Subgroups — #31012700

Order 17152.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562406	C₂	1	2	yes	maximal, complemented
4	#22562348	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562303	C₂	1	2	yes	maximal, complemented
6	#22562254	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562125	C₂	1	2	yes	maximal, complemented
8	C₄×#14523987	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863199	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3334	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
13	D₄×Dic₁₃₄	—	2	1	no	—
14	C₂×#14523987	C₂²	1	1	yes	characteristic
15	C₂×#14523987	C₂²	1	2	yes	—
16	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
17	#16863776	C₂²	1	1	yes	characteristic
18	#16863637	C₂²	1	2	yes	—
19	#16863607	—	2	1	no	—
20	#16863554	—	2	2	no	—
21	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
22	#16863223	C₂²	1	1	yes	characteristic
23	#16863223	C₂²	1	2	yes	—
24	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#16863199	C₂²	1	1	yes	characteristic
26	#16863199	C₂²	1	2	yes	—
27	#16863199	C₂²	1	2	yes	—
28	#16863199	C₂²	1	2	yes	—
29	#16863199	C₂²	1	2	yes	—
30	#16863197	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	#547×C₆₇	—	2	1	no	—
34	#500×C₆₇	C₂²	1	1	yes	characteristic
35	#480×C₆₇	—	2	2	no	—
36	#474×C₆₇	—	2	1	no	—
37	#466×C₆₇	C₂²	1	2	yes	—
38	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₆₇	C₂²	1	2	yes	—
41	C₂²×Dic₁₃₄	—	2	2	no	—
42	C₂×Dic₂₆₈	—	2	4	no	—
43	C₂×Dic₂₆₈	—	2	2	no	—
44	C₂×Dic₂₆₈	—	2	1	no	—
45	C₂×Dic₂₆₈	—	4	2	no	—
46	C₂×Dic₂₆₈	C₂³	1	2	yes	—
47	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
48	#14524043	—	2	1	no	—
49	#14524042	—	4	1	no	—
50	#14524022	—	2	2	no	—
51	#14524020	—	4	2	no	—
52	C₄×Dic₁₃₄	—	2	1	no	—
53	D₄×Dic₆₇	—	4	1	no	—
54	#14523987	—	2	2	no	—
55	#14523987	—	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	#14523968	—	2	1	no	—
62	#14523968	—	2	2	no	—
63	C₂×#41×C₆₇	—	4	1	no	—
64	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₆₇	—	2	1	no	—
67	#123×C₆₇	—	4	2	no	—
68	C₂²⋊Q₈×C₆₇	—	4	1	no	—
69	#121×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	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	#102×C₆₇	—	2	2	no	—
79	#102×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₃₄	—	2	1	no	—
83	C₂×Dic₁₃₄	—	2	2	no	—
84	C₂×Dic₁₃₄	—	4	4	no	—
85	C₂×Dic₁₃₄	—	4	2	no	—
86	C₂×Dic₁₃₄	—	8	2	no	—
87	C₂²×Dic₆₇	—	4	2	no	—
88	Dic₂₆₈	—	2	2	no	—
89	Dic₂₆₈	—	2	1	no	—
90	Dic₂₆₈	—	2	2	no	—
91	Dic₂₆₈	—	4	4	no	—
92	Dic₂₆₈	—	8	1	no	—
93	Dic₂₆₈	#41	1	2	yes	complemented
94	C₄⋊Dic₆₇	—	2	1	no	—
95	C₄⋊Dic₆₇	—	4	1	no	—
96	C₄⋊Dic₆₇	—	4	1	no	—
97	Dic₆₇⋊C₄	—	4	1	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	C₂²⋊Dic₆₇	—	4	2	no	—
100	C₄×Dic₆₇	—	4	1	no	—
101	C₆₇⋊C₁₆	—	2	1	no	—
102	C₆₇⋊C₁₆	—	2	1	no	—
103	C₆₇⋊C₁₆	—	2	2	no	—
104	#41×C₆₇	—	8	4	no	—
105	C₂×Q₈×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	—	2	1	no	—
107	C₂×D₄×C₆₇	—	4	2	no	—
108	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	C₂²×C₄×C₆₇	—	4	1	no	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	D₈×C₆₇	—	2	1	no	—
113	D₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₆₇	—	4	1	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₆₇	—	4	1	no	—
