Subgroups — #31009076

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573311	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569110	C₂	1	1	yes	maximal, characteristic
3	#22562049	C₂	1	4	yes	maximal, complemented
4	#22562044	C₂	1	4	yes	maximal, complemented
5	#22561974	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863181	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863180	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864194	C₄	1	1	yes	characteristic
11	C₂×#14524049	C₂²	1	1	yes	characteristic
12	#16863814	—	2	2	no	—
13	#16863812	C₂²	1	4	yes	complemented
14	C₂×#14523983	C₂²	1	1	yes	characteristic
15	M₃₂×D₆₇	—	2	2	no	—
16	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#16863516	C₄	1	4	yes	—
18	#16863511	C₂²	1	4	yes	—
19	#16863444	C₄	1	1	yes	characteristic
20	#16863413	C₄	1	1	yes	characteristic
21	#16863412	C₄	1	1	yes	characteristic
22	#16863351	C₂²	1	1	yes	characteristic
23	#16863216	C₂²	1	4	yes	complemented
24	#16863181	C₂²	1	2	yes	complemented
25	#16863181	C₂²	1	2	yes	complemented
26	#16863180	C₂²	1	2	yes	complemented
27	#16863180	C₂²	1	2	yes	complemented
28	#16863171	C₂²	1	2	yes	complemented
29	C₁₆×Dic₆₇	C₂²	1	2	yes	complemented
30	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
31	#504×C₆₇	—	2	2	no	—
32	#504×C₆₇	C₂²	1	4	yes	complemented
33	C₂×M₃₂×C₆₇	—	2	2	no	—
34	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
37	#14524074	C₈	1	2	yes	—
38	#14524073	C₈	1	2	yes	—
39	#14524069	C₈	1	2	yes	—
40	#14524068	C₈	1	2	yes	—
41	#14524063	C₂×C₄	1	4	yes	—
42	#14524049	—	2	1	no	—
43	#14524049	—	2	1	no	—
44	#14524049	C₂×C₄	1	2	yes	—
45	#14524049	C₂³	1	2	yes	—
46	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
47	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
48	C₂×#11761611	—	2	1	no	—
49	C₂×C₈×D₆₇	—	2	1	no	—
50	#14524002	C₂×C₄	1	1	yes	characteristic
51	#14523983	—	2	1	no	—
52	#14523983	C₂³	1	2	yes	—
53	#14523982	—	2	4	no	—
54	C₂×C₆₇⋊C₁₆	—	2	1	no	—
55	C₂×C₆₇⋊C₁₆	—	2	1	no	—
56	C₂×C₆₇⋊C₁₆	—	2	1	no	—
57	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₁₆×D₆₇	—	2	4	no	—
60	#14523978	C₂×C₄	1	4	yes	—
61	#14523970	C₂×C₄	1	4	yes	—
62	#14523969	C₂×C₄	1	4	yes	—
63	#14523953	C₂×C₄	1	2	yes	—
64	C₈×Dic₆₇	C₂³	1	2	yes	—
65	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#131×C₆₇	—	2	1	no	—
67	#131×C₆₇	—	2	1	no	—
68	#131×C₆₇	C₂×C₄	1	2	yes	—
69	#131×C₆₇	C₂³	1	2	yes	—
70	C₂×M₁₆×C₆₇	—	2	1	no	—
71	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	—	2	1	no	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	M₃₂×C₆₇	—	2	1	no	—
75	M₃₂×C₆₇	—	2	4	no	—
76	M₃₂×C₆₇	D₄	1	2	yes	—
77	M₃₂×C₆₇	D₄	1	2	yes	—
78	M₃₂×C₆₇	C₂³	1	2	yes	—
79	C₂×C₁₆×C₆₇	—	2	1	no	—
80	C₂×C₁₆×C₆₇	—	2	4	no	—
81	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
82	#11761628	—	2	1	no	—
83	#11761628	—	2	1	no	—
84	#11761628	C₂×C₈	1	2	yes	—
85	#11761628	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
87	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
88	C₂×C₂²⋊D₆₇	—	2	1	no	—
89	C₂×C₄×D₆₇	—	2	1	no	—
90	#11761618	C₂×C₈	1	4	yes	—
91	#11761616	C₂×C₈	1	4	yes	—
92	#11761615	C₂×C₈	1	4	yes	—
93	#11761613	C₂×C₈	1	4	yes	—
94	#11761612	—	2	1	no	—
95	#11761612	C₂×C₈	1	2	yes	—
96	#11761612	C₂×C₈	1	2	yes	—
97	#11761612	C₂²×C₄	1	2	yes	—
98	#11761611	—	2	2	no	—
99	#11761611	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
102	C₈×D₆₇	—	2	2	no	—
103	C₈×D₆₇	—	2	2	no	—
104	C₄⋊Dic₆₇	—	2	1	no	—
105	C₂²⋊Dic₆₇	—	2	1	no	—
106	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
107	C₆₇⋊C₁₆	—	2	2	no	—
108	C₆₇⋊C₁₆	—	2	1	no	—
109	C₆₇⋊C₁₆	—	2	1	no	—
110	#41×C₆₇	—	2	1	no	—
111	#41×C₆₇	—	2	1	no	—
112	#41×C₆₇	C₂×C₈	1	2	yes	—
113	#41×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
115	C₂×D₄×C₆₇	—	2	1	no	—
116	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	M₁₆×C₆₇	—	2	1	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	2	2	no	—
