Subgroups — #31012659

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#22562434	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562394	C₂	1	2	yes	maximal, complemented
4	#22562353	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562307	C₂	1	2	yes	maximal, complemented
6	#22562249	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562137	C₂	1	2	yes	maximal, complemented
8	C₄×#14523986	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863196	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₃₆	C₂²	1	2	yes	—
13	D₄×D₂₆₈	—	2	1	no	—
14	C₂×#14523986	C₂²	1	1	yes	characteristic
15	C₂×#14523986	C₂²	1	2	yes	complemented
16	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
17	#16863773	C₂²	1	1	yes	characteristic
18	#16863632	C₂²	1	2	yes	—
19	#16863592	—	2	1	no	—
20	#16863532	—	2	2	no	—
21	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
22	#16863223	C₂²	1	2	yes	complemented
23	#16863223	C₂²	1	1	yes	characteristic, complemented
24	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
25	#16863200	C₂²	1	1	yes	characteristic, complemented
26	#16863196	C₂²	1	1	yes	characteristic
27	#16863196	C₂²	1	2	yes	—
28	#16863196	C₂²	1	2	yes	complemented
29	#16863196	C₂²	1	2	yes	complemented
30	#16863196	C₂²	1	2	yes	complemented
31	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	complemented
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
33	#554×C₆₇	—	2	1	no	—
34	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#484×C₆₇	—	2	2	no	—
36	#477×C₆₇	—	2	1	no	—
37	#467×C₆₇	C₂²	1	2	yes	complemented
38	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
41	C₂²×D₂₆₈	—	2	2	no	—
42	C₂×D₄×D₆₇	—	4	1	no	—
43	C₂×D₅₃₆	—	2	4	no	—
44	C₂×D₅₃₆	—	2	2	no	—
45	C₂×D₅₃₆	—	2	1	no	—
46	C₂×D₅₃₆	—	4	2	no	—
47	C₂×D₅₃₆	C₂³	1	2	yes	—
48	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
49	#14524040	—	2	1	no	—
50	#14524018	—	4	1	no	—
51	#14524017	—	2	2	no	—
52	#14524013	—	4	2	no	—
53	C₄×D₂₆₈	—	2	1	no	—
54	#14523986	—	2	2	no	—
55	#14523986	—	4	2	no	—
56	C₂×C₆₇⋊C₁₆	—	4	2	no	—
57	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14523977	C₂³	1	1	yes	characteristic
61	#14523965	—	2	1	no	—
62	#14523965	—	2	2	no	—
63	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₆₇	—	2	1	no	—
66	#125×C₆₇	—	4	1	no	—
67	C₂²⋊Q₈×C₆₇	—	2	2	no	—
68	C₂²⋊Q₈×C₆₇	—	4	2	no	—
69	C₄×Q₈×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	—	4	1	no	—
71	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	#107×C₆₇	—	2	2	no	—
74	#107×C₆₇	—	2	1	no	—
75	#107×C₆₇	C₂³	1	2	yes	—
76	#107×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
78	#103×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×D₆₇	—	4	2	no	—
83	C₂×D₂₆₈	—	2	1	no	—
84	C₂×D₂₆₈	—	2	2	no	—
85	C₂×D₂₆₈	—	4	4	no	—
86	C₂×D₂₆₈	—	4	2	no	—
87	C₂×D₂₆₈	—	4	1	no	—
88	C₂×D₂₆₈	—	4	1	no	—
89	C₂×D₂₆₈	—	8	2	no	—
90	C₂×C₂²⋊D₆₇	—	4	2	no	—
91	D₄×D₆₇	—	8	4	no	—
92	C₂×C₄×D₆₇	—	4	1	no	—
93	D₅₃₆	—	2	2	no	—
94	D₅₃₆	—	2	1	no	—
95	D₅₃₆	—	2	2	no	—
96	D₅₃₆	—	4	4	no	—
97	D₅₃₆	—	8	1	no	—
98	D₅₃₆	#41	1	2	yes	—
99	C₄⋊Dic₆₇	—	2	1	no	—
100	(C₂×D₆₇)⋊C₄	—	4	1	no	—
101	(C₂×D₆₇)⋊C₄	—	4	2	no	—
102	C₆₇⋊C₁₆	—	2	1	no	—
103	C₆₇⋊C₁₆	—	2	1	no	—
104	C₆₇⋊C₁₆	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	2	1	no	—
106	C₂×Q₈×C₆₇	—	4	1	no	—
107	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	Q₁₆×C₆₇	—	2	1	no	—
111	Q₁₆×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
126	C₄²×C₆₇	—	4	1	no	—
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₆₇	C₂×D₈	1	2	yes	—
129	C₂²×D₆₇	—	4	1	no	—
130	C₂²×D₆₇	—	4	2	no	—
131	C₂²×D₆₇	—	8	4	no	—
132	C₂²×D₆₇	—	8	4	no	—
133	C₂²×D₆₇	—	16	2	no	—
134	Q₈×C₆₇	—	4	1	no	—
135	Q₈×C₆₇	—	4	1	no	—
136	Q₈×C₆₇	—	16	1	no	—
137	D₄×C₆₇	SD₃₂	1	4	yes	complemented
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	16	2	no	—
144	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₆₇	—	8	4	no	—
149	C₂×Dic₆₇	—	4	1	no	—
150	D₂₆₈	—	2	2	no	—
151	D₂₆₈	—	4	2	no	—
152	D₂₆₈	—	4	1	no	—
153	D₂₆₈	—	4	2	no	—
154	D₂₆₈	—	8	4	no	—
155	D₂₆₈	—	8	2	no	—
156	D₂₆₈	—	16	1	no	—
157	C₄×D₆₇	—	8	2	no	—
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	—
165	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₆₇	—	4	2	no	—
167	C₂×D₆₇	—	8	1	no	—
168	C₂×D₆₇	—	8	1	no	—
169	C₂×D₆₇	—	8	2	no	—
170	C₂×D₆₇	—	8	2	no	—
171	C₂×D₆₇	—	8	2	no	—
172	C₂×D₆₇	—	16	8	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	—	8	2	no	—
177	C₄×C₆₇	—	8	1	no	—
178	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
179	C₄×C₆₇	#546	1	1	yes	characteristic
180	C₄×C₆₇	#577	1	1	yes	characteristic
181	Dic₆₇	—	8	1	no	—
182	#33862	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	—	2	4	no	—
184	C₂×C₆₇	#3294	1	1	yes	characteristic
185	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
186	C₂×C₆₇	#3415	1	1	yes	characteristic
187	D₆₇	—	8	2	no	—
188	D₆₇	—	16	2	no	—