129	C₂³×C₆₇	C₂×D₈	1	2	yes	—
130	Q₈×C₆₇	—	8	2	no	—
131	D₄×C₆₇	—	4	1	no	—
132	D₄×C₆₇	—	4	1	no	—
133	D₄×C₆₇	—	8	4	no	—
134	D₄×C₆₇	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	8	4	no	—
140	C₂×C₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₆₇	—	4	1	no	—
146	C₂×Dic₆₇	—	4	1	no	—
147	C₂×Dic₆₇	—	4	2	no	—
148	C₂×Dic₆₇	—	8	4	no	—
149	Dic₁₃₄	—	2	2	no	—
150	Dic₁₃₄	—	4	1	no	—
151	Dic₁₃₄	—	4	2	no	—
152	Dic₁₃₄	—	4	2	no	—
153	Dic₁₃₄	—	8	4	no	—
154	C₈×C₆₇	—	2	2	no	—
155	C₈×C₆₇	—	2	1	no	—
156	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
157	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	—	8	1	no	—
162	C₂²×C₆₇	—	8	1	no	—
163	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	—
164	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₆₇	—	2	1	no	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	8	1	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
170	C₄×C₆₇	#546	1	1	yes	characteristic
171	C₄×C₆₇	#577	1	1	yes	characteristic
172	Dic₆₇	—	4	2	no	—
173	Dic₆₇	—	8	1	no	—
174	Dic₆₇	—	8	2	no	—
175	#33865	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	4	no	—
177	C₂×C₆₇	—	16	1	no	—
178	C₂×C₆₇	#3294	1	1	yes	characteristic
179	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
180	C₂×C₆₇	#3417	1	1	yes	characteristic
181	#3334	D₆₇	1	1	yes	characteristic
182	D₄×Q₁₆	—	67	1	no	—
183	#2241	—	67	1	no	—
184	#2235	—	67	2	no	—
185	#2225	—	67	1	no	—
186	#2217	—	67	2	no	—
187	#2208	—	67	1	no	—
188	#2185	—	67	2	no	—
189	C₄×SD₃₂	—	67	1	no	—
190	D₄×C₁₆	—	67	1	no	—
191	C₂×#358	—	67	2	no	—
192	C₆₇	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	—	67	2	no	—
194	D₄×Q₈	—	134	1	no	—
195	#547	—	2	1	no	—
196	C₂×SD₃₂	—	67	2	no	—
197	C₂×SD₃₂	—	67	1	no	—
198	C₂²×C₁₆	—	67	2	no	—
199	#500	C₂×D₆₇	1	1	yes	characteristic
200	#494	—	67	1	no	—
201	#480	—	2	2	no	—
202	#474	—	2	1	no	—
203	#467	—	67	2	no	—
204	#466	C₂×D₆₇	1	2	yes	—
205	#462	—	134	1	no	—
206	#451	—	134	2	no	—
207	C₄×Q₁₆	—	67	1	no	—
208	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
209	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
210	C₂×#107	C₂×D₆₇	1	2	yes	—
211	#367	—	67	1	no	—
212	#367	—	67	2	no	—
213	C₄⋊C₁₆	—	67	1	no	—
214	#358	—	67	1	no	—
215	#358	—	67	2	no	—
216	#358	—	67	2	no	—
217	#358	—	67	2	no	—
218	#358	—	67	2	no	—
219	#357	—	67	1	no	—
220	C₂²⋊C₁₆	—	67	2	no	—
221	C₄×C₁₆	—	67	1	no	—
222	C₂×#41	—	4	1	no	—
223	C₂²×Q₈	—	134	2	no	—
224	C₂×Q₁₆	—	268	2	no	—
225	C₂×Q₁₆	—	67	1	no	—
226	C₂×Q₁₆	—	67	2	no	—
227	C₂×Q₁₆	—	134	4	no	—
228	C₂×Q₁₆	—	134	2	no	—
229	C₂×Q₁₆	—	134	1	no	—
230	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	268	1	no	—
234	C₄⋊Q₈	—	134	1	no	—
235	#123	—	4	2	no	—
236	C₂²⋊Q₈	—	4	1	no	—
237	C₂²⋊Q₈	—	268	2	no	—
238	C₂²⋊Q₈	—	134	2	no	—
239	#121	—	2	2	no	—
240	C₄×Q₈	—	134	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	268	1	no	—
243	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	2	2	no	—
245	SD₃₂	—	268	2	no	—
246	SD₃₂	—	134	2	no	—
247	C₂×C₁₆	—	268	2	no	—
248	C₂×C₁₆	—	67	1	no	—
249	C₂×C₁₆	—	67	1	no	—
250	C₂×C₁₆	—	67	2	no	—
251	#107	—	2	2	no	—
252	#107	—	2	1	no	—
253	#107	—	67	1	no	—
254	#107	C₂²×D₆₇	1	1	yes	characteristic
