Subgroups — #30755134

Order 17152.

574 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863770	C₂	1	1	yes	maximal, characteristic
2	#22561162	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561135	C₂	1	1	yes	maximal, characteristic
4	#22560079	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863148	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
7	C₂×#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761615	—	2	1	no	—
9	C₂²×#11761615	—	2	1	no	—
10	C₂×#14524040	C₂²	1	1	yes	characteristic
11	#16863770	—	2	2	no	—
12	#16863770	C₄	1	4	yes	complemented
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14523965	—	2	2	no	—
15	C₂×#14523953	—	2	1	no	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863158	—	2	2	no	—
19	#16863150	—	2	1	no	—
20	#16863148	—	2	2	no	—
21	#16863139	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#332×C₆₇	—	2	2	no	—
26	C₂²×D₂₆₈	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	1	no	—
28	C₂²×D₂₆₈	—	2	1	no	—
29	C₂×#11761615	—	2	4	no	—
30	C₂×#11761615	—	2	4	no	—
31	C₂×#11761615	—	4	2	no	—
32	C₂×#11761615	—	4	2	no	—
33	C₂×#11761615	—	4	2	no	—
34	C₂×#11761615	—	4	2	no	—
35	C₂²×C₆₇⋊C₈	—	2	1	no	—
36	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
37	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
38	#14524040	—	2	1	no	—
39	#14524040	—	2	1	no	—
40	#14524040	—	2	1	no	—
41	#14524040	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
43	C₂×C₄×Dic₆₇	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
46	#14523965	—	4	4	no	—
47	#14523953	—	4	2	no	—
48	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
50	C₆₇⋊#95	—	2	1	no	—
51	C₂²×Q₈×C₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	—	2	1	no	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	1	no	—
55	C₄⋊Q₈×C₆₇	—	2	1	no	—
56	C₄⋊Q₈×C₆₇	—	2	1	no	—
57	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
65	C₂³×D₆₇	—	4	1	no	—
66	C₂×D₂₆₈	—	2	2	no	—
67	C₂×D₂₆₈	—	2	2	no	—
68	C₂×D₂₆₈	—	2	2	no	—
69	C₂×D₂₆₈	—	4	1	no	—
70	C₂×D₂₆₈	—	4	1	no	—
71	C₂×D₂₆₈	—	4	2	no	—
72	C₂×D₂₆₈	—	4	1	no	—
73	C₂×D₂₆₈	—	4	1	no	—
74	C₂×D₂₆₈	—	4	2	no	—
75	C₂×D₂₆₈	—	4	1	no	—
76	C₂×D₂₆₈	—	4	1	no	—
77	C₂×D₂₆₈	—	4	1	no	—
78	C₂×D₂₆₈	—	8	2	no	—
79	C₂²×Dic₆₇	—	2	2	no	—
80	#11761615	—	4	8	no	—
81	#11761615	—	4	8	no	—
82	C₂×C₆₇⋊C₈	—	2	1	no	—
83	C₂×C₆₇⋊C₈	—	2	1	no	—
84	C₂×C₆₇⋊C₈	—	4	1	no	—
85	C₂×C₆₇⋊C₈	—	4	2	no	—
86	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
89	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
90	C₄⋊Dic₆₇	—	2	2	no	—
91	C₄⋊Dic₆₇	—	2	2	no	—
92	C₄×Dic₆₇	—	4	1	no	—
93	C₄×Dic₆₇	—	4	1	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	4	1	no	—
96	C₂×Q₈×C₆₇	—	4	1	no	—
97	C₂×Q₈×C₆₇	—	4	1	no	—
98	C₂×Q₈×C₆₇	—	4	1	no	—
99	C₂×Q₈×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	—	4	1	no	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₂×C₈×C₆₇	—	4	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	#41	1	1	yes	characteristic
117	C₂³×C₆₇	#121	1	1	yes	characteristic
118	C₂²×D₆₇	—	4	2	no	—
119	C₂²×D₆₇	—	8	1	no	—
120	C₂²×D₆₇	—	8	1	no	—
121	C₂²×D₆₇	—	8	2	no	—
122	C₂²×D₆₇	—	8	2	no	—
123	Q₈×C₆₇	—	4	2	no	—
124	Q₈×C₆₇	—	4	2	no	—
125	Q₈×C₆₇	—	8	1	no	—
126	Q₈×C₆₇	—	8	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	8	1	no	—
130	C₂×C₄×C₆₇	—	8	1	no	—
