Subgroups — #30761672

Order 17152.

802 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
2	#22562616	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863167	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₂²	1	2	yes	—
6	C₂×#14524002	C₂²	1	2	yes	—
7	#16863420	—	2	1	no	—
8	#16863420	—	2	1	no	—
9	#16863420	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
11	#16863266	—	2	1	no	—
12	#16863262	—	2	1	no	—
13	#16863256	C₄	1	4	yes	—
14	C₄×C₄⋊Dic₆₇	—	2	2	no	—
15	#16863167	—	2	8	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₆₇	—	2	2	no	—
19	C₄×M₁₆×C₆₇	—	2	2	no	—
20	C₂×#97×C₆₇	C₄	1	2	yes	complemented
21	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	4	no	—
24	C₂×#11761612	—	2	4	no	—
25	C₂×#11761612	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	#14524002	—	2	4	no	—
28	#14524002	—	2	4	no	—
29	C₂×C₄⋊Dic₆₇	—	2	4	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
35	C₆₇⋊#95	—	2	4	no	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
43	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
54	#97×C₆₇	—	2	4	no	—
55	C₄×C₈×C₆₇	—	2	4	no	—
56	C₂²×Dic₆₇	—	2	4	no	—
57	#11761612	—	2	8	no	—
58	#11761612	—	4	2	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	4	4	no	—
62	C₄⋊Dic₆₇	—	4	8	no	—
63	C₄⋊Dic₆₇	—	4	2	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₂²⋊Dic₆₇	—	2	8	no	—
66	C₄×Dic₆₇	—	4	4	no	—
67	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
70	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	M₁₆×C₆₇	—	2	4	no	—
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	complemented
92	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
97	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
98	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₆₇	C₄×D₄	1	2	yes	—
107	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	#95	1	2	yes	—
115	C₂×C₄×C₆₇	#95	1	2	yes	—
116	C₂×C₄×C₆₇	#95	1	2	yes	—
117	C₂×C₄×C₆₇	#95	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#117	1	2	yes	—
124	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	#121	1	2	yes	—
127	C₂×C₄×C₆₇	#121	1	2	yes	—
128	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
131	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#125	1	2	yes	—
133	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×Dic₆₇	—	4	8	no	—
137	C₂×Dic₆₇	—	4	4	no	—
138	C₈×C₆₇	—	2	8	no	—
139	C₆₇⋊C₈	—	4	4	no	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
146	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
147	C₂²×C₆₇	#389	1	2	yes	—
148	C₂²×C₆₇	#390	1	1	yes	characteristic
149	C₂²×C₆₇	#391	1	1	yes	characteristic
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	#344	1	4	yes	complemented
154	C₄×C₆₇	#380	1	2	yes	—
155	C₄×C₆₇	#382	1	1	yes	characteristic
156	C₄×C₆₇	#384	1	1	yes	characteristic
157	C₄×C₆₇	#389	1	2	yes	—
158	C₄×C₆₇	#390	1	1	yes	characteristic
159	C₄×C₆₇	#391	1	1	yes	characteristic
160	Dic₆₇	—	8	4	no	—
161	#10848	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	—	2	2	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#344	1	2	yes	—
168	C₂×#423	—	67	1	no	—
169	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
170	#2285	—	67	1	no	—
171	C₂×#344	—	67	4	no	—
172	C₆₇	#10848	1	1	yes	characteristic, complemented, Sylow
173	C₂²×M₁₆	—	67	2	no	—
174	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
175	C₂×#117	—	67	2	no	—
176	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
177	#423	—	67	4	no	—
178	#423	—	134	1	no	—
179	#423	—	134	1	no	—
180	C₂×C₄⋊C₈	—	134	2	no	—
181	C₄×M₁₆	—	2	2	no	—
182	C₄×M₁₆	—	2	2	no	—
183	C₂×#97	Dic₆₇	1	2	yes	complemented
184	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
185	#384	—	134	1	no	—
186	#382	—	134	1	no	—
187	#380	—	67	4	no	—
188	C₄×C₄⋊C₄	—	134	2	no	—
189	#344	—	134	8	no	—
190	C₂³×C₄	C₄×D₆₇	1	2	yes	—
191	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	67	4	no	—
197	C₂×M₁₆	—	134	4	no	—
198	C₂×M₁₆	—	134	4	no	—
199	C₂×M₁₆	—	134	2	no	—
200	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
201	C₂×M₁₆	D₂₆₈	1	4	yes	—
202	C₂²×C₈	—	134	2	no	—
203	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
204	#117	—	134	4	no	—
205	#117	—	134	4	no	—
206	C₂×C₄⋊C₄	—	134	4	no	—
