Subgroups — #31010514

Order 17152.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569062	C₂	1	1	yes	maximal, characteristic
2	#22562363	C₂	1	2	yes	maximal, complemented
3	#22562363	C₂	1	2	yes	maximal, complemented
4	#22562353	C₂	1	2	yes	maximal, complemented
5	#22562343	C₂	1	2	yes	maximal, complemented
6	#22561995	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561991	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561968	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561924	C₂	1	1	yes	maximal, characteristic
10	C₂×#16863196	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
13	#16863870	—	2	1	no	—
14	C₂×#14523983	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863509	—	2	2	no	—
17	C₄×D₅₃₆	C₂²	1	2	yes	—
18	C₄×D₅₃₆	C₂²	1	2	yes	—
19	#16863434	C₂²	1	1	yes	characteristic
20	#16863402	—	2	1	no	—
21	#16863223	C₂²	1	1	yes	characteristic, complemented
22	#16863223	C₂²	1	2	yes	complemented
23	#16863223	C₂²	1	1	yes	characteristic, complemented
24	#16863221	C₂²	1	1	yes	characteristic, complemented
25	#16863221	C₂²	1	2	yes	complemented
26	#16863221	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	complemented
28	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	complemented
29	#16863196	C₂²	1	2	yes	—
30	#16863196	C₂²	1	2	yes	—
31	#16863196	C₂²	1	2	yes	—
32	#16863196	C₂²	1	2	yes	—
33	#544×C₆₇	—	2	1	no	—
34	#501×C₆₇	—	2	2	no	—
35	#500×C₆₇	C₂²	1	2	yes	complemented
36	#499×C₆₇	C₂²	1	2	yes	complemented
37	#428×C₆₇	—	2	1	no	—
38	#427×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
40	#405×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×D₂₆₈	—	2	1	no	—
42	C₂×D₅₃₆	—	2	1	no	—
43	C₂×D₅₃₆	—	2	2	no	—
44	C₂×D₅₃₆	Q₈	1	4	yes	complemented
45	C₂×D₅₃₆	C₂³	1	2	yes	—
46	C₄×D₂₆₈	—	2	2	no	—
47	C₄×D₂₆₈	—	2	2	no	—
48	#14524005	—	4	1	no	—
49	#14524005	—	4	1	no	—
50	#14524002	—	2	1	no	—
51	C₂²⋊C₄×D₆₇	—	4	2	no	—
52	#14523983	—	2	2	no	—
53	C₂×C₆₇⋊C₁₆	—	2	2	no	—
54	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
57	#14523977	C₂³	1	1	yes	characteristic
58	#14523977	C₂³	1	1	yes	characteristic
59	#14523975	—	2	1	no	—
60	#14523965	—	2	2	no	—
61	#14523965	—	2	2	no	—
62	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₇	—	2	2	no	—
65	C₄⋊Q₈×C₆₇	—	4	1	no	—
66	#125×C₆₇	—	2	2	no	—
67	#125×C₆₇	—	4	1	no	—
68	C₂²⋊Q₈×C₆₇	—	4	2	no	—
69	#117×C₆₇	—	2	1	no	—
70	#117×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	#107×C₆₇	—	2	2	no	—
73	#107×C₆₇	—	2	1	no	—
74	#107×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#107×C₆₇	C₂³	1	1	yes	characteristic
77	#106×C₆₇	—	2	2	no	—
78	#106×C₆₇	—	2	1	no	—
79	#97×C₆₇	D₄	1	2	yes	—
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂³×D₆₇	—	4	1	no	—
82	C₂×D₂₆₈	—	2	4	no	—
83	C₂×D₂₆₈	—	2	2	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×D₂₆₈	—	4	2	no	—
86	C₂×C₄×D₆₇	—	4	2	no	—
87	C₂×C₄×D₆₇	—	4	2	no	—
88	D₅₃₆	—	2	4	no	—
89	D₅₃₆	—	2	2	no	—
90	D₅₃₆	C₂×Q₈	1	4	yes	—
91	C₄⋊Dic₆₇	—	2	1	no	—
92	C₄⋊Dic₆₇	—	2	1	no	—
93	C₂²⋊Dic₆₇	—	4	1	no	—
94	(C₂×D₆₇)⋊C₄	—	4	2	no	—
95	(C₂×D₆₇)⋊C₄	—	4	2	no	—
96	C₄×Dic₆₇	—	4	1	no	—
97	C₆₇⋊C₁₆	—	2	1	no	—
98	C₆₇⋊C₁₆	—	2	1	no	—
99	C₆₇⋊C₁₆	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	4	2	no	—
101	C₂²×C₄×C₆₇	—	4	1	no	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	M₁₆×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₆₇	#41	1	2	yes	—
108	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	—	4	1	no	—
