Subgroups — #31012752

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571931	C₂	1	1	yes	maximal, characteristic
2	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562417	C₂	1	2	yes	maximal, complemented
4	#22562343	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562298	C₂	1	2	yes	maximal, complemented
6	#22562259	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562147	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	#22561914	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	2	yes	—
13	#16864051	—	2	1	no	—
14	C₂×#14523989	C₂²	1	2	yes	—
15	C₂×#14523986	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
17	#16863776	C₂²	1	1	yes	characteristic
18	#16863637	C₂²	1	2	yes	—
19	#16863602	—	2	1	no	—
20	#16863560	—	2	2	no	—
21	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
22	#16863223	C₂²	1	1	yes	characteristic
23	#16863221	C₂²	1	2	yes	—
24	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#16863200	C₂²	1	1	yes	characteristic
26	#16863199	C₂²	1	2	yes	—
27	#16863199	C₂²	1	2	yes	—
28	#16863199	C₂²	1	2	yes	—
29	#16863196	C₂²	1	1	yes	characteristic
30	#16863196	C₂²	1	2	yes	—
31	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	#556×C₆₇	—	2	1	no	—
34	#500×C₆₇	C₂²	1	1	yes	characteristic
35	#485×C₆₇	—	2	2	no	—
36	#475×C₆₇	—	2	1	no	—
37	#467×C₆₇	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	#427×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	—	2	2	no	—
42	#14524065	—	2	4	no	—
43	#14524065	—	4	2	no	—
44	C₂×Dic₂₆₈	—	2	1	no	—
45	C₂×Dic₂₆₈	C₂³	1	2	yes	—
46	C₂×#11761617	—	2	2	no	—
47	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
48	#14524043	—	2	1	no	—
49	#14524025	—	4	2	no	—
50	#14524022	—	2	2	no	—
51	#14524014	—	4	1	no	—
52	C₄×D₂₆₈	—	2	1	no	—
53	C₄⋊C₄×D₆₇	—	4	1	no	—
54	#14523989	—	4	2	no	—
55	#14523986	—	2	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	2	no	—
62	#14523965	—	2	1	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	#126×C₆₇	—	4	2	no	—
67	#124×C₆₇	—	4	1	no	—
68	C₂²⋊Q₈×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	—	2	1	no	—
70	C₄×Q₈×C₆₇	—	4	1	no	—
71	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
72	#117×C₆₇	—	2	2	no	—
73	#107×C₆₇	—	2	1	no	—
74	#107×C₆₇	C₂³	1	2	yes	—
75	#107×C₆₇	C₂³	1	1	yes	characteristic
76	#106×C₆₇	—	2	2	no	—
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	#11761628	—	4	4	no	—
83	#11761628	—	8	2	no	—
84	C₂×Dic₁₃₄	—	2	2	no	—
85	C₂×D₂₆₈	—	2	1	no	—
86	C₂×C₂²⋊D₆₇	—	4	2	no	—
87	C₂×C₄×D₆₇	—	4	1	no	—
88	C₂×C₄×D₆₇	—	4	2	no	—
89	Dic₂₆₈	—	2	2	no	—
90	Dic₂₆₈	—	2	2	no	—
91	Dic₂₆₈	—	8	1	no	—
92	#11761617	—	4	4	no	—
93	D₅₃₆	—	2	1	no	—
94	D₅₃₆	#41	1	2	yes	—
95	C₄⋊Dic₆₇	—	2	1	no	—
96	C₄⋊Dic₆₇	—	4	1	no	—
97	C₄⋊Dic₆₇	—	4	1	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	C₂²⋊Dic₆₇	—	4	2	no	—
100	(C₂×D₆₇)⋊C₄	—	4	1	no	—
101	C₆₇⋊C₁₆	—	2	1	no	—
102	C₆₇⋊C₁₆	—	2	1	no	—
103	C₆₇⋊C₁₆	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	Q₁₆×C₆₇	—	2	1	no	—
109	Q₁₆×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	—	4	1	no	—
120	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