189	#3338	D₆₇	1	1	yes	characteristic, complemented
190	D₄×D₈	—	67	1	no	—
191	#2240	—	67	1	no	—
192	#2233	—	67	2	no	—
193	#2226	—	67	1	no	—
194	#2217	—	67	2	no	—
195	#2207	—	67	1	no	—
196	#2187	—	67	2	no	—
197	C₄×SD₃₂	—	67	1	no	—
198	D₄×C₁₆	—	67	1	no	—
199	C₂×#357	—	67	2	no	—
200	C₆₇	#33862	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₈	—	67	2	no	—
202	#554	—	2	1	no	—
203	D₄²	—	134	1	no	—
204	C₂×SD₃₂	—	67	2	no	—
205	C₂×SD₃₂	—	67	1	no	—
206	C₂²×C₁₆	—	67	2	no	—
207	#500	C₂×D₆₇	1	1	yes	characteristic, complemented
208	#493	—	67	1	no	—
209	#484	—	2	2	no	—
210	#477	—	2	1	no	—
211	#467	C₂×D₆₇	1	2	yes	complemented
212	#466	—	67	2	no	—
213	#459	—	134	1	no	—
214	#447	—	134	2	no	—
215	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
216	C₄×D₈	—	67	1	no	—
217	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
218	C₂×#107	C₂×D₆₇	1	2	yes	complemented
219	#367	—	67	1	no	—
220	#367	—	67	2	no	—
221	C₄⋊C₁₆	—	67	1	no	—
222	#358	—	67	1	no	—
223	#357	—	67	1	no	—
224	#357	—	67	2	no	—
225	#357	—	67	2	no	—
226	#357	—	67	2	no	—
227	#357	—	67	2	no	—
228	C₂²⋊C₁₆	—	67	2	no	—
229	C₄×C₁₆	—	67	1	no	—
230	C₂²×D₄	—	268	1	no	—
231	C₂²×D₄	—	134	2	no	—
232	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
233	C₂×D₈	—	268	2	no	—
234	C₂×D₈	—	67	1	no	—
235	C₂×D₈	—	67	2	no	—
236	C₂×D₈	—	134	4	no	—
237	C₂×D₈	—	134	2	no	—
238	C₂×D₈	—	134	1	no	—
239	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
240	C₄⋊Q₈	—	2	1	no	—
241	#127	—	134	1	no	—
242	#125	—	4	1	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	C₂²⋊Q₈	—	4	2	no	—
245	#121	—	268	1	no	—
246	#121	—	134	2	no	—
247	#120	—	268	2	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×D₄	—	4	1	no	—
250	C₄×D₄	—	134	1	no	—
251	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	2	2	no	—
253	SD₃₂	—	268	2	no	—
254	SD₃₂	—	134	2	no	—
255	C₂×C₁₆	—	268	2	no	—
256	C₂×C₁₆	—	67	1	no	—
257	C₂×C₁₆	—	67	1	no	—
258	C₂×C₁₆	—	67	2	no	—
259	#107	—	2	2	no	—
260	#107	—	2	1	no	—
261	#107	—	67	1	no	—
262	#107	C₂²×D₆₇	1	1	yes	characteristic
263	#107	C₂²×D₆₇	1	2	yes	—
264	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
265	#103	—	2	1	no	—
266	#103	—	2	2	no	—
267	#102	—	134	1	no	—
268	#102	—	134	2	no	—
269	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
270	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	C₂⁴	—	268	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×D₄	—	268	4	no	—
275	C₂×D₄	—	268	2	no	—
276	C₂×D₄	—	268	2	no	—
277	C₂×D₄	—	268	1	no	—
278	C₂×D₄	—	268	1	no	—
279	C₂×D₄	—	536	2	no	—
280	C₂×D₄	—	536	4	no	—
281	C₂×D₄	—	134	2	no	—
282	C₂×D₄	—	134	1	no	—
283	C₂×D₄	#11761613	1	1	yes	characteristic
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	268	1	no	—
286	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
287	Q₁₆	—	2	1	no	—
288	Q₁₆	—	2	1	no	—
289	D₈	—	268	4	no	—
290	D₈	—	536	1	no	—
291	D₈	—	67	2	no	—
292	D₈	—	134	2	no	—
293	D₈	—	134	2	no	—
294	D₈	—	134	1	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
299	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	134	1	no	—
307	C₄⋊C₄	#11761613	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	268	1	no	—
310	C₂²⋊C₄	—	268	2	no	—
311	C₂²⋊C₄	#11761613	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
314	C₁₆	—	134	1	no	—
315	C₁₆	—	134	1	no	—
316	C₁₆	—	134	2	no	—
317	C₂³	—	268	1	no	—
318	C₂³	—	268	2	no	—
319	C₂³	—	536	4	no	—
320	C₂³	—	536	4	no	—
321	C₂³	—	1072	2	no	—
322	C₂³	C₂×#11761613	1	2	yes	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	16	1	no	—
326	D₄	—	268	2	no	—
327	D₄	—	268	1	no	—
328	D₄	—	268	2	no	—
329	D₄	—	536	4	no	—
330	D₄	—	536	4	no	—
331	D₄	—	536	2	no	—
332	D₄	—	1072	1	no	—
333	D₄	—	134	2	no	—
334	D₄	#14523986	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	268	1	no	—
341	C₂×C₄	—	16	2	no	—
342	C₂×C₄	—	536	2	no	—
343	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761613	1	2	yes	—
345	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	1	no	—
349	C₈	#41×D₆₇	1	1	yes	characteristic
350	C₈	#14524089	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	268	2	no	—
354	C₂²	—	536	1	no	—
355	C₂²	—	536	1	no	—
356	C₂²	—	536	2	no	—
357	C₂²	—	536	2	no	—
358	C₂²	—	536	2	no	—
359	C₂²	—	1072	8	no	—
360	C₂²	C₂×#14523986	1	4	yes	—
361	C₂²	C₂²×#11761613	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	1	no	—
367	C₄	—	536	1	no	—
368	C₄	C₂×#14523986	1	2	yes	—
369	C₄	#16864034	1	1	yes	characteristic
370	C₄	#16864204	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	536	2	no	—
373	C₂	—	1072	2	no	—
374	C₂	#22572016	1	1	yes	characteristic
375	C₂	C₂²×#14523986	1	1	yes	characteristic
376	C₂	#22573348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3338	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	D₄×D₅₃₆	C₂
hypercenter	#3338	D₆₇
hypocenter	C₆₇	#33862
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3338