122	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
130	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₁₆×C₆₇	—	2	2	no	—
133	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
134	C₂³×C₆₇	—	2	1	no	—
135	C₂³×C₆₇	#131	1	1	yes	characteristic
136	C₂²×D₆₇	—	2	1	no	—
137	Q₈×C₆₇	—	2	1	no	—
138	Q₈×C₆₇	C₂²×C₈	1	2	yes	—
139	D₄×C₆₇	—	2	2	no	—
140	D₄×C₆₇	—	2	2	no	—
141	D₄×C₆₇	—	2	1	no	—
142	D₄×C₆₇	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
152	C₂²⋊D₆₇	—	2	2	no	—
153	C₂²⋊D₆₇	—	2	2	no	—
154	C₂×Dic₆₇	—	2	1	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	D₂₆₈	—	2	1	no	—
157	D₂₆₈	C₂²×C₈	1	2	yes	—
158	C₄×D₆₇	—	2	2	no	—
159	C₄×D₆₇	—	2	2	no	—
160	Dic₁₃₄	—	2	1	no	—
161	Dic₁₃₄	C₂²×C₈	1	2	yes	—
162	C₈×C₆₇	—	2	2	no	—
163	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
164	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
165	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₆₇⋊C₈	—	2	2	no	—
168	C₆₇⋊C₈	C₂²×C₈	1	4	yes	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	4	1	no	—
171	C₂²×C₆₇	—	8	1	no	—
172	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
173	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
174	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
175	C₂×D₆₇	—	2	2	no	—
176	C₂×D₆₇	—	8	1	no	—
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
179	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
180	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
182	Dic₆₇	—	2	2	no	—
183	Dic₆₇	—	4	2	no	—
184	#33442	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	—	2	1	no	—
186	C₂×C₆₇	—	2	1	no	—
187	C₂×C₆₇	—	2	1	no	—
188	C₂×C₆₇	—	4	2	no	—
189	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
190	D₆₇	—	4	2	no	—
191	#3407	—	67	1	no	—
192	#3407	D₆₇	1	1	yes	characteristic, complemented
193	#2955	—	67	1	no	—
194	#2171	—	67	4	no	—
195	#2170	—	67	4	no	—
196	#2153	—	67	1	no	—
197	C₂×#350	—	67	1	no	—
198	C₂×#350	—	67	1	no	—
199	#2108	—	67	1	no	—
200	C₆₇	#33442	1	1	yes	characteristic, complemented, Sylow
201	#575	—	67	1	no	—
202	C₂×#131	—	67	1	no	—
203	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
204	#504	—	2	2	no	—
205	#504	—	67	4	no	—
206	#504	—	134	2	no	—
207	#504	C₂×D₆₇	1	4	yes	complemented
208	C₂×M₃₂	—	2	2	no	—
209	C₂×M₃₂	—	67	1	no	—
210	C₂×M₃₂	—	134	2	no	—
211	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
212	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
213	C₂²×C₁₆	—	67	1	no	—
214	#444	—	67	4	no	—
215	#443	—	67	4	no	—
216	#430	—	67	1	no	—
217	#421	—	67	1	no	—
218	#421	—	67	1	no	—
219	#405	—	67	1	no	—
220	#364	—	67	4	no	—
221	#350	—	67	2	no	—
222	#350	—	67	2	no	—
223	#350	—	67	2	no	—
224	#350	—	67	2	no	—
225	#346	—	67	2	no	—
226	C₄×C₁₆	—	67	2	no	—
227	C₂×#41	—	67	1	no	—
228	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
229	#137	—	67	2	no	—
230	#137	—	67	2	no	—
231	AGL(1,ℤ/8)	—	67	2	no	—
232	AGL(1,ℤ/8)	—	67	2	no	—
233	#135	—	67	4	no	—
234	#131	—	2	1	no	—
235	#131	—	2	1	no	—
236	#131	—	67	2	no	—
237	#131	—	67	2	no	—
238	#131	—	134	1	no	—
239	#131	—	134	1	no	—
240	#131	C₄×D₆₇	1	2	yes	—
241	#131	C₂²×D₆₇	1	2	yes	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	67	1	no	—
244	C₂×M₁₆	—	67	1	no	—
245	C₂×M₁₆	—	134	1	no	—
246	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	134	1	no	—
249	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
250	#117	—	67	1	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	2	4	no	—
253	M₃₂	—	67	2	no	—
254	M₃₂	—	134	1	no	—
255	M₃₂	—	134	4	no	—
256	M₃₂	C₂²⋊D₆₇	1	2	yes	—
257	M₃₂	C₂²⋊D₆₇	1	2	yes	—
258	M₃₂	C₂²×D₆₇	1	2	yes	—
259	C₂×C₁₆	—	2	1	no	—
260	C₂×C₁₆	—	2	4	no	—
261	C₂×C₁₆	—	67	1	no	—
262	C₂×C₁₆	—	67	1	no	—
263	C₂×C₁₆	—	134	1	no	—
264	C₂×C₁₆	—	134	1	no	—
265	C₂×C₁₆	—	134	1	no	—
266	C₂×C₁₆	—	134	4	no	—
267	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
268	#108	—	67	4	no	—
269	#104	—	67	4	no	—
270	#104	—	67	4	no	—