121	C₂²⋊C₄×C₆₇	—	4	2	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
124	C₄²×C₆₇	—	4	2	no	—
125	C₄²×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₆₇	C₂×D₈	1	2	yes	—
128	C₂²×D₆₇	—	4	1	no	—
129	Q₈×C₆₇	—	4	1	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	8	2	no	—
132	D₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₆₇	—	8	4	no	—
143	C₂×Dic₆₇	—	4	1	no	—
144	C₂×Dic₆₇	—	4	2	no	—
145	D₂₆₈	—	2	2	no	—
146	D₂₆₈	—	4	1	no	—
147	C₄×D₆₇	—	8	4	no	—
148	C₄×D₆₇	—	8	2	no	—
149	Dic₁₃₄	—	4	2	no	—
150	Dic₁₃₄	—	4	2	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	—	2	1	no	—
153	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₆₇	—	2	2	no	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₆₇	—	4	2	no	—
160	C₂×D₆₇	—	8	1	no	—
161	C₂×D₆₇	—	8	1	no	—
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	8	2	no	—
165	C₄×C₆₇	—	8	1	no	—
166	C₄×C₆₇	—	8	1	no	—
167	C₄×C₆₇	#508	1	2	yes	—
168	C₄×C₆₇	#546	1	1	yes	characteristic
169	C₄×C₆₇	#577	1	1	yes	characteristic
170	Dic₆₇	—	8	1	no	—
171	Dic₆₇	—	8	2	no	—
172	#33869	—	67	1	no	maximal, Sylow
173	C₂×C₆₇	—	4	2	no	—
174	C₂×C₆₇	#3294	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
176	C₂×C₆₇	#3417	1	1	yes	characteristic
177	D₆₇	—	8	2	no	—
178	#3342	D₆₇	1	1	yes	characteristic, complemented
179	#3283	—	67	1	no	—
180	#2241	—	67	1	no	—
181	#2237	—	67	2	no	—
182	#2224	—	67	1	no	—
183	#2216	—	67	2	no	—
184	#2209	—	67	1	no	—
185	#2189	—	67	2	no	—
186	C₄×SD₃₂	—	67	1	no	—
187	D₄×C₁₆	—	67	1	no	—
188	#2138	—	67	2	no	—
189	C₆₇	#33869	1	1	yes	characteristic, complemented, Sylow
190	C₂×#135	—	67	2	no	—
191	#556	—	2	1	no	—
192	#547	—	134	1	no	—
193	C₂×Q₃₂	—	67	2	no	—
194	C₂×SD₃₂	—	67	1	no	—
195	C₂²×C₁₆	—	67	2	no	—
196	#500	C₂×D₆₇	1	1	yes	characteristic
197	#494	—	67	1	no	—
198	#485	—	2	2	no	—
199	#475	—	2	1	no	—
200	#467	—	67	2	no	—
201	#467	C₂×D₆₇	1	2	yes	—
202	#461	—	134	1	no	—
203	#452	—	134	2	no	—
204	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
205	C₄×D₈	—	67	1	no	—
206	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
207	#427	C₂×D₆₇	1	2	yes	—
208	#367	—	67	1	no	—
209	#366	—	67	2	no	—
210	C₄⋊C₁₆	—	67	1	no	—
211	#358	—	67	1	no	—
212	#358	—	67	2	no	—
213	#358	—	67	2	no	—
214	#358	—	67	2	no	—
215	#357	—	67	1	no	—
216	#357	—	67	2	no	—
217	C₂²⋊C₁₆	—	67	2	no	—
218	C₄×C₁₆	—	67	1	no	—
219	C₂×#41	—	134	2	no	—
220	#135	—	268	2	no	—
221	#135	—	134	4	no	—
222	C₂×Q₁₆	—	67	2	no	—
223	C₂×Q₁₆	—	134	1	no	—
224	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
225	C₂×SD₁₆	—	134	2	no	—
226	C₂×D₈	—	67	1	no	—
227	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	134	1	no	—
230	#126	—	4	2	no	—
231	#124	—	4	1	no	—
232	#123	—	268	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	134	2	no	—
235	#121	—	268	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	4	1	no	—
238	C₄×D₄	—	134	1	no	—
239	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
240	#117	—	2	2	no	—
241	C₂×C₄⋊C₄	—	268	1	no	—
242	Q₃₂	—	268	2	no	—
243	SD₃₂	—	134	2	no	—
244	C₂×C₁₆	—	268	2	no	—
245	C₂×C₁₆	—	67	1	no	—
246	C₂×C₁₆	—	67	1	no	—
247	C₂×C₁₆	—	67	2	no	—
248	#107	—	2	1	no	—
