Subgroups — #97×D₁₈₈ #26214739

Order 12032.

1274 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₈₈	C₂	1	2	yes	maximal
2	C₂×#97×D₄₇	C₂	1	2	yes	maximal, complemented
3	C₄²×D₁₈₈	C₂	1	1	yes	maximal, characteristic
4	#20243255	C₂	1	2	yes	maximal, complemented
5	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243165	C₂	1	4	yes	maximal, complemented
7	#20243151	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	4	yes	—
10	C₂×C₄×D₁₈₈	C₄	1	2	yes	—
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	2	yes	—
13	C₈×D₁₈₈	C₄	1	16	yes	complemented
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×#13093934	C₂²	1	4	yes	—
18	#97×D₄₇	—	2	8	no	—
19	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄×(C₂×D₄₇)⋊C₄	C₄	1	4	yes	—
22	C₄×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	#15688878	C₂²	1	2	yes	—
24	#15688877	C₂²	1	4	yes	—
25	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×D₄₇	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₂×C₈×D₄₇	—	2	8	no	—
35	C₂×C₈×D₄₇	—	2	8	no	—
36	C₂×C₈×D₄₇	C₂×C₄	1	16	yes	complemented
37	C₄×D₁₈₈	—	2	8	no	—
38	C₄×D₁₈₈	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
45	C₄²×D₄₇	—	2	4	no	—
46	C₄²×D₄₇	—	2	4	no	—
47	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
48	(C₂×D₄₇)⋊C₈	C₂×C₄	1	16	yes	complemented
49	#13093935	—	2	4	no	—
50	#13093934	—	2	8	no	—
51	C₄₇⋊#95	C₂×C₄	1	2	yes	—
52	C₄₇⋊#95	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#97×C₄₇	—	2	4	no	—
61	#97×C₄₇	—	2	2	no	—
62	#97×C₄₇	D₄	1	4	yes	complemented
63	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
64	C₂³×D₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
72	C₂×D₁₈₈	C₄²	1	4	yes	—
73	C₂×D₁₈₈	M₁₆	1	8	yes	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	2	2	no	—
76	C₂×C₄×D₄₇	—	2	4	no	—
77	C₂×C₄×D₄₇	—	2	8	no	—
78	C₂×C₄×D₄₇	—	2	8	no	—
79	C₂×C₄×D₄₇	—	4	4	no	—
80	C₂×C₄×D₄₇	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₄₇	#41	1	4	yes	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	4	2	no	—
90	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	#41	1	2	yes	—
93	(C₂×D₄₇)⋊C₄	—	2	8	no	—
94	(C₂×D₄₇)⋊C₄	C₄²	1	8	yes	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	C₄²	1	4	yes	—
97	C₄×Dic₄₇	—	2	8	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
103	C₈×D₄₇	—	2	32	no	—
104	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂²×D₄₇	—	2	2	no	—
106	C₂²×D₄₇	—	2	2	no	—
107	C₂²×D₄₇	—	2	4	no	—
108	C₂²×D₄₇	C₂×C₄²	1	4	yes	—
109	C₂²×D₄₇	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
130	D₁₈₈	#97	1	16	yes	direct product factor
131	C₄×D₄₇	—	2	8	no	—
132	C₄×D₄₇	—	2	8	no	—
133	C₄×D₄₇	—	4	16	no	—
134	C₈×C₄₇	—	2	4	no	—
135	C₈×C₄₇	C₄×D₄	1	8	yes	complemented
136	C₄₇⋊C₈	—	2	8	no	—
137	D₄×#97	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
141	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
