Subgroups — #30864031

Order 17152.

738 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	#22570554	C₂	1	1	yes	maximal, characteristic
2	#22569146	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568927	C₂	1	1	yes	maximal, characteristic
4	#22561186	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560917	C₂	1	2	yes	maximal, complemented
6	#22560910	C₂	1	2	yes	maximal, complemented
7	#22560349	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560054	C₂	1	2	yes	maximal, complemented
9	#22560047	C₂	1	2	yes	maximal, complemented
10	C₄×#14523954	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1962	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₂²	1	2	yes	—
13	#16863953	C₂²	1	1	yes	characteristic
14	#16863764	C₄	1	2	yes	complemented
15	#16863760	C₄	1	2	yes	complemented
16	#16863755	C₄	1	4	yes	complemented
17	#16863525	—	2	4	no	—
18	#16863468	—	2	4	no	—
19	#16863424	C₂²	1	2	yes	—
20	#16863420	C₂²	1	1	yes	characteristic
21	#16863392	C₂²	1	2	yes	—
22	#16863392	C₂²	1	2	yes	—
23	#16863363	C₂²	1	2	yes	—
24	#16863354	C₂²	1	2	yes	—
25	C₂×#14523954	C₂²	1	1	yes	characteristic
26	C₂×#14523954	C₂²	1	1	yes	characteristic
27	C₂×#14523954	C₂²	1	1	yes	characteristic
28	C₂×#14523954	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	#16863160	C₂²	1	2	yes	—
32	#16863160	C₂²	1	2	yes	—
33	#16863156	C₂²	1	2	yes	—
34	#16863156	C₂²	1	2	yes	—
35	#532×C₆₇	C₂²	1	1	yes	characteristic
36	#423×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#104×C₆₇	C₂²	1	2	yes	—
38	C₂×#104×C₆₇	C₂²	1	2	yes	—
39	C₄³×C₆₇	C₂²	1	1	yes	characteristic
40	#14524050	—	2	4	no	—
41	#14524050	—	2	4	no	—
42	C₂×#11761612	—	2	2	no	—
43	C₂×#11761612	C₂³	1	2	yes	—
44	C₂²×C₆₇⋊C₈	—	2	2	no	—
45	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	#14524043	C₂×C₄	1	2	yes	—
50	#14524035	C₂×C₄	1	2	yes	—
51	#14524022	—	2	2	no	—
52	C₄×Dic₁₃₄	—	2	2	no	—
53	#14524002	C₂³	1	2	yes	—
54	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	#14523967	C₂×C₄	1	4	yes	—
57	#14523967	C₂×C₄	1	4	yes	—
58	#14523964	C₂×C₄	1	4	yes	—
59	#14523964	C₂×C₄	1	4	yes	—
60	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
61	#14523954	—	2	2	no	—
62	#14523954	—	2	2	no	—
63	#14523954	C₂×C₄	1	2	yes	—
64	#14523954	C₂×C₄	1	2	yes	—
65	#14523954	D₄	1	4	yes	—
66	#14523953	—	2	2	no	—
67	C₄×C₆₇⋊C₈	—	2	2	no	—
68	C₄×C₆₇⋊C₈	—	2	2	no	—
69	C₈×Dic₆₇	—	2	2	no	—
70	C₂×#41×C₆₇	C₂³	1	2	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
74	#124×C₆₇	C₂×C₄	1	2	yes	—
75	#121×C₆₇	—	2	2	no	—
76	C₄×D₄×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	#104×C₆₇	—	2	4	no	—
83	#104×C₆₇	—	2	4	no	—
84	C₂×Dic₁₃₄	—	2	2	no	—
85	#11761612	—	2	2	no	—
86	#11761612	—	2	2	no	—
87	#11761612	—	4	4	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	1	no	—
93	C₂×C₆₇⋊C₈	—	2	1	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₂×C₆₇⋊C₈	—	4	4	no	—
96	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
100	C₂×C₆₇⋊C₈	#41	1	2	yes	—
101	C₂×C₆₇⋊C₈	#41	1	2	yes	—
102	C₄⋊Dic₆₇	—	2	2	no	—
103	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
104	Dic₆₇⋊C₄	—	2	4	no	—
105	C₂²⋊Dic₆₇	—	2	2	no	—
106	C₄×Dic₆₇	—	2	2	no	—
107	#41×C₆₇	—	2	2	no	—
108	#41×C₆₇	—	2	2	no	—
109	#41×C₆₇	—	2	2	no	—