249	#107	—	67	1	no	—
250	#107	C₂²×D₆₇	1	2	yes	—
251	#107	C₂²×D₆₇	1	1	yes	characteristic
252	#106	—	2	2	no	—
253	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
254	#103	—	2	1	no	—
255	#103	—	2	2	no	—
256	#103	—	134	2	no	—
257	#102	—	134	1	no	—
258	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
259	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
260	#41	—	268	4	no	—
261	#41	—	536	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	134	2	no	—
265	C₂×D₄	—	268	2	no	—
266	C₂×D₄	—	134	1	no	—
267	C₂×D₄	#11761613	1	1	yes	characteristic
268	C₂²×C₄	—	268	1	no	—
269	C₂²×C₄	—	268	2	no	—
270	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	—	536	1	no	—
274	Q₁₆	—	134	2	no	—
275	Q₁₆	—	134	2	no	—
276	SD₁₆	—	268	4	no	—
277	D₈	—	67	2	no	—
278	D₈	—	134	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
283	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	268	1	no	—
290	C₄⋊C₄	—	268	2	no	—
291	C₄⋊C₄	—	268	1	no	—
292	C₄⋊C₄	—	134	1	no	—
293	C₄⋊C₄	#11761613	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	268	2	no	—
297	C₂²⋊C₄	—	268	1	no	—
298	C₂²⋊C₄	#11761613	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
302	C₁₆	—	134	1	no	—
303	C₁₆	—	134	1	no	—
304	C₁₆	—	134	2	no	—
305	C₂³	—	268	1	no	—
306	C₂³	C₂×#11761613	1	2	yes	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	8	2	no	—
310	Q₈	—	268	2	no	—
311	Q₈	—	268	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	268	1	no	—
314	D₄	—	536	4	no	—
315	D₄	—	134	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	268	1	no	—
322	C₂×C₄	—	268	2	no	—
323	C₂×C₄	—	536	4	no	—
324	C₂×C₄	—	536	2	no	—
325	C₂×C₄	C₂×#11761613	1	2	yes	—
326	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
327	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	#41×D₆₇	1	1	yes	characteristic
332	C₈	#14524089	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	268	2	no	—
337	C₂²	—	536	1	no	—
338	C₂²	—	536	1	no	—
339	C₂²	C₂²×#11761613	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	536	1	no	—
346	C₄	—	536	2	no	—
347	C₄	#16863826	1	2	yes	—
348	C₄	#16864034	1	1	yes	characteristic
349	C₄	#16864204	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	536	2	no	—
352	C₂	#22572016	1	1	yes	characteristic
353	C₂	C₂×#16863826	1	1	yes	characteristic
354	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	C₆₇×#3342	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22571931	C₂
hypercenter	#3342	D₆₇
hypocenter	C₆₇	#33869
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #22562438	C₂	semidirect
3. #22562417	C₂	semidirect
4. #22562343	C₂	semidirect
5. #22562298	C₂	semidirect
6. #22562259	C₂	semidirect
7. #22562147	C₂	semidirect
8. C₄×#14523986	C₂	semidirect
9. D₄×C₆₇⋊C₁₆	C₂	semidirect
10. #22561914	C₂	semidirect
11. C₆₇×#3342	C₂	semidirect
178. #3342	D₆₇	semidirect
189. C₆₇	#33869	semidirect

Maximal subgroups

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

subgroup	index
#22571931	2
#22562438	2
#22562417	2
#22562343	2
#22562298	2
#22562259	2
#22562147	2
C₄×#14523986	2
D₄×C₆₇⋊C₁₆	2
#22561914	2
C₆₇×#3342	2
#33869	67