Subgroups — #108×Dic₁₄₂ #32016344

Order 18176.

658 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	M₁₆×Dic₁₄₂	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
3	#22589115	C₂	1	2	yes	maximal
4	#22589114	C₂	1	4	yes	maximal
5	#22588512	C₂	1	1	yes	maximal, characteristic
6	#22588510	C₂	1	2	yes	maximal
7	#108×Dic₇₁	C₂	1	2	yes	maximal
8	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
10	#17986129	—	2	2	no	—
11	#17986127	—	2	4	no	—
12	C₈×Dic₁₄₂	—	2	2	no	—
13	C₈×Dic₁₄₂	C₄	1	4	yes	—
14	C₈×Dic₁₄₂	C₄	1	2	yes	—
15	C₈×Dic₁₄₂	C₄	1	2	yes	—
16	C₂×#14556064	C₂²	1	2	yes	—
17	C₂×#14556064	C₂²	1	2	yes	—
18	C₂×#14556063	C₂²	1	4	yes	—
19	#17986023	C₂²	1	2	yes	—
20	#17986022	C₂²	1	4	yes	—
21	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×Dic₇₁⋊C₈	C₂²	1	2	yes	—
23	M₁₆×Dic₇₁	C₂²	1	4	yes	—
24	C₂×C₈×Dic₇₁	C₂²	1	2	yes	—
25	#17985763	C₂²	1	4	yes	—
26	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₇₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
30	#341×C₇₁	C₂²	1	2	yes	—
31	C₂²×Dic₁₄₂	Q₈	1	1	yes	characteristic
32	C₂×#11765849	C₂³	1	4	yes	—
33	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
34	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
38	C₄×Dic₁₄₂	D₄	1	2	yes	—
39	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₇₁	Q₈	1	1	yes	characteristic
41	C₂×Dic₇₁⋊C₄	Q₈	1	2	yes	—
42	C₂×C₄×Dic₇₁	C₂³	1	2	yes	—
43	#14556064	—	2	4	no	—
44	#14556064	—	4	2	no	—
45	#14556063	—	4	4	no	—
46	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
48	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
49	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
50	Dic₇₁⋊C₈	—	2	4	no	—
51	Dic₇₁⋊C₈	—	2	4	no	—
52	Dic₇₁⋊C₈	C₂×C₄	1	4	yes	—
53	Dic₇₁⋊C₈	C₂×C₄	1	4	yes	—
54	#14556038	—	2	4	no	—
55	C₈×Dic₇₁	—	2	4	no	—
56	C₈×Dic₇₁	C₂×C₄	1	4	yes	—
57	C₈×Dic₇₁	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
63	#108×C₇₁	—	2	2	no	—
64	#108×C₇₁	Q₈	1	4	yes	complemented
65	#97×C₇₁	—	2	2	no	—
66	C₄×C₈×C₇₁	—	2	2	no	—
67	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
69	C₂×Dic₁₄₂	—	2	4	no	—
70	C₂×Dic₁₄₂	C₄⋊C₄	1	1	yes	characteristic
71	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
72	C₂×Dic₁₄₂	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×Dic₇₁	C₂×Q₈	1	2	yes	—
74	C₂²×Dic₇₁	C₂×Q₈	1	2	yes	—
75	#11765849	—	2	4	no	—
76	#11765849	—	2	4	no	—
77	C₂×C₇₁⋊C₈	—	2	2	no	—
78	C₂×C₇₁⋊C₈	—	2	2	no	—
79	C₂×C₇₁⋊C₈	—	2	4	no	—
80	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
81	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
82	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
83	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
84	Dic₇₁⋊C₄	C₄⋊C₄	1	2	yes	—
85	Dic₇₁⋊C₄	C₄⋊C₄	1	2	yes	—
86	Dic₇₁⋊C₄	C₄⋊C₄	1	2	yes	—
87	Dic₇₁⋊C₄	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
89	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
90	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
91	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₇₁	—	2	2	no	—
96	M₁₆×C₇₁	—	2	2	no	—
97	M₁₆×C₇₁	—	2	2	no	—
98	M₁₆×C₇₁	C₂×Q₈	1	4	yes	—
99	C₂×C₈×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	—	2	1	no	—