131	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₆₇	#137	1	2	yes	—
143	C₂×Dic₆₇	—	4	2	no	—
144	C₂×Dic₆₇	—	4	2	no	—
145	C₂×Dic₆₇	—	4	2	no	—
146	D₂₆₈	—	4	2	no	—
147	D₂₆₈	—	4	2	no	—
148	D₂₆₈	—	4	4	no	—
149	D₂₆₈	—	8	2	no	—
150	D₂₆₈	—	8	2	no	—
151	D₂₆₈	—	8	1	no	—
152	D₂₆₈	—	8	1	no	—
153	D₂₆₈	—	8	1	no	—
154	D₂₆₈	—	8	1	no	—
155	C₈×C₆₇	—	4	2	no	—
156	C₈×C₆₇	—	8	1	no	—
157	C₆₇⋊C₈	—	2	2	no	—
158	C₆₇⋊C₈	—	8	1	no	—
159	C₆₇⋊C₈	#102	1	4	yes	complemented
160	C₂²×C₆₇	#390	1	1	yes	characteristic
161	C₂²×C₆₇	#459	1	1	yes	characteristic
162	C₂²×C₆₇	#460	1	1	yes	characteristic
163	C₂²×C₆₇	#468	1	2	yes	—
164	C₂²×C₆₇	#471	1	2	yes	—
165	C₂×D₆₇	—	8	2	no	—
166	C₂×D₆₇	—	8	2	no	—
167	C₂×D₆₇	—	8	2	no	—
168	C₂×D₆₇	—	16	2	no	—
169	C₂×D₆₇	—	16	2	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
174	C₄×C₆₇	#418	1	1	yes	characteristic
175	C₄×C₆₇	#440	1	2	yes	—
176	C₄×C₆₇	#459	1	1	yes	characteristic
177	C₄×C₆₇	#460	1	1	yes	characteristic
178	C₄×C₆₇	#496	1	1	yes	characteristic
179	C₄×C₆₇	#497	1	1	yes	characteristic
180	Dic₆₇	—	8	2	no	—
181	#9744	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	#1685	1	2	yes	—
183	C₂×C₆₇	#1686	1	2	yes	—
184	C₂×C₆₇	#1973	1	1	yes	characteristic
185	C₂×C₆₇	#1977	1	1	yes	characteristic
186	C₂×C₆₇	#1982	1	1	yes	characteristic
187	D₆₇	—	16	2	no	—
188	C₂×#492	—	67	1	no	—
189	#1988	—	67	1	no	—
190	#1982	—	67	1	no	—
191	#1770	—	67	1	no	—
192	C₂×#335	—	67	1	no	—
193	C₂×#332	D₆₇	1	1	yes	characteristic, complemented
194	C₂×#331	—	67	1	no	—
195	C₆₇	#9744	1	1	yes	characteristic, complemented, Sylow
196	C₂²×SD₁₆	—	134	1	no	—
197	C₂²×SD₁₆	—	134	1	no	—
198	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
199	C₂×#127	—	67	1	no	—
200	#492	—	67	4	no	—
201	#492	—	134	2	no	—
202	C₂×C₄⋊C₈	—	67	1	no	—
203	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
205	C₂×#102	—	134	2	no	—
206	C₂×#97	—	134	1	no	—
207	C₂×C₄×C₈	—	67	1	no	—
208	C₄×C₄⋊C₄	—	67	1	no	—
209	#340	—	134	2	no	—
210	#336	—	134	1	no	—
211	#335	—	134	2	no	—
212	#332	—	2	2	no	—
213	#331	—	134	2	no	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×D₄	—	134	1	no	—
216	C₂²×D₄	—	134	1	no	—
217	C₂²×D₄	—	134	1	no	—
218	C₂×SD₁₆	—	268	2	no	—
219	C₂×SD₁₆	—	268	2	no	—
220	C₂×SD₁₆	—	268	2	no	—
221	C₂×SD₁₆	—	268	2	no	—
222	C₂×SD₁₆	—	134	4	no	—
223	C₂×SD₁₆	—	134	4	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	67	1	no	—
227	C₂²×C₈	—	67	1	no	—
228	C₂²×C₈	—	134	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	C₄×D₆₇	1	2	yes	—
233	#127	—	67	2	no	—
234	#127	—	134	1	no	—
235	#127	—	134	1	no	—
236	#127	—	134	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	67	2	no	—
240	C₂×C₄²	—	134	1	no	—
241	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	134	1	no	—
246	C₄⋊C₈	—	134	1	no	—
247	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
248	#102	—	268	4	no	—
249	#97	—	268	2	no	—
250	C₄×C₈	—	67	2	no	—
251	C₄×C₈	—	67	2	no	—
252	#95	—	134	1	no	—
253	C₂⁴	—	268	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×D₄	—	268	1	no	—
261	C₂×D₄	—	268	1	no	—
262	C₂×D₄	—	268	1	no	—
263	C₂×D₄	—	268	1	no	—
264	C₂×D₄	—	268	1	no	—
265	C₂×D₄	—	268	1	no	—
266	C₂×D₄	—	268	2	no	—
267	C₂×D₄	—	268	1	no	—
268	C₂×D₄	—	268	2	no	—
269	C₂×D₄	—	536	2	no	—
270	C₂×D₄	—	134	2	no	—
271	C₂×D₄	—	134	2	no	—