113	C₄⋊C₄×C₆₇	—	4	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	—	4	2	no	—
116	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
117	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
118	C₂²⋊C₄×C₆₇	—	4	1	no	—
119	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
120	C₄²×C₆₇	—	4	1	no	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₆₇	—	4	2	no	—
124	C₂²×D₆₇	—	4	4	no	—
125	C₂²×D₆₇	—	16	1	no	—
126	C₂²×D₆₇	—	16	1	no	—
127	Q₈×C₆₇	—	16	2	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	16	1	no	—
132	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#137	1	2	yes	—
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₆₇	—	4	1	no	—
139	C₂×Dic₆₇	—	4	1	no	—
140	D₂₆₈	—	2	4	no	—
141	D₂₆₈	—	4	2	no	—
142	D₂₆₈	—	4	4	no	—
143	C₄×D₆₇	—	16	2	no	—
144	C₄×D₆₇	—	16	2	no	—
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	#474	1	2	yes	—
152	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₆₇	—	4	4	no	—
154	C₂×D₆₇	—	8	2	no	—
155	C₂×D₆₇	—	8	2	no	—
156	C₂×D₆₇	—	16	4	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	8	2	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	#474	1	2	yes	—
163	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
164	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
165	Dic₆₇	—	8	1	no	—
166	Dic₆₇	—	8	1	no	—
167	#33631	—	67	1	no	maximal, Sylow
168	C₂×C₆₇	—	2	2	no	—
169	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
170	C₂×C₆₇	#3415	1	1	yes	characteristic
171	C₂×C₆₇	#3416	1	1	yes	characteristic
172	D₆₇	—	8	4	no	—
173	#3164	D₆₇	1	1	yes	characteristic, complemented
174	#2948	—	67	1	no	—
175	#2228	—	67	2	no	—
176	#2228	—	67	2	no	—
177	#2226	—	67	2	no	—
178	#2224	—	67	2	no	—
179	#2159	—	67	1	no	—
180	#2158	—	67	1	no	—
181	#2151	—	67	1	no	—
182	#2140	—	67	1	no	—
183	C₂×#357	—	67	1	no	—
184	C₆₇	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	—	67	1	no	—
186	#544	—	2	1	no	—
187	#518	—	134	1	no	—
188	C₂×M₃₂	—	67	1	no	—
189	C₂²×C₁₆	—	67	1	no	—
190	#501	—	2	2	no	—
191	#500	C₂×D₆₇	1	2	yes	complemented
192	#499	C₂×D₆₇	1	2	yes	complemented
193	#442	—	134	2	no	—
194	C₄×D₈	—	67	2	no	—
195	C₄×D₈	—	67	2	no	—
196	#428	—	2	1	no	—
197	#427	—	67	1	no	—
198	#427	C₂×D₆₇	1	1	yes	characteristic
199	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
200	#418	—	134	1	no	—
201	#405	C₂×D₆₇	1	1	yes	characteristic
202	#367	—	67	1	no	—
203	#367	—	67	1	no	—
204	#367	—	67	2	no	—
205	#366	—	67	1	no	—
206	#366	—	67	1	no	—
207	#366	—	67	2	no	—
208	C₄⋊C₁₆	—	67	2	no	—
209	C₄⋊C₁₆	—	67	2	no	—
210	#357	—	67	2	no	—
211	#357	—	67	2	no	—
212	#357	—	67	2	no	—
213	#357	—	67	2	no	—
214	C₂²×D₄	—	134	1	no	—
215	C₂×D₈	—	67	4	no	—
216	C₂×D₈	—	67	2	no	—
217	C₂×D₈	—	134	1	no	—
218	C₂×D₈	—	134	2	no	—
219	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	1	no	—
225	C₂²⋊Q₈	—	4	2	no	—
226	C₄×D₄	—	268	1	no	—
227	C₄×D₄	—	268	1	no	—
228	C₄×D₄	—	134	2	no	—
229	C₄×D₄	—	134	2	no	—
230	#117	—	2	1	no	—
231	#117	—	134	1	no	—
232	#117	C₂²⋊D₆₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	268	2	no	—
235	M₃₂	—	134	2	no	—
236	C₂×C₁₆	—	67	1	no	—
237	C₂×C₁₆	—	67	2	no	—
238	C₂×C₁₆	—	67	1	no	—
239	C₂×C₁₆	—	134	2	no	—
240	#107	—	2	1	no	—
241	#107	—	2	2	no	—
242	#107	—	67	1	no	—
243	#107	—	67	1	no	—
244	#107	C₂²×D₆₇	1	2	yes	—
245	#107	C₂²×D₆₇	1	1	yes	characteristic
246	#107	C₂²×D₆₇	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	2	no	—