142	C₂²×C₄₇	#568	1	2	yes	—
143	C₂×D₄₇	—	2	4	no	—
144	C₂×D₄₇	—	2	8	no	—
145	C₂×D₄₇	—	2	8	no	—
146	C₂×D₄₇	—	2	4	no	—
147	C₂×D₄₇	C₂×#97	1	8	yes	complemented
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	C₂×#97	1	4	yes	complemented
152	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
153	C₄×C₄₇	C₄×#41	1	2	yes	—
154	Dic₄₇	—	2	4	no	—
155	Dic₄₇	—	4	4	no	—
156	C₂×C₈×D₄	—	47	2	no	—
157	C₂²×#97	—	47	2	no	—
158	C₄²×D₄	—	47	1	no	—
159	#1770	—	47	2	no	—
160	#1767	—	47	1	no	—
161	#1752	—	47	4	no	—
162	#1749	—	47	2	no	—
163	C₄×#97	D₄₇	1	1	yes	characteristic, complemented
164	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
165	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
166	C₂×C₄₇	#2875	1	1	yes	characteristic
167	C₂×C₄₇	D₄×M₁₆	1	2	yes	—
168	C₂×C₄₇	#2970	1	2	yes	—
169	D₄₇	—	2	8	no	—
170	C₂³×C₈	—	47	4	no	—
171	C₂×C₄×D₄	—	47	2	no	—
172	C₂×C₄×D₄	—	47	1	no	—
173	C₂²×C₄²	—	47	2	no	—
174	C₈×D₄	—	47	16	no	—
175	C₂×C₄⋊C₈	—	47	2	no	—
176	C₂×C₂²⋊C₈	—	47	4	no	—
177	C₂×#97	—	47	4	no	—
178	C₂×#97	—	47	2	no	—
179	C₂×#97	—	94	8	no	—
180	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
181	C₂×#97	C₂×D₄₇	1	2	yes	—
182	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
184	C₄×C₄⋊C₄	—	47	2	no	—
185	C₄×C₄⋊C₄	—	47	1	no	—
186	C₄×C₂²⋊C₄	—	47	4	no	—
187	C₄×C₂²⋊C₄	—	47	2	no	—
188	C₄³	C₂×D₄₇	1	1	yes	characteristic
189	#336	—	47	4	no	—
190	#336	—	47	2	no	—
191	C₄₇	D₄×#97	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	47	1	no	—
193	C₂³×C₄	—	47	2	no	—
194	C₂³×C₄	—	47	4	no	—
195	C₂²×C₈	—	47	4	no	—
196	C₂²×C₈	—	47	16	no	—
197	C₂²×C₈	—	94	8	no	—
198	C₂²×C₈	—	94	8	no	—
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
201	C₄×D₄	—	47	8	no	—
202	C₄×D₄	—	94	8	no	—
203	C₂×C₄⋊C₄	—	47	1	no	—
204	C₂×C₄⋊C₄	—	47	2	no	—
205	C₂×C₂²⋊C₄	—	47	2	no	—
206	C₂×C₂²⋊C₄	—	47	4	no	—
207	C₂×C₄²	—	47	4	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	—	94	4	no	—
210	C₂×C₄²	—	94	4	no	—
211	C₂×C₄²	C₄×D₄₇	1	2	yes	—
212	C₂×C₄²	C₄×D₄₇	1	2	yes	—
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	47	8	no	—
217	C₂²⋊C₈	—	47	16	no	—
218	#97	—	2	4	no	—
219	#97	—	2	2	no	—
220	#97	—	94	8	no	—
221	#97	—	94	4	no	—
222	#97	D₁₈₈	1	4	yes	direct product factor
223	C₄×C₈	C₄×D₄₇	1	8	yes	complemented
224	#95	—	47	4	no	—
225	#95	—	47	2	no	—
226	C₂⁴	—	47	2	no	—
227	C₂×D₄	—	47	4	no	—
228	C₂×D₄	—	47	8	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	47	8	no	—
231	C₂²×C₄	—	47	4	no	—
232	C₂²×C₄	—	94	8	no	—
233	C₂²×C₄	—	94	2	no	—
234	C₂²×C₄	—	94	2	no	—
235	C₂²×C₄	—	94	4	no	—
236	C₂²×C₄	—	94	8	no	—
237	C₂²×C₄	—	188	4	no	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	47	8	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	94	32	no	—
249	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
250	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
251	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
