Subgroups — #31009184

Order 17152.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863837	C₂	1	1	yes	maximal, characteristic
2	#22569086	C₂	1	1	yes	maximal, characteristic
3	#22562078	C₂	1	2	yes	maximal, complemented
4	#22562071	C₂	1	2	yes	maximal, complemented
5	C₄×#14523989	C₂	1	2	yes	maximal, complemented
6	C₄×#14523986	C₂	1	2	yes	maximal, complemented
7	#22561998	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561914	C₂	1	1	yes	maximal, characteristic
9	C₂×#16863200	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561806	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	#16863876	—	2	1	no	—
14	#16863837	C₄	1	8	yes	complemented
15	C₂×#14523989	C₂²	1	2	yes	—
16	C₂×#14523986	C₂²	1	2	yes	—
17	C₂×#14523983	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	#16863498	—	2	2	no	—
20	C₄×Dic₂₆₈	C₂²	1	2	yes	—
21	C₄×D₅₃₆	C₂²	1	2	yes	—
22	C₂×#14523977	C₂²	1	1	yes	characteristic
23	#16863398	—	2	1	no	—
24	#16863223	C₂²	1	2	yes	—
25	#16863221	C₂²	1	2	yes	—
26	#16863200	C₂²	1	2	yes	—
27	#16863200	C₂²	1	2	yes	—
28	#16863199	C₂²	1	1	yes	characteristic
29	#16863199	C₂²	1	1	yes	characteristic
30	#16863196	C₂²	1	1	yes	characteristic
31	#16863196	C₂²	1	1	yes	characteristic
32	#16863172	C₂²	1	2	yes	—
33	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
34	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	#519×C₆₇	—	2	1	no	—
36	#441×C₆₇	—	2	2	no	—
37	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
39	#427×C₆₇	C₂²	1	1	yes	characteristic
40	#419×C₆₇	—	2	1	no	—
41	#405×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#11761628	—	2	1	no	—
43	#14524065	—	2	2	no	—
44	#14524065	C₂×C₄	1	4	yes	—
45	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
46	C₂×#11761617	—	2	1	no	—
47	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
48	C₄×Dic₁₃₄	—	2	2	no	—
49	C₄×D₂₆₈	—	2	2	no	—
50	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
51	#14524001	—	4	1	no	—
52	C₂×C₄⋊Dic₆₇	—	2	1	no	—
53	C₄⋊C₄×D₆₇	—	4	1	no	—
54	#14523989	C₂×C₄	1	8	yes	complemented
55	#14523986	C₂×C₄	1	8	yes	complemented
56	#14523983	C₂×C₄	1	4	yes	complemented
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	#14523977	—	2	1	no	—
62	#14523977	C₂³	1	2	yes	—
63	#14523968	—	2	2	no	—
64	#14523965	—	2	2	no	—
65	C₂²×Q₈×C₆₇	—	2	1	no	—
66	C₂×Q₁₆×C₆₇	—	2	1	no	—
67	C₂×Q₁₆×C₆₇	—	2	2	no	—
68	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
69	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	—	2	2	no	—
73	C₄×Q₈×C₆₇	—	2	2	no	—
74	C₄×Q₈×C₆₇	—	4	1	no	—
75	C₄×Q₈×C₆₇	—	4	1	no	—
76	#117×C₆₇	—	2	1	no	—
77	#117×C₆₇	—	4	2	no	—
78	#117×C₆₇	D₄	1	1	yes	characteristic
79	#107×C₆₇	C₂³	1	1	yes	characteristic
80	#107×C₆₇	C₂³	1	1	yes	characteristic
81	#106×C₆₇	—	2	1	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#97×C₆₇	D₄	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	#11761628	—	2	4	no	—
87	#11761628	—	4	2	no	—
88	C₂×Dic₁₃₄	—	2	1	no	—
89	C₂×D₂₆₈	—	2	1	no	—
90	C₂×C₂²⋊D₆₇	—	4	1	no	—
91	C₂²×Dic₆₇	—	4	1	no	—
92	C₂×C₄×D₆₇	—	4	1	no	—
93	C₂×C₄×D₆₇	—	4	2	no	—
94	Dic₂₆₈	—	2	1	no	—
95	Dic₂₆₈	C₂²×C₄	1	2	yes	—
96	#11761617	—	2	4	no	—
97	D₅₃₆	—	2	1	no	—
98	D₅₃₆	C₂²×C₄	1	2	yes	—
99	C₄⋊Dic₆₇	—	2	2	no	—
100	C₄⋊Dic₆₇	—	4	1	no	—
101	Dic₆₇⋊C₄	—	4	2	no	—
102	(C₂×D₆₇)⋊C₄	—	4	2	no	—
103	C₄×Dic₆₇	—	4	2	no	—
104	C₆₇⋊C₁₆	—	2	1	no	—
105	C₆₇⋊C₁₆	—	2	1	no	—
106	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	—
107	C₂×Q₈×C₆₇	—	2	4	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	—	4	1	no	—
110	C₂×Q₈×C₆₇	—	4	2	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₆₇	—	2	4	no	—
114	Q₁₆×C₆₇	—	2	2	no	—
115	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
