Subgroups — #30856298

Order 17152.

954 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863444	C₂	1	2	yes	maximal
2	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
3	#22568772	C₂	1	2	yes	maximal
4	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
5	#22560382	C₂	1	4	yes	maximal, complemented
6	#22560379	C₂	1	4	yes	maximal, complemented
7	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	2	yes	—
9	C₂²×#11761612	C₂²	1	1	yes	characteristic
10	C₂×#14524002	C₂²	1	1	yes	characteristic
11	#16863444	—	2	8	no	—
12	C₂×#14523979	—	2	4	no	—
13	C₂×#14523979	C₂²	1	2	yes	—
14	C₂×#14523979	C₂²	1	2	yes	—
15	C₂×#14523978	C₂²	1	8	yes	—
16	#16863424	—	2	4	no	—
17	#16863422	—	2	4	no	—
18	#16863419	C₂²	1	8	yes	—
19	#16863351	C₄	1	4	yes	—
20	#16863351	C₄	1	4	yes	—
21	C₂×#14523953	C₂²	1	2	yes	—
22	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₆₇	—	2	4	no	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	—
27	C₂×#108×C₆₇	C₂²	1	2	yes	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	4	no	—
30	C₂×#11761612	—	2	4	no	—
31	C₂×#11761612	C₂×C₄	1	2	yes	—
32	C₂×#11761612	C₂×C₄	1	2	yes	—
33	C₂×#11761612	D₄	1	4	yes	—
34	C₂×#11761612	Q₈	1	4	yes	—
35	C₂×#11761612	C₂³	1	4	yes	—
36	C₂²×C₆₇⋊C₈	—	2	2	no	—
37	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
38	#14524002	C₂×C₄	1	2	yes	—
39	#14524002	C₂×C₄	1	2	yes	—
40	#14524002	D₄	1	2	yes	—
41	#14524002	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₆₇	Q₈	1	2	yes	—
43	C₂×C₂²⋊Dic₆₇	Q₈	1	2	yes	—
44	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
45	#14523979	—	2	4	no	—
46	#14523979	—	2	4	no	—
47	#14523978	—	2	16	no	—
48	Dic₆₇⋊C₈	—	2	16	no	—
49	#14523953	C₂×C₄	1	8	yes	—
50	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂×M₁₆×C₆₇	—	2	4	no	—
54	C₂×M₁₆×C₆₇	—	2	4	no	—
55	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₆₇	D₄	1	4	yes	—
60	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	#108×C₆₇	—	2	4	no	—
64	#108×C₆₇	—	2	4	no	—
65	C₂²×Dic₆₇	C₂×Q₈	1	4	yes	—
66	#11761612	—	2	4	no	—
67	#11761612	—	2	4	no	—
68	#11761612	—	4	4	no	—
69	#11761612	C₄⋊C₄	1	8	yes	—
70	#11761612	C₄⋊C₄	1	8	yes	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	4	4	no	—
73	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
75	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
76	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
77	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
78	C₂²⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
79	C₂²⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
80	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
81	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
82	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	—	2	4	no	—
95	M₁₆×C₆₇	—	4	4	no	—
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	4	4	no	—
98	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
99	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
106	C₂³×C₆₇	—	4	2	no	—
107	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
110	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
111	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	#127	1	2	yes	—
119	C₂×C₄×C₆₇	#127	1	2	yes	—
120	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
132	C₈×C₆₇	—	4	4	no	—
133	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₆₇⋊C₈	—	4	4	no	—
136	C₆₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	#384	1	2	yes	—
143	C₂²×C₆₇	#384	1	2	yes	—
144	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₄×C₆₇	#384	1	2	yes	—
148	C₄×C₆₇	#384	1	2	yes	—
149	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
152	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₆₇	—	2	8	no	—
154	#20807	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	2	2	no	—
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
160	C₂×C₆₇	#2916	1	2	yes	—
161	C₂×#430	—	67	2	no	—
162	C₂²×#108	—	67	1	no	—
163	C₂²×#108	D₆₇	1	1	yes	characteristic
164	#2905	—	67	2	no	—
165	C₂×#405	—	67	1	no	—
166	#1849	—	67	4	no	—
167	#1848	—	67	4	no	—
168	C₆₇	#20807	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	67	1	no	—
170	C₂²×M₁₆	—	67	2	no	—
171	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
172	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
173	C₂×#117	—	67	1	no	—
174	#430	—	134	8	no	—
