Subgroups — #31008942

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22562013	C₂	1	2	yes	maximal, complemented
3	#22562013	C₂	1	2	yes	maximal, complemented
4	#22562009	C₂	1	2	yes	maximal, complemented
5	#22562005	C₂	1	2	yes	maximal, complemented
6	#22561964	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561823	C₂	1	1	yes	maximal, characteristic
9	#22561808	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561805	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₄	1	1	yes	characteristic
13	#41×Dic₆₇	C₄	1	1	yes	characteristic
14	C₂×#14523983	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863354	C₄	1	1	yes	characteristic
17	#16863354	C₄	1	2	yes	—
18	#16863351	C₂²	1	1	yes	characteristic
19	#16863351	C₂²	1	2	yes	—
20	M₁₆×Dic₆₇	C₄	1	2	yes	—
21	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
24	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
26	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
27	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
30	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
31	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
32	#16863171	C₂²	1	2	yes	—
33	#16863171	C₂²	1	2	yes	—
34	C₁₆×Dic₆₇	C₂²	1	2	yes	—
35	C₁₆×Dic₆₇	C₂²	1	2	yes	—
36	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
40	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
41	#14524050	—	2	4	no	—
42	C₂×#11761612	C₂×C₄	1	2	yes	—
43	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
46	Q₈×Dic₆₇	C₈	1	1	yes	characteristic
47	Q₈×Dic₆₇	C₈	1	1	yes	characteristic
48	D₄×Dic₆₇	C₈	1	2	yes	—
49	D₄×Dic₆₇	C₈	1	1	yes	characteristic
50	D₄×Dic₆₇	C₈	1	2	yes	—
51	D₄×Dic₆₇	C₈	1	1	yes	characteristic
52	#14524002	C₂×C₄	1	2	yes	—
53	#14524002	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
56	#14523983	—	2	2	no	—
57	C₂×C₆₇⋊C₁₆	—	2	2	no	—
58	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
61	#14523953	C₂×C₄	1	2	yes	—
62	#14523953	C₂×C₄	1	2	yes	—
63	#14523953	C₂×C₄	1	2	yes	—
64	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
65	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
67	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	2	yes	—
70	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
71	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
72	#131×C₆₇	—	2	2	no	—
73	#131×C₆₇	—	2	2	no	—
74	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	M₃₂×C₆₇	—	2	2	no	—
79	C₂×C₁₆×C₆₇	—	2	2	no	—
80	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
84	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
85	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
86	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
87	#11761612	—	2	4	no	—
88	#11761612	M₁₆	1	4	yes	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	4	no	—
91	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
94	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
97	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
98	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
99	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
100	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
101	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
102	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
104	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
107	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₆₇⋊C₁₆	—	2	1	no	—