116	M₁₆×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	#41	1	2	yes	—
121	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	4	2	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
127	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
128	C₂²⋊C₄×C₆₇	—	4	1	no	—
129	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
130	C₄²×C₆₇	—	4	1	no	—
131	C₄²×C₆₇	—	4	2	no	—
132	C₄²×C₆₇	—	4	2	no	—
133	C₄²×C₆₇	C₂×D₄	1	2	yes	—
134	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₆₇	—	4	1	no	—
136	Q₈×C₆₇	—	2	4	no	—
137	Q₈×C₆₇	—	4	2	no	—
138	Q₈×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#135	1	2	yes	—
148	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₆₇	—	4	4	no	—
150	C₂×Dic₆₇	—	4	1	no	—
151	C₂×Dic₆₇	—	4	2	no	—
152	C₂×Dic₆₇	—	16	1	no	—
153	D₂₆₈	—	2	2	no	—
154	D₂₆₈	—	4	1	no	—
155	C₄×D₆₇	—	4	4	no	—
156	C₄×D₆₇	—	16	2	no	—
157	Dic₁₃₄	—	2	2	no	—
158	Dic₁₃₄	—	4	1	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	C₄×D₄	1	2	yes	—
161	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	C₄×D₈	1	2	yes	—
166	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₆₇	—	4	2	no	—
168	C₂×D₆₇	—	8	1	no	—
169	C₂×D₆₇	—	8	1	no	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	4	4	no	—
173	C₄×C₆₇	—	8	1	no	—
174	C₄×C₆₇	—	8	1	no	—
175	C₄×C₆₇	C₄×D₈	1	2	yes	—
176	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
177	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
178	Dic₆₇	—	4	2	no	—
179	Dic₆₇	—	8	2	no	—
180	#33455	—	67	1	no	maximal, Sylow
181	C₂×C₆₇	—	2	2	no	—
182	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₆₇	#3416	1	1	yes	characteristic
184	C₂×C₆₇	#3417	1	1	yes	characteristic
185	D₆₇	—	8	2	no	—
186	C₂×#510	—	67	1	no	—
187	#2951	—	67	1	no	—
188	#2950	D₆₇	1	1	yes	characteristic, complemented
189	#2176	—	67	2	no	—
190	#2175	—	67	2	no	—
191	C₄×Q₃₂	—	67	2	no	—
192	C₄×SD₃₂	—	67	2	no	—
193	#2159	—	67	1	no	—
194	#2138	—	67	1	no	—
195	C₂×#358	—	67	1	no	—
196	#2108	—	67	1	no	—
197	C₆₇	#33455	1	1	yes	characteristic, complemented, Sylow
198	C₂×#135	—	67	1	no	—
199	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
200	#520	—	134	1	no	—
201	#519	—	2	1	no	—
202	#510	—	67	8	no	—
203	C₂×Q₃₂	—	67	2	no	—
204	C₂×SD₃₂	—	67	2	no	—
205	C₂×M₃₂	—	67	1	no	—
206	C₂²×C₁₆	—	67	1	no	—
207	#441	—	2	2	no	—
208	#440	—	134	2	no	—
209	C₄×Q₁₆	—	67	2	no	—
210	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
211	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
212	C₄×D₈	—	67	2	no	—
213	#427	C₂×D₆₇	1	1	yes	characteristic
214	C₂×#107	—	67	1	no	—
215	#419	—	2	1	no	—
216	#417	—	134	1	no	—
217	#405	C₂×D₆₇	1	1	yes	characteristic
218	#367	—	67	2	no	—
219	#366	—	67	2	no	—
220	#358	—	67	2	no	—
221	#358	—	67	2	no	—
222	#358	—	67	1	no	—
223	#358	—	67	1	no	—
224	#357	—	67	1	no	—
225	#357	—	67	1	no	—
226	#346	—	67	2	no	—
227	C₄×C₁₆	—	67	2	no	—
228	C₂×#41	—	134	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	67	4	no	—
231	#135	—	134	2	no	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	—	67	1	no	—
235	C₂×Q₁₆	C₄×D₆₇	1	4	yes	complemented
236	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
237	C₂×SD₁₆	—	134	1	no	—
238	C₂×D₈	—	67	1	no	—
239	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	4	1	no	—
244	C₄×Q₈	—	4	1	no	—
245	C₄×Q₈	—	134	2	no	—
246	C₄×D₄	—	268	2	no	—
247	C₄×D₄	—	134	2	no	—
248	#117	—	2	1	no	—
249	#117	—	4	2	no	—
250	#117	—	268	1	no	—
251	#117	C₂²⋊D₆₇	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	268	1	no	—
253	C₂×C₄⋊C₄	—	134	1	no	—
254	Q₃₂	—	67	8	no	—
255	SD₃₂	—	67	8	no	—
256	M₃₂	—	67	4	no	—
257	C₂×C₁₆	—	67	2	no	—
258	C₂×C₁₆	—	67	1	no	—
259	C₂×C₁₆	—	67	1	no	—