249	#106	—	134	1	no	—
250	#102	—	134	2	no	—
251	#102	—	134	2	no	—
252	#97	C₂²⋊D₆₇	1	2	yes	—
253	C₄×C₈	C₂²×D₆₇	1	2	yes	—
254	C₂⁴	—	268	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	268	1	no	—
257	C₂×D₄	—	268	2	no	—
258	C₂×D₄	—	134	4	no	—
259	C₂×D₄	—	134	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	268	2	no	—
262	C₂²×C₄	—	268	2	no	—
263	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
264	D₈	—	67	4	no	—
265	D₈	—	134	4	no	—
266	D₈	—	134	2	no	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₆₇	1	2	yes	—
271	C₂×C₈	#11761629	1	2	yes	—
272	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	134	1	no	—
281	C₄⋊C₄	—	134	1	no	—
282	C₄⋊C₄	#11761613	1	1	yes	characteristic
283	C₄⋊C₄	#11761613	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	268	1	no	—
286	C₂²⋊C₄	—	268	2	no	—
287	C₂²⋊C₄	—	268	2	no	—
288	C₂²⋊C₄	#11761613	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	268	1	no	—
291	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
292	C₁₆	—	134	1	no	—
293	C₁₆	—	134	1	no	—
294	C₁₆	—	134	2	no	—
295	C₂³	—	268	2	no	—
296	C₂³	—	268	4	no	—
297	C₂³	—	1072	1	no	—
298	C₂³	—	1072	1	no	—
299	C₂³	C₂×#11761613	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	268	4	no	—
302	D₄	—	268	2	no	—
303	D₄	—	134	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	268	1	no	—
308	C₂×C₄	—	268	1	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	1072	2	no	—
311	C₂×C₄	—	1072	2	no	—
312	C₂×C₄	#14524019	1	2	yes	—
313	C₂×C₄	C₂×#11761613	1	2	yes	—
314	C₂×C₄	C₂×#11761613	1	2	yes	—
315	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
316	C₂×C₄	#14524074	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#14524019	1	2	yes	—
320	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
321	C₈	C₂×#11761629	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	268	4	no	—
325	C₂²	—	536	2	no	—
326	C₂²	—	536	2	no	—
327	C₂²	—	1072	4	no	—
328	C₂²	#16863673	1	2	yes	—
329	C₂²	C₂²×#11761613	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	536	1	no	—
336	C₄	—	536	1	no	—
337	C₄	#16863673	1	2	yes	—
338	C₄	C₂×#14524019	1	1	yes	characteristic
339	C₄	C₂×#14524074	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	536	4	no	—
342	C₂	C₂×#16863673	1	1	yes	characteristic
343	C₂	#22573348	1	1	yes	characteristic
344	C₂	#22573353	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3164	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	#3164	D₆₇
hypocenter	C₆₇	#33631
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #22562363	C₂	semidirect
3. #22562363	C₂	semidirect
4. #22562353	C₂	semidirect
5. #22562343	C₂	semidirect
6. #22561995	C₂	semidirect
7. #22561991	C₂	semidirect
8. #22561968	C₂	semidirect
11. C₆₇×#3164	C₂	semidirect
21. #16863223	C₂²	semidirect
22. #16863223	C₂²	semidirect
23. #16863223	C₂²	semidirect
24. #16863221	C₂²	semidirect
25. #16863221	C₂²	semidirect
26. #16863221	C₂²	semidirect
27. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
28. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
35. #500×C₆₇	C₂²	semidirect
36. #499×C₆₇	C₂²	semidirect
44. C₂×D₅₃₆	Q₈	semidirect
173. #3164	D₆₇	semidirect
184. C₆₇	#33631	semidirect
191. #500	C₂×D₆₇	semidirect
192. #499	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569062	2
#22562363	2
#22562353	2
#22562343	2
#22561995	2
#22561991	2
#22561968	2
#22561924	2
C₂×#16863196	2
C₆₇×#3164	2
#33631	67