252	C₂×C₈	#10625350	1	4	yes	—
253	C₄⋊C₄	—	47	4	no	—
254	C₄⋊C₄	—	94	4	no	—
255	C₂²⋊C₄	—	47	8	no	—
256	C₂²⋊C₄	—	94	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	94	8	no	—
261	C₄²	—	94	2	no	—
262	C₄²	—	94	2	no	—
263	C₄²	C₂×C₄×D₄₇	1	4	yes	—
264	C₄²	C₂×D₁₈₈	1	2	yes	—
265	C₄²	#10625350	1	2	yes	—
266	C₂³	—	47	4	no	—
267	C₂³	—	47	8	no	—
268	C₂³	—	94	2	no	—
269	C₂³	—	94	2	no	—
270	C₂³	—	94	4	no	—
271	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
272	D₄	—	47	16	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	47	4	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	94	8	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	188	16	no	—
288	C₂×C₄	C₄²×D₄₇	1	2	yes	—
289	C₂×C₄	C₄²×D₄₇	1	2	yes	—
290	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
291	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
293	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
294	C₂×C₄	C₂×#10625350	1	2	yes	—
295	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
296	C₈	—	2	4	no	—
297	C₈	—	94	8	no	—
298	C₈	C₄×D₁₈₈	1	8	yes	complemented
299	C₂²	—	47	8	no	—
300	C₂²	—	94	4	no	—
301	C₂²	—	94	8	no	—
302	C₂²	—	94	4	no	—
303	C₂²	—	94	8	no	—
304	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
305	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
306	C₂²	C₄×#10625350	1	1	yes	characteristic
307	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
308	C₂²	#15689889	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	94	4	no	—
313	C₄	—	188	4	no	—
314	C₄	#97×D₄₇	1	4	yes	complemented
315	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
316	C₄	C₄×#10625350	1	2	yes	—
317	C₂	—	94	8	no	—
318	C₂	C₄²×D₁₈₈	1	1	yes	characteristic
319	C₂	C₂×#97×D₄₇	1	1	yes	characteristic
320	C₂	#20251813	1	1	yes	characteristic
321	C₂	M₁₆×D₁₈₈	1	2	yes	—
322	C₂	#20252403	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	D₄×#97
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#97×D₄₇	C₂	semidirect
4. #20243255	C₂	semidirect
5. #20243241	C₂	semidirect
6. #20243165	C₂	semidirect
7. #20243151	C₂	semidirect
8. C₄×#97×C₄₇	C₂	semidirect
13. C₈×D₁₈₈	C₄	semidirect
36. C₂×C₈×D₄₇	C₂×C₄	semidirect
47. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
48. (C₂×D₄₇)⋊C₈	C₂×C₄	semidirect
62. #97×C₄₇	D₄	semidirect
63. C₄×C₈×C₄₇	C₂×C₄	semidirect
130. D₁₈₈	#97	direct (N × H)
135. C₈×C₄₇	C₄×D₄	semidirect
147. C₂×D₄₇	C₂×#97	semidirect
151. C₄×C₄₇	C₂×#97	semidirect
163. C₄×#97	D₄₇	semidirect
191. C₄₇	D₄×#97	semidirect
222. #97	D₁₈₈	direct (N × H)
223. C₄×C₈	C₄×D₄₇	semidirect
298. C₈	C₄×D₁₈₈	semidirect
314. C₄	#97×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₈₈	2
C₂×#97×D₄₇	2
C₄²×D₁₈₈	2
#20243255	2
#20243241	2
#20243165	2
#20243151	2
C₄×#97×C₄₇	2
D₄×#97	47

Subgroups — #97×D188 #26214739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #97×D₁₈₈ #26214739