255	#107	C₂²×D₆₇	1	2	yes	—
256	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
257	#103	—	134	1	no	—
258	#103	—	134	2	no	—
259	#102	—	2	1	no	—
260	#102	—	2	2	no	—
261	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
262	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
263	#41	—	8	4	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	268	4	no	—
266	C₂×Q₈	—	268	2	no	—
267	C₂×Q₈	—	536	2	no	—
268	C₂×Q₈	—	134	2	no	—
269	C₂×Q₈	—	134	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	#11761613	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	268	2	no	—
276	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
277	Q₁₆	—	268	4	no	—
278	Q₁₆	—	536	1	no	—
279	Q₁₆	—	67	2	no	—
280	Q₁₆	—	134	2	no	—
281	Q₁₆	—	134	2	no	—
282	Q₁₆	—	134	1	no	—
283	D₈	—	2	1	no	—
284	D₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
289	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	268	1	no	—
295	C₄⋊C₄	—	268	1	no	—
296	C₄⋊C₄	—	268	1	no	—
297	C₄⋊C₄	—	268	2	no	—
298	C₄⋊C₄	—	134	1	no	—
299	C₄⋊C₄	#11761613	1	1	yes	characteristic
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	268	2	no	—
303	C₂²⋊C₄	#11761613	1	2	yes	—
304	C₄²	—	268	1	no	—
305	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
306	C₁₆	—	134	1	no	—
307	C₁₆	—	134	1	no	—
308	C₁₆	—	134	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	C₂×#11761613	1	2	yes	—
311	Q₈	—	8	2	no	—
312	Q₈	—	268	1	no	—
313	Q₈	—	268	2	no	—
314	Q₈	—	268	2	no	—
315	Q₈	—	536	4	no	—
316	Q₈	—	134	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	8	4	no	—
320	D₄	#14523987	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	268	1	no	—
328	C₂×C₄	—	268	1	no	—
329	C₂×C₄	—	268	2	no	—
330	C₂×C₄	—	536	4	no	—
331	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
332	C₂×C₄	C₂×#11761613	1	2	yes	—
333	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	#41×D₆₇	1	1	yes	characteristic
338	C₈	#14524089	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	C₂×#14523987	1	4	yes	—
345	C₂²	C₂²×#11761613	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	268	2	no	—
351	C₄	—	536	1	no	—
352	C₄	—	536	2	no	—
353	C₄	C₂×#14523987	1	2	yes	—
354	C₄	#16864034	1	1	yes	characteristic
355	C₄	#16864204	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	16	1	no	—
358	C₂	#22572016	1	1	yes	characteristic
359	C₂	C₂²×#14523987	1	1	yes	characteristic
360	C₂	#22573360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3334	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	#3334	D₆₇
hypocenter	C₆₇	#33865
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₂₆₈	C₂	semidirect
2. #22562438	C₂	semidirect
3. #22562406	C₂	semidirect
4. #22562348	C₂	semidirect
5. #22562303	C₂	semidirect
6. #22562254	C₂	semidirect
7. #22562125	C₂	semidirect
8. C₄×#14523987	C₂	semidirect
9. D₄×C₆₇⋊C₁₆	C₂	semidirect
10. C₂×#16863199	C₂	semidirect
93. Dic₂₆₈	#41	semidirect
134. D₄×C₆₇	SD₃₂	semidirect
192. C₆₇	#33865	semidirect
320. D₄	#14523987	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₂₆₈	2
#22562438	2
#22562406	2
#22562348	2
#22562303	2
#22562254	2
#22562125	2
C₄×#14523987	2
D₄×C₆₇⋊C₁₆	2
C₂×#16863199	2
C₆₇×#3334	2
#33865	67