271	#97	—	67	2	no	—
272	C₄×C₈	—	67	2	no	—
273	#41	—	2	1	no	—
274	#41	—	2	1	no	—
275	#41	—	67	2	no	—
276	#41	—	67	2	no	—
277	#41	—	134	1	no	—
278	#41	—	134	1	no	—
279	#41	C₈×D₆₇	1	2	yes	—
280	#41	C₂×C₄×D₆₇	1	2	yes	—
281	C₂×Q₈	—	67	1	no	—
282	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	67	1	no	—
285	C₂×D₄	—	134	1	no	—
286	C₂×D₄	C₈×D₆₇	1	1	yes	characteristic
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	134	1	no	—
289	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
290	Q₁₆	—	67	4	no	—
291	SD₁₆	—	67	4	no	—
292	SD₁₆	—	67	4	no	—
293	D₈	—	67	4	no	—
294	M₁₆	—	2	1	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	67	2	no	—
298	M₁₆	—	67	2	no	—
299	M₁₆	—	67	2	no	—
300	M₁₆	—	134	1	no	—
301	M₁₆	—	134	2	no	—
302	M₁₆	—	134	2	no	—
303	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	67	2	no	—
308	C₂×C₈	—	134	1	no	—
309	C₂×C₈	—	134	2	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
314	C₂×C₈	#11761628	1	1	yes	characteristic
315	C₂×C₈	#11761628	1	1	yes	characteristic
316	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
317	C₄⋊C₄	—	134	1	no	—
318	C₂²⋊C₄	—	134	1	no	—
319	C₄²	—	67	2	no	—
320	C₁₆	—	2	2	no	—
321	C₁₆	—	134	2	no	—
322	C₁₆	—	134	1	no	—
323	C₁₆	—	134	1	no	—
324	C₁₆	C₂×C₂²⋊D₆₇	1	4	yes	—
325	C₂³	—	2	1	no	—
326	C₂³	—	134	1	no	—
327	C₂³	#14524049	1	1	yes	characteristic
328	Q₈	—	2	1	no	—
329	Q₈	—	67	2	no	—
330	Q₈	—	134	1	no	—
331	Q₈	C₂×C₈×D₆₇	1	2	yes	—
332	D₄	—	2	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	1	no	—
335	D₄	—	67	2	no	—
336	D₄	—	134	2	no	—
337	D₄	—	134	2	no	—
338	D₄	—	134	1	no	—
339	D₄	C₂×C₈×D₆₇	1	2	yes	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	—	134	1	no	—
345	C₂×C₄	—	134	2	no	—
346	C₂×C₄	—	134	2	no	—
347	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
348	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
351	C₂×C₄	#14524049	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	67	4	no	—
355	C₈	—	134	2	no	—
356	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
357	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
358	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
359	C₈	C₂×#11761628	1	1	yes	characteristic
360	C₂²	—	2	2	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	536	1	no	—
364	C₂²	—	134	2	no	—
365	C₂²	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
366	C₂²	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
367	C₂²	C₂×#14524049	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	268	2	no	—
370	C₄	—	134	2	no	—
371	C₄	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
372	C₄	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
373	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
374	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	268	2	no	—
380	C₂	C₂×C₈×C₂²⋊D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3407	C₂
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	#16864194	C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#3407	D₆₇
hypocenter	C₆₇	#33442
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #3407

Complements

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

N	H ≅ G/N	type
1. #22573311	C₂	semidirect
3. #22562049	C₂	semidirect
4. #22562044	C₂	semidirect
5. #22561974	C₂	semidirect
6. C₂×#16863181	C₂	semidirect
7. C₂×#16863180	C₂	semidirect
8. #22561808	C₂	semidirect
9. C₆₇×#3407	C₂	semidirect
13. #16863812	C₂²	semidirect
23. #16863216	C₂²	semidirect
24. #16863181	C₂²	semidirect
25. #16863181	C₂²	semidirect
26. #16863180	C₂²	semidirect
27. #16863180	C₂²	semidirect
28. #16863171	C₂²	semidirect
29. C₁₆×Dic₆₇	C₂²	semidirect
32. #504×C₆₇	C₂²	semidirect
192. #3407	D₆₇	semidirect
200. C₆₇	#33442	semidirect
207. #504	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573311	2
#22569110	2
#22562049	2
#22562044	2
#22561974	2
C₂×#16863181	2
C₂×#16863180	2
#22561808	2
C₆₇×#3407	2
#33442	67