110	#41×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	#41	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₄²×C₆₇	—	2	1	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	4	4	no	—
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₄²×C₆₇	#41	1	2	yes	—
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	2	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	8	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₆₇	—	2	4	no	—
159	Dic₁₃₄	—	8	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₆₇⋊C₈	—	2	2	no	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₆₇⋊C₈	—	2	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₆₇⋊C₈	#124	1	4	yes	—
166	C₂²×C₆₇	—	2	2	no	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	4	2	no	—
169	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₆₇	#529	1	1	yes	characteristic
171	C₂²×C₆₇	#529	1	1	yes	characteristic
172	C₄×C₆₇	—	2	4	no	—
173	C₄×C₆₇	—	2	2	no	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	4	2	no	—
176	C₄×C₆₇	—	4	2	no	—
177	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
178	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
179	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
180	C₄×C₆₇	#532	1	1	yes	characteristic
181	Dic₆₇	—	4	4	no	—
182	#21662	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	—	2	1	no	—
185	C₂×C₆₇	—	8	2	no	—
186	C₂×C₆₇	C₄×#124	1	1	yes	characteristic
187	C₂×C₆₇	#2956	1	2	yes	—
188	#3135	—	67	1	no	—
189	#2961	—	67	1	no	—
190	#2932	—	67	1	no	—
191	#1994	—	67	1	no	—
192	#1962	D₆₇	1	1	yes	characteristic
193	#1938	—	67	2	no	—
194	#1937	—	67	2	no	—
195	#1838	—	67	1	no	—
196	#1765	—	67	2	no	—
197	#1764	—	67	2	no	—
198	C₄×#97	—	67	1	no	—
199	C₆₇	#21662	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	67	2	no	—
201	#533	—	67	1	no	—
202	#532	C₂×D₆₇	1	1	yes	characteristic
203	#491	—	67	2	no	—
204	#490	—	67	2	no	—
205	#489	—	67	4	no	—
206	#446	—	134	4	no	—
207	#435	—	134	4	no	—
208	#424	—	67	2	no	—
209	#423	—	67	1	no	—
210	#423	C₂×D₆₇	1	1	yes	characteristic
211	C₂×#104	C₂×D₆₇	1	2	yes	—
212	C₂×#104	C₂×D₆₇	1	2	yes	—
213	#416	—	67	2	no	—
214	#416	—	67	2	no	—
215	#408	—	67	2	no	—
216	#405	—	67	2	no	—
217	C₂×#97	—	67	1	no	—
218	C₂×#97	—	67	1	no	—
219	C₂×#97	—	67	1	no	—
220	C₂×#97	—	67	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	C₂×C₄×C₈	—	67	1	no	—
223	C₄³	C₂×D₆₇	1	1	yes	characteristic
224	#341	—	67	2	no	—
225	#341	—	67	2	no	—
226	#339	—	67	2	no	—
227	#339	—	67	2	no	—
228	C₂×#41	C₂²×D₆₇	1	2	yes	—
229	#131	—	134	4	no	—
230	#131	—	134	4	no	—
231	C₂×M₁₆	—	67	2	no	—
232	C₂×M₁₆	—	134	2	no	—
233	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
234	C₂²×C₈	—	67	1	no	—
235	C₂²×C₈	—	67	1	no	—
236	C₂²×C₈	—	67	1	no	—
237	C₂²×C₈	—	67	1	no	—
238	C₂²×C₈	—	134	2	no	—
239	C₄⋊Q₈	—	67	2	no	—
240	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
241	#127	C₄×D₆₇	1	1	yes	characteristic
242	#125	—	67	2	no	—
243	#124	C₄×D₆₇	1	2	yes	—
244	C₂²⋊Q₈	—	134	2	no	—
245	#121	—	2	2	no	—
246	C₄×Q₈	—	134	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	—	67	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	134	4	no	—
255	C₄⋊C₈	—	134	2	no	—
256	#104	—	2	4	no	—
257	#104	—	2	4	no	—
258	#103	—	67	4	no	—