272	C₂×D₄	—	134	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	134	2	no	—
275	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
277	C₂²×C₄	#11761628	1	1	yes	characteristic
278	SD₁₆	—	268	8	no	—
279	SD₁₆	—	268	8	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	268	1	no	—
286	C₂×C₈	—	268	2	no	—
287	C₂×C₈	—	67	2	no	—
288	C₂×C₈	—	67	2	no	—
289	C₂×C₈	—	67	2	no	—
290	C₂×C₈	—	67	2	no	—
291	C₂×C₈	—	134	1	no	—
292	C₂×C₈	—	134	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	134	2	no	—
298	C₄⋊C₄	—	134	2	no	—
299	C₄²	—	268	1	no	—
300	C₄²	—	268	1	no	—
301	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₄²	D₄×D₆₇	1	1	yes	characteristic
303	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
304	C₄²	#11761627	1	1	yes	characteristic
305	C₂³	—	268	2	no	—
306	C₂³	—	536	1	no	—
307	C₂³	—	536	1	no	—
308	C₂³	—	536	2	no	—
309	C₂³	—	536	2	no	—
310	C₂³	#14524016	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	1	no	—
314	Q₈	—	8	1	no	—
315	D₄	—	268	4	no	—
316	D₄	—	268	2	no	—
317	D₄	—	268	2	no	—
318	D₄	—	536	2	no	—
319	D₄	—	536	2	no	—
320	D₄	—	536	1	no	—
321	D₄	—	536	1	no	—
322	D₄	—	536	1	no	—
323	D₄	—	536	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	268	2	no	—
329	C₂×C₄	—	268	2	no	—
330	C₂×C₄	—	268	2	no	—
331	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
332	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
334	C₂×C₄	#14524015	1	1	yes	characteristic
335	C₂×C₄	#14524029	1	1	yes	characteristic
336	C₂×C₄	#14524039	1	1	yes	characteristic
337	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
338	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
339	C₂×C₄	#14524067	1	1	yes	characteristic
340	C₂×C₄	#14524070	1	2	yes	—
341	C₂×C₄	#14524070	1	1	yes	characteristic
342	C₂×C₄	#14524075	1	2	yes	—
343	C₈	—	4	2	no	—
344	C₈	—	8	1	no	—
345	C₈	—	536	1	no	—
346	C₈	—	67	4	no	—
347	C₈	—	134	2	no	—
348	C₂²	—	536	2	no	—
349	C₂²	—	536	2	no	—
350	C₂²	—	536	2	no	—
351	C₂²	—	1072	2	no	—
352	C₂²	—	1072	2	no	—
353	C₂²	#16863296	1	1	yes	characteristic
354	C₂²	#16863590	1	1	yes	characteristic
355	C₂²	#16863595	1	1	yes	characteristic
356	C₂²	#16863645	1	2	yes	—
357	C₂²	#16863662	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	8	2	no	—
361	C₄	—	536	2	no	—
362	C₄	#102×D₆₇	1	1	yes	characteristic
363	C₄	#16863399	1	1	yes	characteristic
364	C₄	#16863499	1	2	yes	—
365	C₄	#16863589	1	1	yes	characteristic
366	C₄	#16863594	1	1	yes	characteristic
367	C₄	#16863782	1	1	yes	characteristic
368	C₄	#16863789	1	1	yes	characteristic
369	C₂	—	1072	2	no	—
370	C₂	#22559677	1	2	yes	—
371	C₂	#22559684	1	2	yes	—
372	C₂	#22561087	1	1	yes	characteristic
373	C₂	#22561113	1	1	yes	characteristic
374	C₂	#22561134	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524016
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#332×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#332	D₆₇
hypocenter	C₆₇	#9744
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #22561162	C₂	semidirect
4. #22560079	C₂	semidirect
5. C₂×#16863148	C₂	semidirect
7. C₂×#332×C₆₇	C₂	semidirect
12. #16863770	C₄	semidirect
159. C₆₇⋊C₈	#102	semidirect
193. C₂×#332	D₆₇	semidirect
195. C₆₇	#9744	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863770	2
#22561162	2
#22561135	2
#22560079	2
C₂×#16863148	2
C₂×#16863139	2
C₂×#332×C₆₇	2
#9744	67