101	C₂×C₈×C₇₁	—	2	1	no	—
102	C₂×C₈×C₇₁	—	2	2	no	—
103	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₇₁	#41	1	2	yes	—
107	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	—	2	2	no	—
113	C₂×C₄×C₇₁	—	4	1	no	—
114	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×Dic₇₁	—	2	4	no	—
125	C₂×Dic₇₁	—	4	2	no	—
126	C₂×Dic₇₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₇₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₇₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₇₁	C₂×C₄⋊C₄	1	2	yes	—
130	Dic₁₄₂	—	2	2	no	—
131	Dic₁₄₂	—	2	4	no	—
132	Dic₁₄₂	#108	1	4	yes	direct product factor
133	C₈×C₇₁	—	2	2	no	—
134	C₈×C₇₁	—	2	4	no	—
135	C₈×C₇₁	—	4	2	no	—
136	C₈×C₇₁	C₄×Q₈	1	4	yes	—
137	C₇₁⋊C₈	—	2	4	no	—
138	C₇₁⋊C₈	—	4	4	no	—
139	C₂²×C₇₁	—	2	1	no	—
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₇₁	#557	1	1	yes	characteristic
143	C₂²×C₇₁	Q₈²	1	1	yes	characteristic
144	C₄×C₇₁	—	2	1	no	—
145	C₄×C₇₁	—	2	1	no	—
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	C₂×#108	1	2	yes	—
148	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₇₁	#520	1	1	yes	characteristic
150	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
151	C₄×C₇₁	#550	1	1	yes	characteristic
152	Dic₇₁	—	2	2	no	—
153	Dic₇₁	—	2	2	no	—
154	Dic₇₁	—	2	2	no	—
155	Dic₇₁	C₂×#108	1	4	yes	—
156	Q₈×#108	—	71	1	no	maximal, Sylow
157	C₂×C₇₁	—	2	1	no	—
158	C₂×C₇₁	—	2	1	no	—
159	C₂×C₇₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₇₁	C₂²×#108	1	1	yes	characteristic
161	C₂×C₇₁	#2916	1	1	yes	characteristic
162	Q₈×M₁₆	—	71	2	no	—
163	C₂×C₈×Q₈	—	71	1	no	—
164	#1950	—	71	2	no	—
165	#1950	—	71	4	no	—
166	#1848	—	71	2	no	—
167	#1848	—	71	1	no	—
168	C₄×#108	—	71	2	no	—
169	C₄×#108	D₇₁	1	1	yes	characteristic
170	C₇₁	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
171	C₂×C₄×Q₈	—	71	1	no	—
172	#446	—	142	2	no	—
173	#446	—	142	4	no	—
174	C₈×Q₈	—	71	2	no	—
175	C₈×Q₈	—	71	4	no	—
176	C₈×Q₈	—	71	2	no	—
177	C₈×Q₈	—	142	2	no	—
178	C₂×#108	—	71	4	no	—
179	C₂×#108	—	71	2	no	—
180	C₂×#108	—	71	2	no	—
181	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
182	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
183	#423	—	71	4	no	—
184	#423	—	71	2	no	—
185	C₂×C₄⋊C₈	—	71	2	no	—
186	C₂×C₄⋊C₈	—	71	1	no	—
187	C₄×M₁₆	—	71	4	no	—
188	C₄×M₁₆	C₂×D₇₁	1	2	yes	—
189	C₂×C₄×C₈	—	71	2	no	—
190	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
191	#341	—	71	4	no	—
192	#341	C₂×D₇₁	1	2	yes	—
193	C₂²×Q₈	—	71	1	no	—
194	C₂×M₁₆	—	71	4	no	—
195	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
196	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
197	C₂²×C₈	—	71	2	no	—
198	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
200	C₄×Q₈	—	71	1	no	—
201	C₄×Q₈	—	71	2	no	—
202	C₄×Q₈	—	71	1	no	—
203	C₄×Q₈	—	71	1	no	—
204	C₄×Q₈	—	71	2	no	—
205	C₄×Q₈	—	71	1	no	—
206	C₂×C₄⋊C₄	—	71	2	no	—
207	C₂×C₄⋊C₄	—	71	1	no	—
208	C₂×C₄²	—	71	2	no	—
209	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
210	#108	—	2	2	no	—
211	#108	—	284	2	no	—
212	#108	—	284	4	no	—
213	#108	—	142	4	no	—
214	#108	Dic₁₄₂	1	4	yes	direct product factor
215	C₄⋊C₈	—	71	4	no	—
216	C₄⋊C₈	—	71	4	no	—
217	C₄⋊C₈	—	71	2	no	—
