Subgroups — #31008946

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562016	C₂	1	4	yes	maximal, complemented
2	#22562012	C₂	1	2	yes	maximal, complemented
3	#22562008	C₂	1	2	yes	maximal, complemented
4	#22561965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561823	C₂	1	2	yes	maximal
7	#22561806	C₂	1	2	yes	maximal, complemented
8	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523983	C₂²	1	2	yes	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
14	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
17	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
18	#16863172	C₂²	1	4	yes	—
19	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
20	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
23	#405×C₆₇	C₄	1	2	yes	—
24	#405×C₆₇	C₄	1	1	yes	characteristic
25	#405×C₆₇	C₂²	1	1	yes	characteristic
26	#405×C₆₇	C₂²	1	2	yes	—
27	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	#14523983	—	2	4	no	—
32	C₂×C₆₇⋊C₁₆	—	2	4	no	—
33	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
34	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
35	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
36	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
37	#131×C₆₇	—	2	4	no	—
38	#131×C₆₇	—	2	2	no	—
39	#131×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
49	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
50	C₄×D₄×C₆₇	C₈	1	2	yes	—
51	C₄×D₄×C₆₇	C₈	1	1	yes	characteristic
52	C₄×D₄×C₆₇	C₈	1	2	yes	—
53	C₄×D₄×C₆₇	C₈	1	1	yes	characteristic
54	#117×C₆₇	C₂×C₄	1	2	yes	—
55	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	#97×C₆₇	C₂×C₄	1	2	yes	—
59	#97×C₆₇	C₂×C₄	1	2	yes	—
60	#97×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₆₇⋊C₁₆	—	2	2	no	—
69	C₆₇⋊C₁₆	—	2	2	no	—
70	C₆₇⋊C₁₆	—	2	4	no	—
71	#41×C₆₇	—	2	2	no	—
72	#41×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
75	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	M₁₆×C₆₇	—	2	2	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	—	2	4	no	—
85	M₁₆×C₆₇	M₁₆	1	4	yes	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₇	#41	1	2	yes	—
96	C₂×C₈×C₆₇	#41	1	2	yes	—
97	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
105	C₄²×C₆₇	C₂×C₈	1	2	yes	—
106	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₆₇	C₂×C₈	1	2	yes	—
108	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
113	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
114	Q₈×C₆₇	—	2	2	no	—
115	D₄×C₆₇	—	2	4	no	—
116	D₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	#117	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₆₇	—	2	2	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	#117	1	2	yes	—
138	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
139	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	#431	1	2	yes	—
147	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	2	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	#431	1	2	yes	—
155	C₄×C₆₇	#504	1	2	yes	—
156	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
157	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
158	#33423	—	67	1	no	maximal, Sylow
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	—	4	2	no	—
161	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
162	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
163	C₂×C₆₇	#3407	1	1	yes	characteristic
164	C₄×#131	D₆₇	1	1	yes	characteristic
165	#2164	—	67	4	no	—
166	#2163	—	67	2	no	—
167	#2162	—	67	2	no	—
168	#2150	—	67	1	no	—
169	C₂×C₄⋊C₁₆	—	67	1	no	—
170	#2113	—	67	2	no	—
171	#2108	—	67	2	no	—
172	C₆₇	#33423	1	1	yes	characteristic, complemented, Sylow
173	C₂×#131	Dic₆₇	1	1	yes	characteristic
174	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
175	C₄×#41	Dic₆₇	1	1	yes	characteristic
176	C₂×M₃₂	—	67	2	no	—
177	C₂²×C₁₆	—	67	2	no	—
178	#405	Dic₆₇	1	1	yes	characteristic
179	#405	Dic₆₇	1	2	yes	—
180	#405	C₂×D₆₇	1	1	yes	characteristic
181	#405	C₂×D₆₇	1	2	yes	—
182	C₄×M₁₆	Dic₆₇	1	2	yes	—
183	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
186	C₄⋊C₁₆	—	67	1	no	—
187	C₄⋊C₁₆	—	67	1	no	—
188	C₄⋊C₁₆	—	67	2	no	—
189	C₄⋊C₁₆	—	67	2	no	—
190	C₄⋊C₁₆	—	67	2	no	—
191	C₂²⋊C₁₆	—	67	4	no	—
192	C₂²⋊C₁₆	—	67	4	no	—
193	#346	—	67	4	no	—
194	C₄×C₁₆	—	67	4	no	—
195	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
196	#131	—	2	4	no	—
197	#131	—	2	2	no	—
198	#131	—	2	2	no	—
199	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
200	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
202	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
205	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
206	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
208	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
209	C₄×D₄	C₆₇⋊C₈	1	2	yes	—
210	C₄×D₄	C₆₇⋊C₈	1	1	yes	characteristic
211	C₄×D₄	C₆₇⋊C₈	1	2	yes	—
212	C₄×D₄	C₆₇⋊C₈	1	1	yes	characteristic
213	#117	C₂×Dic₆₇	1	2	yes	—
214	#117	C₂×Dic₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
216	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
217	M₃₂	—	134	4	no	—
218	C₂×C₁₆	—	67	2	no	—
219	C₂×C₁₆	—	67	4	no	—
220	C₂×C₁₆	—	67	2	no	—
221	C₂×C₁₆	—	134	4	no	—
222	#97	C₂×Dic₆₇	1	2	yes	—
223	#97	C₂×Dic₆₇	1	2	yes	—
224	#97	C₂×Dic₆₇	1	2	yes	—
225	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
226	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
227	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
228	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₆₇	1	2	yes	—
230	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
235	C₂×D₄	C₂×C₆₇⋊C₈	1	2	yes	—
236	C₂×D₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
238	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
240	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	#11761612	1	4	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
253	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
256	C₂×C₈	#11761628	1	2	yes	—
257	C₂×C₈	#11761628	1	2	yes	—
258	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
259	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
260	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
261	C₄⋊C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
262	C₄⋊C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
263	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
264	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
265	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
266	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
267	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
268	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
269	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
270	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
271	C₄²	C₂²×Dic₆₇	1	2	yes	—
272	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
273	C₁₆	—	134	2	no	—
274	C₁₆	—	134	2	no	—
275	C₁₆	—	134	4	no	—
276	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
277	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	#14524002	1	2	yes	—
288	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
290	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
291	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761612	1	2	yes	—
297	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	#14524002	1	2	yes	—
302	C₈	C₂×#11761612	1	4	yes	—
303	C₈	C₂×#11761628	1	1	yes	characteristic
304	C₈	C₂×#11761628	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#16863449	1	2	yes	—
311	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#16863449	1	2	yes	—
319	C₄	#16863813	1	2	yes	—
320	C₄	C₂×#14524002	1	1	yes	characteristic
321	C₄	C₂²×#11761612	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#16863449	1	1	yes	characteristic
325	C₂	C₂×#16863813	1	1	yes	characteristic
326	C₂	#22573312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×#131×C₆₇	C₂
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#33423
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #22562016	C₂	semidirect
2. #22562012	C₂	semidirect
3. #22562008	C₂	semidirect
4. #22561965	C₂	semidirect
5. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
7. #22561806	C₂	semidirect
172. C₆₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#22562016	2
#22562012	2
#22562008	2
#22561965	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22561823	2
#22561806	2
C₄×#131×C₆₇	2
#33423	67