260	C₂×C₁₆	—	134	2	no	—
261	#107	—	67	2	no	—
262	#107	—	134	1	no	—
263	#107	C₂²×D₆₇	1	1	yes	characteristic
264	#107	C₂²×D₆₇	1	1	yes	characteristic
265	#106	—	2	1	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	134	2	no	—
269	#102	—	134	2	no	—
270	#97	C₂²⋊D₆₇	1	2	yes	—
271	C₄×C₈	C₂²×D₆₇	1	2	yes	—
272	#41	—	268	2	no	—
273	#41	—	134	4	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	134	1	no	—
279	C₂×D₄	—	268	1	no	—
280	C₂×D₄	—	134	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	268	1	no	—
283	C₂²×C₄	—	268	1	no	—
284	C₂²×C₄	—	268	2	no	—
285	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
286	Q₁₆	—	2	4	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	67	2	no	—
289	Q₁₆	—	134	1	no	—
290	Q₁₆	C₂×C₄×D₆₇	1	4	yes	—
291	SD₁₆	—	134	4	no	—
292	D₈	—	67	2	no	—
293	D₈	—	134	1	no	—
294	M₁₆	—	2	2	no	—
295	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
297	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
298	C₂×C₈	#11761628	1	2	yes	—
299	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	268	1	no	—
305	C₄⋊C₄	—	268	2	no	—
306	C₄⋊C₄	—	134	2	no	—
307	C₄⋊C₄	#11761613	1	1	yes	characteristic
308	C₄⋊C₄	#11761613	1	1	yes	characteristic
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	268	2	no	—
311	C₂²⋊C₄	#11761613	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	4	2	no	—
315	C₄²	—	268	2	no	—
316	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
317	C₁₆	—	67	4	no	—
318	C₁₆	—	134	1	no	—
319	C₁₆	—	134	1	no	—
320	C₂³	—	268	1	no	—
321	C₂³	C₂×#11761613	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	268	1	no	—
326	Q₈	—	134	2	no	—
327	D₄	—	268	4	no	—
328	D₄	—	268	1	no	—
329	D₄	—	134	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	268	1	no	—
335	C₂×C₄	—	268	4	no	—
336	C₂×C₄	—	268	2	no	—
337	C₂×C₄	—	1072	2	no	—
338	C₂×C₄	—	1072	1	no	—
339	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
340	C₂×C₄	C₂×#11761613	1	2	yes	—
341	C₂×C₄	C₂×#11761613	1	2	yes	—
342	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
343	C₂×C₄	#14524063	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
347	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
348	C₈	C₂×#11761628	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	268	2	no	—
352	C₂²	—	536	1	no	—
353	C₂²	—	536	1	no	—
354	C₂²	C₄×#11761613	1	2	yes	—
355	C₂²	C₂²×#11761613	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	268	2	no	—
362	C₄	—	536	2	no	—
363	C₄	C₄×#11761613	1	2	yes	—
364	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
365	C₄	C₂×#14524063	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	536	2	no	—
368	C₂	C₂×C₄×#11761613	1	1	yes	characteristic
369	C₂	#22573353	1	1	yes	characteristic
370	C₂	#22573360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2950	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#14524065	C₂²
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2950	D₆₇
hypocenter	C₆₇	#33455
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #22562078	C₂	semidirect
4. #22562071	C₂	semidirect
5. C₄×#14523989	C₂	semidirect
6. C₄×#14523986	C₂	semidirect
7. #22561998	C₂	semidirect
9. C₂×#16863200	C₂	semidirect
10. #22561806	C₂	semidirect
11. C₆₇×#2950	C₂	semidirect
14. #16863837	C₄	semidirect
54. #14523989	C₂×C₄	semidirect
55. #14523986	C₂×C₄	semidirect
56. #14523983	C₂×C₄	semidirect
68. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
188. #2950	D₆₇	semidirect
197. C₆₇	#33455	semidirect
235. C₂×Q₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863837	2
#22569086	2
#22562078	2
#22562071	2
C₄×#14523989	2
C₄×#14523986	2
#22561998	2
#22561914	2
C₂×#16863200	2
#22561806	2
C₆₇×#2950	2
#33455	67