109	C₆₇⋊C₁₆	—	2	1	no	—
110	C₆₇⋊C₁₆	—	2	2	no	—
111	#41×C₆₇	—	2	2	no	—
112	#41×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
114	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
115	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	4	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	#41	1	2	yes	—
125	C₂×C₈×C₆₇	#41	1	2	yes	—
126	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₁₆×C₆₇	—	2	1	no	—
128	C₁₆×C₆₇	—	2	1	no	—
129	C₁₆×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
131	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
132	Q₈×C₆₇	—	2	2	no	—
133	D₄×C₆₇	—	2	4	no	—
134	D₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	#117	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₆₇	—	2	2	no	—
143	C₂×Dic₆₇	—	2	4	no	—
144	C₂×Dic₆₇	—	4	1	no	—
145	C₂×Dic₆₇	—	4	2	no	—
146	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
148	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
150	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
151	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	#117	1	2	yes	—
154	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₆₇⋊C₈	—	2	1	no	—
157	C₆₇⋊C₈	—	2	1	no	—
158	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	#431	1	2	yes	—
165	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	#431	1	2	yes	—
168	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
169	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
170	Dic₆₇	—	2	1	no	—
171	Dic₆₇	—	2	2	no	—
172	Dic₆₇	—	2	1	no	—
173	Dic₆₇	—	2	1	no	—
174	Dic₆₇	—	2	2	no	—
175	Dic₆₇	#504	1	2	yes	—
176	#33423	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	—	2	2	no	—
178	C₂×C₆₇	—	4	2	no	—
179	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
181	C₂×C₆₇	#3407	1	1	yes	characteristic
182	C₄×#131	—	67	1	no	—
183	#2164	—	67	2	no	—
184	#2164	—	67	2	no	—
185	#2163	—	67	2	no	—
186	#2162	—	67	2	no	—
187	#2150	—	67	1	no	—
188	C₂×C₄⋊C₁₆	—	67	1	no	—
189	#2113	—	67	1	no	—
190	#2113	D₆₇	1	1	yes	characteristic
191	#2108	—	67	1	no	—
192	#2108	—	67	1	no	—
193	C₆₇	#33423	1	1	yes	characteristic, complemented, Sylow
194	C₂×#131	—	67	1	no	—
195	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
196	C₄×#41	—	67	1	no	—
197	C₂×M₃₂	—	67	1	no	—
198	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
199	C₂²×C₁₆	—	67	1	no	—
200	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
201	#405	—	67	1	no	—
202	#405	—	67	2	no	—
203	#405	—	67	1	no	—
204	#405	—	67	2	no	—
205	C₄×M₁₆	—	67	1	no	—
206	C₄×M₁₆	—	67	2	no	—
207	C₂×C₄×C₈	—	67	1	no	—
208	C₂×C₄×C₈	—	67	2	no	—
209	C₄⋊C₁₆	—	67	1	no	—
210	C₄⋊C₁₆	—	67	1	no	—
211	C₄⋊C₁₆	—	67	1	no	—
212	C₄⋊C₁₆	—	67	1	no	—
213	C₄⋊C₁₆	—	67	2	no	—
214	C₄⋊C₁₆	—	67	2	no	—
215	C₂²⋊C₁₆	—	67	2	no	—
216	C₂²⋊C₁₆	—	67	2	no	—
217	C₂²⋊C₁₆	C₂×D₆₇	1	2	yes	—
218	C₂²⋊C₁₆	C₂×D₆₇	1	2	yes	—
219	#346	—	67	2	no	—
220	#346	—	67	2	no	—
221	C₄×C₁₆	—	67	2	no	—
222	C₄×C₁₆	—	67	2	no	—
223	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	134	4	no	—
227	C₂×M₁₆	—	67	1	no	—
228	C₂×M₁₆	—	67	2	no	—
229	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
230	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
231	C₂²×C₈	—	67	2	no	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄×Q₈	—	67	1	no	—
236	C₄×Q₈	—	67	1	no	—
237	C₄×D₄	—	67	2	no	—
238	C₄×D₄	—	67	1	no	—
239	C₄×D₄	—	67	2	no	—
240	C₄×D₄	—	67	1	no	—
241	#117	—	67	2	no	—
