Subgroups — C₄×#15688864 #26107197

Order 12032.

532 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094016	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688864	C₂	1	2	yes	maximal
4	#20241270	C₂	1	2	yes	maximal
5	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	C₂²	1	1	yes	characteristic
7	C₂×#13093935	—	2	1	no	—
8	C₂×#13093935	C₂²	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
10	#15689021	C₄	1	1	yes	characteristic
11	#15689021	C₄	1	1	yes	characteristic
12	#15689007	—	2	1	no	—
13	#15689007	—	2	1	no	—
14	#15688989	C₂²	1	1	yes	characteristic
15	C₄×Dic₄₇⋊C₄	—	2	1	no	—
16	C₄×Dic₄₇⋊C₄	—	2	1	no	—
17	C₄×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
18	C₄×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
19	#15688864	C₄	1	8	yes	direct product factor
20	C₂×#97×C₄₇	—	2	1	no	—
21	C₂×#97×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13094016	—	2	2	no	—
26	#13094016	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₄₇⋊C₄	—	2	1	no	—
32	C₂×Dic₄₇⋊C₄	—	2	1	no	—
33	C₂×Dic₄₇⋊C₄	—	2	1	no	—
34	C₂×Dic₄₇⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	#13093935	—	2	4	no	—
38	#13093935	—	2	2	no	—
39	#13093935	C₂×C₄	1	4	yes	—
40	C₄×C₄₇⋊C₈	—	2	4	no	—
41	C₄₇⋊#95	—	2	1	no	—
42	C₄₇⋊#95	—	2	1	no	—
43	C₄₇⋊#95	—	2	1	no	—
44	C₄₇⋊#95	—	2	1	no	—
45	C₄₇⋊#95	—	2	1	no	—
46	C₄₇⋊#95	—	2	1	no	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#97×C₄₇	—	2	4	no	—
56	#97×C₄₇	—	2	2	no	—
57	#97×C₄₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	1	no	—
67	C₂²×Dic₄₇	—	2	1	no	—
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂²×Dic₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	#41	1	2	yes	—
81	C₄⋊Dic₄₇	—	2	1	no	—
82	C₄⋊Dic₄₇	—	2	1	no	—
83	C₄⋊Dic₄₇	—	4	1	no	—
84	C₄⋊Dic₄₇	—	4	1	no	—
85	C₄⋊Dic₄₇	C₄²	1	2	yes	—
86	C₄⋊Dic₄₇	C₄²	1	2	yes	—
87	Dic₄₇⋊C₄	—	2	4	no	—
88	Dic₄₇⋊C₄	—	2	4	no	—
89	Dic₄₇⋊C₄	—	4	2	no	—
90	Dic₄₇⋊C₄	—	4	2	no	—
91	C₄×Dic₄₇	—	4	2	no	—
92	C₄×Dic₄₇	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	#101	1	2	yes	—
111	C₂×C₄×C₄₇	#104	1	2	yes	—
112	C₂×C₄×C₄₇	#104	1	2	yes	—
113	C₂×C₄×C₄₇	#104	1	2	yes	—
114	C₂×C₄×C₄₇	#104	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	4	1	no	—
122	C₂×Dic₄₇	—	4	1	no	—
123	C₂×Dic₄₇	—	4	1	no	—
124	C₂×Dic₄₇	—	4	1	no	—
125	C₂×Dic₄₇	—	4	1	no	—
126	C₂×Dic₄₇	—	4	1	no	—
127	C₂×Dic₄₇	—	4	1	no	—
128	C₂×Dic₄₇	—	4	1	no	—
129	C₂×Dic₄₇	—	4	1	no	—
130	C₂×Dic₄₇	—	4	1	no	—
131	C₈×C₄₇	—	2	8	no	—
132	C₄₇⋊C₈	—	2	8	no	—
133	C₄×#330	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
136	C₂²×C₄₇	#413	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	4	1	no	—
148	C₄×C₄₇	#330	1	4	yes	complemented
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	8	1	no	—
152	Dic₄₇	—	8	1	no	—
153	C₄×#125	—	47	1	no	—
154	C₄×#97	—	47	1	no	—
155	C₄×#97	D₄₇	1	1	yes	characteristic
156	C₂×#330	—	47	2	no	—
157	#1295	—	47	2	no	—
158	C₂×C₄₇	C₂×#330	1	1	yes	characteristic
159	C₂×C₄₇	#1524	1	1	yes	characteristic
160	C₂×C₄₇	#1524	1	1	yes	characteristic
161	C₂×C₄₇	#1526	1	1	yes	characteristic
162	C₂×C₄₇	C₄×#101	1	1	yes	characteristic
163	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
164	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
165	C₂×#125	—	47	1	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×#97	—	47	2	no	—
168	C₂×#97	—	94	1	no	—
169	C₂×#97	C₂×D₄₇	1	2	yes	—
170	C₂×C₄×C₈	—	2	1	no	—
171	C₂×C₄×C₈	—	94	1	no	—
172	#389	—	47	1	no	—
173	#389	—	47	1	no	—
174	#387	—	94	1	no	—
175	#387	—	94	1	no	—
