—
149	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
150	#2285	—	47	1	no	—
151	#1295	—	47	4	no	—
152	C₂×C₄₇	—	4	2	no	—
153	C₂×C₄₇	#1525	1	2	yes	—
154	C₂×C₄₇	#1526	1	2	yes	—
155	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
156	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
158	C₂²×M₁₆	—	2	1	no	—
159	C₂²×M₁₆	—	94	1	no	—
160	C₂×#117	—	47	2	no	—
161	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
162	C₄×M₁₆	—	2	2	no	—
163	C₄×M₁₆	—	47	4	no	—
164	C₄×M₁₆	—	94	2	no	—
165	C₄×M₁₆	Dic₄₇	1	4	yes	complemented
166	C₂×#97	—	47	2	no	—
167	C₂×#97	C₂×D₄₇	1	2	yes	—
168	C₂×C₄×C₈	—	2	1	no	—
169	C₂×C₄×C₈	—	94	1	no	—
170	#384	—	47	2	no	—
171	#382	—	47	2	no	—
172	#380	—	94	1	no	—
173	#380	—	94	1	no	—
174	C₄×C₄⋊C₄	—	94	1	no	—
175	C₄×C₄⋊C₄	—	94	1	no	—
176	C₄₇	#8028	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
178	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
179	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	8	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	94	8	no	—
183	C₂×M₁₆	—	94	4	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	94	2	no	—
186	#117	—	188	4	no	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₄⋊C₄	—	94	2	no	—
189	C₂×C₄⋊C₄	—	94	2	no	—
190	C₂×C₂²⋊C₄	—	94	2	no	—
191	C₂×C₄²	—	94	2	no	—
192	C₂×C₄²	Dic₉₄	1	2	yes	—
193	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₄₇	1	2	yes	—
195	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
196	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#97	—	2	2	no	—
201	#97	—	47	4	no	—
202	#97	—	94	2	no	—
203	#97	C₂×Dic₄₇	1	4	yes	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	—	94	4	no	—
206	#95	—	94	2	no	—
207	#95	—	94	2	no	—
208	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
215	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
216	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
217	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	M₁₆	—	2	16	no	—
227	M₁₆	—	94	16	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	94	8	no	—
231	C₂×C₈	—	94	4	no	—
232	C₄⋊C₄	—	188	4	no	—
233	C₄⋊C₄	—	188	4	no	—
234	C₄⋊C₄	—	188	4	no	—
235	C₂²⋊C₄	—	188	4	no	—
236	C₄²	—	188	4	no	—
237	C₄²	C₄×Dic₄₇	1	2	yes	—
238	C₄²	Dic₄₇⋊C₄	1	2	yes	—
239	C₄²	Dic₄₇⋊C₄	1	2	yes	—
240	C₄²	C₄⋊Dic₄₇	1	2	yes	—
241	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
242	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
243	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
244	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₄₇⋊#95	1	2	yes	—
252	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	C₄₇⋊#95	1	2	yes	—
267	C₂×C₄	C₄₇⋊#95	1	2	yes	—
268	C₂×C₄	C₄₇⋊#95	1	2	yes	—
269	C₂×C₄	#13093940	1	2	yes	—
270	C₂×C₄	#13093942	1	2	yes	—
271	C₂×C₄	#13093950	1	4	yes	—
272	C₂×C₄	#13093952	1	4	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₈	—	94	8	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
287	C₂²	C₂×#13093940	1	1	yes	characteristic
288	C₂²	C₂×#13093942	1	1	yes	characteristic
289	C₂²	C₂×#13093950	1	2	yes	—
290	C₂²	C₂×#13093952	1	2	yes	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	376	4	no	—
298	C₄	#15688883	1	4	yes	—
299	C₄	#15688884	1	4	yes	—
300	C₄	#15688891	1	4	yes	—
301	C₂	—	4	2	no	—
302	C₂	#20241979	1	2	yes	—
303	C₂	#20241982	1	2	yes	—
304	C₂	C₂×#15688883	1	1	yes	characteristic
305	C₂	C₂×#15688884	1	1	yes	characteristic
306	C₂	C₂×#15688891	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#8028
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20241270	C₂	semidirect
8. C₄×#10625329	C₄	semidirect
20. C₄×M₁₆×C₄₇	C₄	semidirect
165. C₄×M₁₆	Dic₄₇	semidirect
176. C₄₇	#8028	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
#20245795	2
#20241270	2
C₂×C₄×M₁₆×C₄₇	2
#8028	47