242	#117	—	67	1	no	—
243	C₂×C₄²	—	67	2	no	—
244	C₂×C₄²	—	67	1	no	—
245	M₃₂	—	2	2	no	—
246	M₃₂	—	134	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	67	1	no	—
249	C₂×C₁₆	—	67	2	no	—
250	C₂×C₁₆	—	67	1	no	—
251	C₂×C₁₆	—	134	2	no	—
252	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
254	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
255	#97	—	67	2	no	—
256	#97	—	67	2	no	—
257	#97	—	67	2	no	—
258	C₄×C₈	—	67	2	no	—
259	C₄×C₈	—	67	1	no	—
260	C₄×C₈	—	67	2	no	—
261	C₄×C₈	—	67	1	no	—
262	C₄×C₈	—	67	2	no	—
263	C₄×C₈	—	67	1	no	—
264	C₄×C₈	—	67	1	no	—
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
268	C₂×D₄	C₈×D₆₇	1	2	yes	—
269	C₂×D₄	C₈×D₆₇	1	1	yes	characteristic
270	C₂²×C₄	—	67	2	no	—
271	C₂²×C₄	—	67	1	no	—
272	C₂²×C₄	—	67	2	no	—
273	C₂²×C₄	—	67	1	no	—
274	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
275	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	67	4	no	—
279	M₁₆	—	134	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	67	2	no	—
283	C₂×C₈	—	67	1	no	—
284	C₂×C₈	—	67	1	no	—
285	C₂×C₈	—	134	4	no	—
286	C₂×C₈	—	134	2	no	—
287	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
289	C₂×C₈	#11761627	1	2	yes	—
290	C₂×C₈	#11761627	1	2	yes	—
291	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
292	C₄⋊C₄	—	67	2	no	—
293	C₄⋊C₄	—	67	2	no	—
294	C₄⋊C₄	—	67	1	no	—
295	C₄⋊C₄	—	67	1	no	—
296	C₂²⋊C₄	—	67	2	no	—
297	C₂²⋊C₄	—	67	2	no	—
298	C₂²⋊C₄	—	67	2	no	—
299	C₄²	—	67	2	no	—
300	C₄²	—	67	1	no	—
301	C₄²	—	67	2	no	—
302	C₄²	—	67	1	no	—
303	C₄²	—	67	1	no	—
304	C₄²	—	67	2	no	—
305	C₄²	—	67	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	134	1	no	—
310	C₁₆	—	134	1	no	—
311	C₁₆	—	134	2	no	—
312	C₂³	C₂×C₈×D₆₇	1	2	yes	—
313	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	268	1	no	—
320	C₂×C₄	—	268	2	no	—
321	C₂×C₄	—	67	1	no	—
322	C₂×C₄	—	67	1	no	—
323	C₂×C₄	—	67	2	no	—
324	C₂×C₄	—	67	1	no	—
325	C₂×C₄	—	67	2	no	—
326	C₂×C₄	—	67	1	no	—
327	C₂×C₄	—	134	2	no	—
328	C₂×C₄	—	134	4	no	—
329	C₂×C₄	#14523999	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
332	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	67	4	no	—
336	C₈	—	134	1	no	—
337	C₈	—	134	1	no	—
338	C₈	#14523999	1	2	yes	—
339	C₈	C₂×#11761627	1	1	yes	characteristic
340	C₈	C₂×#11761627	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	#16863446	1	2	yes	—
347	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	67	2	no	—
350	C₄	—	134	1	no	—
351	C₄	—	134	2	no	—
352	C₄	—	134	1	no	—
353	C₄	—	134	1	no	—
354	C₄	—	134	2	no	—
355	C₄	#16863446	1	2	yes	—
356	C₄	C₂×#14523999	1	1	yes	characteristic
357	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	4	2	no	—
360	C₂	C₂×#16863446	1	1	yes	characteristic
361	C₂	#504×D₆₇	1	1	yes	characteristic
362	C₂	#22573310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2113	C₂
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2113	D₆₇
hypocenter	C₆₇	#33423
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #22562013	C₂	semidirect
3. #22562013	C₂	semidirect
4. #22562009	C₂	semidirect
5. #22562005	C₂	semidirect
6. #22561964	C₂	semidirect
7. C₂×Dic₆₇⋊C₁₆	C₂	semidirect
9. #22561808	C₂	semidirect
10. #22561805	C₂	semidirect
193. C₆₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₆₇	2
#22562013	2
#22562009	2
#22562005	2
#22561964	2
C₂×Dic₆₇⋊C₁₆	2
#22561823	2
#22561808	2
#22561805	2
C₆₇×#2113	2
#33423	67