176	#382	—	47	1	no	—
177	C₄×C₄⋊C₄	—	47	1	no	—
178	C₄×C₄⋊C₄	—	47	1	no	—
179	C₄×C₄⋊C₄	—	94	1	no	—
180	C₄×C₄⋊C₄	—	94	1	no	—
181	C₄³	C₂×D₄₇	1	1	yes	characteristic
182	#330	—	47	8	no	—
183	C₄₇	C₄×#330	1	1	yes	characteristic, complemented, Sylow
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	94	2	no	—
186	#125	—	47	4	no	—
187	#125	—	94	2	no	—
188	C₂×C₄⋊C₄	—	47	1	no	—
189	C₂×C₄⋊C₄	—	47	1	no	—
190	C₂×C₄⋊C₄	—	47	1	no	—
191	C₂×C₄⋊C₄	—	47	1	no	—
192	C₂×C₄⋊C₄	—	94	1	no	—
193	C₂×C₄⋊C₄	—	94	1	no	—
194	C₂×C₄⋊C₄	—	94	1	no	—
195	C₂×C₄⋊C₄	—	94	1	no	—
196	C₂×C₄²	—	94	1	no	—
197	C₂×C₄²	—	94	1	no	—
198	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
201	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	#97	—	2	4	no	—
206	#97	—	2	2	no	—
207	#97	—	47	4	no	—
208	#97	—	94	4	no	—
209	#97	—	94	2	no	—
210	#97	C₄×D₄₇	1	4	yes	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	—	94	4	no	—
213	#95	—	94	1	no	—
214	#95	—	94	1	no	—
215	#95	—	94	1	no	—
216	#95	—	94	1	no	—
217	#95	—	94	1	no	—
218	#95	—	94	1	no	—
219	C₂²×C₄	—	94	1	no	—
220	C₂²×C₄	—	94	1	no	—
221	C₂²×C₄	—	94	1	no	—
222	C₂²×C₄	—	94	1	no	—
223	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₄⋊C₄	—	47	2	no	—
237	C₄⋊C₄	—	47	2	no	—
238	C₄⋊C₄	—	94	1	no	—
239	C₄⋊C₄	—	94	1	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	—	94	4	no	—
242	C₄⋊C₄	—	188	1	no	—
243	C₄⋊C₄	—	188	1	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	188	2	no	—
253	C₄²	—	188	2	no	—
254	C₄²	C₂×C₄×D₄₇	1	2	yes	—
255	C₄²	#10625350	1	2	yes	—
256	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	188	1	no	—
273	C₂×C₄	—	188	1	no	—
274	C₂×C₄	—	188	1	no	—
275	C₂×C₄	—	188	1	no	—
276	C₂×C₄	—	188	1	no	—
277	C₂×C₄	—	188	1	no	—
278	C₂×C₄	—	188	1	no	—
279	C₂×C₄	—	188	1	no	—
280	C₂×C₄	—	188	1	no	—
281	C₂×C₄	—	188	1	no	—
282	C₂×C₄	#13093942	1	2	yes	—
283	C₂×C₄	#13093950	1	2	yes	—
284	C₂×C₄	#13093950	1	2	yes	—
285	C₂×C₄	#13093952	1	2	yes	—
286	C₂×C₄	#13093952	1	2	yes	—
287	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
288	C₂×C₄	#13093981	1	1	yes	characteristic
289	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
290	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
291	C₈	—	2	8	no	—
292	C₈	—	94	8	no	—
293	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
294	C₂²	C₂×#13093942	1	1	yes	characteristic
295	C₂²	#15689105	1	1	yes	characteristic
296	C₂²	C₂×#13093950	1	1	yes	characteristic
297	C₂²	C₂×#13093950	1	1	yes	characteristic
298	C₂²	C₂×#13093952	1	1	yes	characteristic
299	C₂²	C₂×#13093952	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	376	1	no	—
308	C₄	—	376	1	no	—
309	C₄	—	188	2	no	—
310	C₄	—	188	2	no	—
311	C₄	#15688864	1	4	yes	direct product factor
312	C₂	C₂×#15688864	1	1	yes	characteristic
313	C₂	#20241974	1	1	yes	characteristic
314	C₂	#20241977	1	1	yes	characteristic
315	C₂	#20241982	1	1	yes	characteristic
316	C₂	C₄×#13093942	1	1	yes	characteristic
317	C₂	C₄×#13093952	1	1	yes	characteristic
318	C₂	C₄×#13093950	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	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#330
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
19. #15688864	C₄	direct (N × H)
148. C₄×C₄₇	#330	semidirect
183. C₄₇	C₄×#330	semidirect
311. C₄	#15688864	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13094016	2
C₄×#13093935	2
C₂×#15688864	2
#20241270	2
C₄×#97×C₄₇	2
C₄×#330	47

Subgroups — C4×#15688864 #26107197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688864 #26107197