259	#103	—	67	4	no	—
260	#102	—	67	4	no	—
261	#102	—	67	4	no	—
262	C₂²⋊C₈	—	134	4	no	—
263	#97	—	67	2	no	—
264	#97	—	67	2	no	—
265	#97	—	67	4	no	—
266	#97	—	134	2	no	—
267	#97	—	134	2	no	—
268	#97	—	134	2	no	—
269	C₄×C₈	—	134	2	no	—
270	C₄×C₈	—	134	2	no	—
271	C₄×C₈	—	134	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	C₂×Q₈	—	134	2	no	—
277	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	#11761627	1	2	yes	—
285	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	268	4	no	—
289	M₁₆	—	134	2	no	—
290	M₁₆	—	134	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	268	4	no	—
293	C₂×C₈	—	67	2	no	—
294	C₂×C₈	—	67	2	no	—
295	C₂×C₈	—	67	2	no	—
296	C₂×C₈	—	67	2	no	—
297	C₂×C₈	—	67	2	no	—
298	C₂×C₈	—	67	2	no	—
299	C₂×C₈	—	134	2	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	—	134	1	no	—
302	C₂×C₈	—	134	1	no	—
303	C₂×C₈	—	134	1	no	—
304	C₂×C₈	—	134	1	no	—
305	C₂×C₈	—	134	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	67	4	no	—
308	C₄⋊C₄	—	134	4	no	—
309	C₄⋊C₄	—	134	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	134	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	4	4	no	—
315	C₄²	—	134	2	no	—
316	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
318	C₄²	D₄×D₆₇	1	2	yes	—
319	C₄²	#11761627	1	2	yes	—
320	C₄²	#11761627	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	C₂×#11761627	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	536	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	134	4	no	—
340	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
341	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
344	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
345	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
347	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
348	C₈	—	4	2	no	—
349	C₈	—	268	2	no	—
350	C₈	—	67	4	no	—
351	C₈	—	134	2	no	—
352	C₈	—	134	2	no	—
353	C₈	—	134	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
358	C₂²	#16863927	1	1	yes	characteristic
359	C₂²	#16863927	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	268	4	no	—
366	C₄	C₄×#11761627	1	1	yes	characteristic
367	C₄	C₄×#11761627	1	1	yes	characteristic
368	C₄	#124×D₆₇	1	1	yes	characteristic
369	C₄	#16863944	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	#22562756	1	1	yes	characteristic
374	C₂	#22569115	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1962	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1962	D₆₇
hypocenter	C₆₇	#21662
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

Complements

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

N	H ≅ G/N	type
2. #22569146	C₂	semidirect
4. #22561186	C₂	semidirect
5. #22560917	C₂	semidirect
6. #22560910	C₂	semidirect
7. #22560349	C₂	semidirect
8. #22560054	C₂	semidirect
9. #22560047	C₂	semidirect
10. C₄×#14523954	C₂	semidirect
14. #16863764	C₄	semidirect
15. #16863760	C₄	semidirect
16. #16863755	C₄	semidirect
199. C₆₇	#21662	semidirect

Maximal subgroups

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

subgroup	index
#22570554	2
#22569146	2
#22568927	2
#22561186	2
#22560917	2
#22560910	2
#22560349	2
#22560054	2
#22560047	2
C₄×#14523954	2
C₆₇×#1962	2
#21662	67