175	C₂×#108	—	2	4	no	—
176	C₂×#108	—	67	8	no	—
177	C₂×#108	—	67	2	no	—
178	C₂×#108	—	67	2	no	—
179	C₂×#108	—	134	4	no	—
180	C₂×#108	C₂×D₆₇	1	2	yes	—
181	C₂×#108	C₂×D₆₇	1	2	yes	—
182	#424	—	134	4	no	—
183	#424	—	134	4	no	—
184	#423	—	67	8	no	—
185	#405	—	67	4	no	—
186	#405	—	67	4	no	—
187	C₂×#97	—	67	2	no	—
188	C₂×C₄×C₈	—	67	2	no	—
189	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	67	2	no	—
194	C₂×M₁₆	—	67	2	no	—
195	C₂×M₁₆	—	67	4	no	—
196	C₂×M₁₆	—	67	4	no	—
197	C₂×M₁₆	—	67	4	no	—
198	C₂×M₁₆	—	134	4	no	—
199	C₂×M₁₆	—	134	4	no	—
200	C₂×M₁₆	—	134	2	no	—
201	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	67	2	no	—
204	C₂²×C₈	—	134	2	no	—
205	C₂²×C₈	Dic₁₃₄	1	4	yes	—
206	C₂²×C₈	C₄×D₆₇	1	2	yes	—
207	C₂²×C₈	C₄×D₆₇	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
209	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
211	#117	—	67	2	no	—
212	#117	—	67	2	no	—
213	#117	—	67	2	no	—
214	#117	—	67	2	no	—
215	C₂×C₄⋊C₄	—	67	2	no	—
216	C₂×C₂²⋊C₄	—	67	2	no	—
217	C₂×C₄²	—	67	2	no	—
218	#108	—	2	4	no	—
219	#108	—	2	4	no	—
220	#108	—	134	16	no	—
221	#108	—	134	4	no	—
222	#108	—	134	4	no	—
223	C₄⋊C₈	—	134	16	no	—
224	#97	—	67	8	no	—
225	C₄×C₈	—	67	8	no	—
226	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
227	C₂²×C₄	—	67	4	no	—
228	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₆₇	1	2	yes	—
231	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
234	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
235	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
236	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
237	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	4	no	—
241	M₁₆	—	268	4	no	—
242	M₁₆	—	67	8	no	—
243	M₁₆	—	67	8	no	—
244	M₁₆	—	134	4	no	—
245	M₁₆	—	134	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	268	4	no	—
249	C₂×C₈	—	67	4	no	—
250	C₂×C₈	—	67	4	no	—
251	C₂×C₈	—	67	4	no	—
252	C₂×C₈	—	134	4	no	—
253	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
254	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
255	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
257	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
258	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
259	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
260	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
261	C₄⋊C₄	—	67	4	no	—
262	C₄⋊C₄	—	67	4	no	—
263	C₂²⋊C₄	—	67	4	no	—
264	C₂²⋊C₄	—	67	4	no	—
265	C₄²	—	67	4	no	—
266	C₄²	—	67	4	no	—
267	C₂³	—	4	2	no	—
268	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
269	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
270	C₂³	#14524041	1	2	yes	—
271	C₂³	#14524042	1	2	yes	—
272	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
273	C₂×C₄	—	268	4	no	—
274	C₂×C₄	—	67	8	no	—
275	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
278	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
279	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
280	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
281	C₂×C₄	#14524039	1	2	yes	—
282	C₂×C₄	#14524039	1	2	yes	—
283	C₂×C₄	#14524041	1	2	yes	—
284	C₂×C₄	#14524042	1	2	yes	—
285	C₂×C₄	#14524042	1	2	yes	—
286	C₂×C₄	#14524042	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₈	—	268	4	no	—
295	C₈	—	67	8	no	—
296	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
297	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#16863265	1	2	yes	—
304	C₂²	#16863265	1	2	yes	—
305	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
306	C₂²	C₂×#14524041	1	1	yes	characteristic
307	C₂²	C₂×#14524042	1	1	yes	characteristic
308	C₄	—	134	8	no	—
309	C₄	#16863265	1	2	yes	—
310	C₄	#16863265	1	2	yes	—
311	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
312	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
313	C₄	C₂×#14524039	1	1	yes	characteristic
314	C₄	C₂×#14524042	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	C₂×#16863265	1	1	yes	characteristic
320	C₂	#22568817	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	#20807
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
5. #22560382	C₂	semidirect
6. #22560379	C₂	semidirect
168. C₆₇	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863444	2
C₂²×#14523979	2
#22568772	2
C₂×#16863351	2
#22560382	2
#22560379	2
C₂²×#108×C₆₇	2
#20807	67