218	C₄⋊C₈	—	71	2	no	—
219	C₄⋊C₈	—	142	4	no	—
220	C₄⋊C₈	—	142	4	no	—
221	C₄⋊C₈	—	142	2	no	—
222	C₄⋊C₈	—	142	2	no	—
223	#97	—	2	2	no	—
224	#97	—	142	4	no	—
225	C₄×C₈	—	2	2	no	—
226	C₄×C₈	—	71	4	no	—
227	C₄×C₈	—	71	4	no	—
228	C₄×C₈	—	142	4	no	—
229	C₄×C₈	C₄×D₇₁	1	2	yes	—
230	C₄×C₈	C₄×D₇₁	1	2	yes	—
231	C₂×Q₈	—	71	1	no	—
232	C₂×Q₈	—	71	2	no	—
233	C₂×Q₈	—	71	1	no	—
234	C₂×Q₈	—	142	4	no	—
235	C₂²×C₄	—	71	2	no	—
236	C₂²×C₄	—	71	2	no	—
237	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	142	4	no	—
244	M₁₆	—	142	4	no	—
245	M₁₆	C₂×Dic₁₄₂	1	4	yes	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	71	4	no	—
251	C₂×C₈	—	142	4	no	—
252	C₂×C₈	—	142	2	no	—
253	C₂×C₈	—	142	2	no	—
254	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
256	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
257	C₂×C₈	#11765865	1	2	yes	—
258	C₄⋊C₄	—	71	2	no	—
259	C₄⋊C₄	—	71	2	no	—
260	C₄⋊C₄	—	71	2	no	—
261	C₄⋊C₄	—	71	2	no	—
262	C₄⋊C₄	—	71	2	no	—
263	C₄⋊C₄	—	71	1	no	—
264	C₄⋊C₄	—	71	1	no	—
265	C₄²	—	71	2	no	—
266	C₄²	—	71	2	no	—
267	C₄²	—	71	2	no	—
268	C₄²	—	71	2	no	—
269	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
271	C₄²	D₄×D₇₁	1	1	yes	characteristic
272	C₄²	D₄×D₇₁	1	1	yes	characteristic
273	C₂³	C₂×Q₈×D₇₁	1	1	yes	characteristic
274	Q₈	—	71	4	no	—
275	Q₈	—	142	4	no	—
276	Q₈	—	142	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	284	2	no	—
280	C₂×C₄	—	71	2	no	—
281	C₂×C₄	—	71	2	no	—
282	C₂×C₄	—	71	2	no	—
283	C₂×C₄	—	71	2	no	—
284	C₂×C₄	—	142	4	no	—
285	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
286	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
287	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
288	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
292	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
293	C₂×C₄	#14556174	1	1	yes	characteristic
294	C₂×C₄	#14556175	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₈	—	284	4	no	—
299	C₈	—	142	4	no	—
300	C₈	C₄×Dic₁₄₂	1	4	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂×C₄⋊C₄×D₇₁	1	1	yes	characteristic
304	C₂²	#17986727	1	1	yes	characteristic
305	C₂²	Q₈×Dic₁₄₂	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	71	4	no	—
310	C₄	—	142	2	no	—
311	C₄	—	142	2	no	—
312	C₄	—	142	2	no	—
313	C₄	#108×D₇₁	1	2	yes	—
314	C₄	C₂×C₄×Dic₁₄₂	1	1	yes	characteristic
315	C₄	#17986479	1	1	yes	characteristic
316	C₄	D₄×Dic₁₄₂	1	1	yes	characteristic
317	C₄	#17986670	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	C₄⋊C₄×Dic₁₄₂	1	1	yes	characteristic
321	C₂	C₂×#108×D₇₁	1	1	yes	characteristic
322	C₂	#22596948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₁	C₂²×Q₈
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	Q₈×#108
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
64. #108×C₇₁	Q₈	semidirect
132. Dic₁₄₂	#108	direct (N × H)
170. C₇₁	Q₈×#108	semidirect
214. #108	Dic₁₄₂	direct (N × H)

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₄₂	2
C₂×C₈×Dic₁₄₂	2
#22589115	2
#22589114	2
#22588512	2
#22588510	2
#108×Dic₇₁	2
C₄×#108×C₇₁	2
Q₈×#108	71

Subgroups — #108×Dic142 #32016344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×Dic₁₄₂ #32016344