Complements

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

N	H ≅ G/N	type
2. #22562434	C₂	semidirect
3. #22562394	C₂	semidirect
4. #22562353	C₂	semidirect
5. #22562307	C₂	semidirect
6. #22562249	C₂	semidirect
7. #22562137	C₂	semidirect
8. C₄×#14523986	C₂	semidirect
9. D₄×C₆₇⋊C₁₆	C₂	semidirect
10. C₂×#16863196	C₂	semidirect
11. C₆₇×#3338	C₂	semidirect
15. C₂×#14523986	C₂²	semidirect
16. C₂²×C₆₇⋊C₁₆	C₂²	semidirect
22. #16863223	C₂²	semidirect
23. #16863223	C₂²	semidirect
24. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
25. #16863200	C₂²	semidirect
28. #16863196	C₂²	semidirect
29. #16863196	C₂²	semidirect
30. #16863196	C₂²	semidirect
31. C₂²⋊(C₆₇⋊C₁₆)	C₂²	semidirect
32. C₄×C₆₇⋊C₁₆	C₂²	semidirect
34. #500×C₆₇	C₂²	semidirect
37. #467×C₆₇	C₂²	semidirect
38. C₄×Q₁₆×C₆₇	C₂²	semidirect
40. C₂×#107×C₆₇	C₂²	semidirect
137. D₄×C₆₇	SD₃₂	semidirect
189. #3338	D₆₇	semidirect
200. C₆₇	#33862	semidirect
207. #500	C₂×D₆₇	semidirect
211. #467	C₂×D₆₇	semidirect
215. C₄×Q₁₆	C₂×D₆₇	semidirect
218. C₂×#107	C₂×D₆₇	semidirect
334. D₄	#14523986	semidirect

Maximal subgroups

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

subgroup	index
D₄×D₅₃₆	2
#22562434	2
#22562394	2
#22562353	2
#22562307	2
#22562249	2
#22562137	2
C₄×#14523986	2
D₄×C₆₇⋊C₁₆	2
C₂×#16863196	2
C₆₇×#3338	2
#33862	67