207	C₂×C₄⋊C₄	—	134	2	no	—
208	C₂×C₂²⋊C₄	—	67	4	no	—
209	C₂×C₄²	—	134	2	no	—
210	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
211	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₆₇	1	2	yes	—
213	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
214	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
218	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
219	C₄⋊C₈	—	134	4	no	—
220	C₄⋊C₈	—	134	4	no	—
221	#97	—	2	4	no	—
222	C₄×C₈	—	2	4	no	—
223	#95	—	134	4	no	—
224	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	134	4	no	—
227	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
228	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
229	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
230	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
231	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
232	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
239	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
241	C₂²×C₄	#11761628	1	2	yes	—
242	C₂²×C₄	#11761628	1	2	yes	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	4	no	—
247	M₁₆	—	268	2	no	—
248	M₁₆	—	268	2	no	—
249	M₁₆	—	134	8	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	268	4	no	—
253	C₂×C₈	—	134	4	no	—
254	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	complemented
255	C₄⋊C₄	—	268	8	no	—
256	C₄⋊C₄	—	268	2	no	—
257	C₄⋊C₄	—	268	2	no	—
258	C₂²⋊C₄	—	134	8	no	—
259	C₄²	—	268	4	no	—
260	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
261	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
262	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
263	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
264	C₄²	Dic₆₇⋊C₄	1	2	yes	—
265	C₄²	Dic₆₇⋊C₄	1	2	yes	—
266	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
267	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
268	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
269	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
272	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
273	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
274	C₂³	C₄×D₂₆₈	1	2	yes	—
275	C₂³	C₄×Dic₁₃₄	1	2	yes	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	268	8	no	—
283	C₂×C₄	—	268	4	no	—
284	C₂×C₄	C₆₇⋊#95	1	2	yes	—
285	C₂×C₄	C₆₇⋊#95	1	2	yes	—
286	C₂×C₄	C₆₇⋊#95	1	2	yes	—
287	C₂×C₄	C₆₇⋊#95	1	2	yes	—
288	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
293	C₂×C₄	#14524002	1	2	yes	—
294	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
295	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
296	C₂×C₄	#14524017	1	2	yes	—
297	C₂×C₄	#14524017	1	2	yes	—
298	C₂×C₄	#14524022	1	2	yes	—
299	C₂×C₄	#14524022	1	2	yes	—
300	C₂×C₄	#14524031	1	1	yes	characteristic
301	C₂×C₄	#14524031	1	1	yes	characteristic
302	C₂×C₄	#14524035	1	2	yes	—
303	C₂×C₄	#14524040	1	1	yes	characteristic
304	C₂×C₄	#14524043	1	2	yes	—
305	C₂×C₄	#14524043	1	1	yes	characteristic
306	C₈	—	2	8	no	—
307	C₈	—	268	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
314	C₂²	C₄×C₄⋊Dic₆₇	1	2	yes	—
315	C₂²	#16863294	1	2	yes	—
316	C₂²	#16863298	1	1	yes	characteristic
317	C₂²	#16863302	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	536	4	no	—
322	C₄	#16863167	1	4	yes	complemented
323	C₄	#16863256	1	2	yes	—
324	C₄	#16863262	1	1	yes	characteristic
325	C₄	#16863266	1	1	yes	characteristic
326	C₄	#16863294	1	2	yes	—
327	C₄	#16863298	1	1	yes	characteristic
328	C₄	#16863302	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	2	no	—
332	C₂	—	2	1	no	—
333	C₂	#22558439	1	1	yes	characteristic
334	C₂	C₂×#16863167	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₇
commutator	C₂³×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#10848
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
9. #16863420	C₄	semidirect
20. C₂×#97×C₆₇	C₄	semidirect
21. C₂×C₄×C₈×C₆₇	C₄	semidirect
91. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
153. C₄×C₆₇	#344	semidirect
172. C₆₇	#10848	semidirect
183. C₂×#97	Dic₆₇	semidirect
184. C₂×C₄×C₈	Dic₆₇	semidirect
254. C₂×C₈	C₄⋊Dic₆₇	semidirect
322. C₄	#16863167	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
#22562616	2
C₂×#16863167	2
C₂×C₄×M₁₆×C₆₇	2
#10848	67