Subgroups — #30873434

Order 17152.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568867	C₂	1	2	yes	maximal
2	#22560963	C₂	1	2	yes	maximal, complemented
3	#22560810	C₂	1	4	yes	maximal, complemented
4	#22560308	C₂	1	2	yes	maximal
5	#22560247	C₂	1	1	yes	maximal, characteristic
6	#22560095	C₂	1	2	yes	maximal, complemented
7	#22559985	C₂	1	1	yes	maximal, characteristic
8	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863457	—	2	4	no	—
10	#16863453	—	2	2	no	—
11	#16863449	—	2	2	no	—
12	C₂×#14523979	C₂²	1	2	yes	—
13	#16863423	C₂²	1	4	yes	—
14	#16863360	C₂²	1	2	yes	—
15	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#16863352	C₂²	1	4	yes	—
17	C₂×#14523953	C₂²	1	2	yes	—
18	C₄×C₂²⋊Dic₆₇	C₂²	1	1	yes	characteristic
19	#16863161	C₂²	1	2	yes	—
20	#16863155	C₂²	1	4	yes	—
21	#16863155	C₂²	1	4	yes	—
22	#16863150	C₂²	1	2	yes	—
23	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₆₇	C₂²	1	2	yes	—
25	#427×C₆₇	C₄	1	1	yes	characteristic, complemented
26	#427×C₆₇	C₄	1	1	yes	characteristic, complemented
27	#427×C₆₇	C₄	1	2	yes	complemented
28	#427×C₆₇	C₂²	1	2	yes	—
29	#427×C₆₇	C₂²	1	1	yes	characteristic
30	#427×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#107×C₆₇	C₄	1	1	yes	characteristic, complemented
32	C₂×#107×C₆₇	C₄	1	1	yes	characteristic, complemented
33	C₂×#106×C₆₇	C₄	1	2	yes	complemented
34	C₂×#11761612	C₂³	1	4	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
36	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
38	#14523979	—	2	4	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
40	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
43	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
44	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
45	#14523954	—	2	4	no	—
46	#14523953	—	2	4	no	—
47	C₄×C₆₇⋊C₈	—	2	4	no	—
48	C₆₇⋊#95	—	2	1	no	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	—	2	4	no	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	#117×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	C₂×C₄	1	2	yes	—
60	#117×C₆₇	C₂×C₄	1	2	yes	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	#117×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	#107×C₆₇	C₂×C₄	1	2	yes	—
68	#107×C₆₇	C₂×C₄	1	2	yes	—
69	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
70	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
71	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
72	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
73	#106×C₆₇	C₂×C₄	1	2	yes	—
74	#106×C₆₇	C₂×C₄	1	2	yes	—
75	#106×C₆₇	C₂×C₄	1	2	yes	—
76	#106×C₆₇	C₂×C₄	1	2	yes	—
77	C₂²×Dic₆₇	—	2	2	no	—
78	#11761612	—	2	4	no	—
79	#11761612	—	2	4	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	2	4	no	—
83	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
84	C₂²⋊Dic₆₇	—	4	2	no	—
85	C₄×Dic₆₇	—	2	2	no	—
86	C₄×Dic₆₇	—	2	2	no	—
87	C₂⁴×C₆₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	#41	1	4	yes	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	4	1	no	—
125	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
147	C₂×Dic₆₇	—	4	2	no	—
148	C₂×Dic₆₇	—	4	2	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	C₈×C₆₇	—	2	1	no	—
151	C₈×C₆₇	—	2	1	no	—
152	C₈×C₆₇	#117	1	2	yes	—
153	C₈×C₆₇	#117	1	2	yes	—
154	C₆₇⋊C₈	—	2	4	no	—
155	C₆₇⋊C₈	—	4	4	no	—
156	C₂²×C₆₇	—	2	1	no	—
157	C₂²×C₆₇	—	2	1	no	—
158	C₂²×C₆₇	—	2	1	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	4	4	no	—
164	C₂²×C₆₇	#444	1	2	yes	—
165	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₆₇	#552	1	2	yes	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	4	2	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
174	C₄×C₆₇	#519	1	1	yes	characteristic
175	C₄×C₆₇	#547	1	2	yes	—
176	Dic₆₇	—	8	2	no	—
177	#22607	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	—	2	2	no	—
179	C₂×C₆₇	—	2	2	no	—
180	C₂×C₆₇	—	4	1	no	—
181	C₂×C₆₇	#2344	1	1	yes	characteristic
182	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
183	C₂×C₆₇	#2957	1	1	yes	characteristic
184	#2924	—	67	2	no	—
185	C₂×#427	D₆₇	1	1	yes	characteristic
186	#1946	—	67	2	no	—
187	#1922	—	67	4	no	—
188	#1826	—	67	2	no	—
189	#1810	—	67	1	no	—
190	#1773	—	67	2	no	—
191	#1752	—	67	1	no	—
192	C₆₇	#22607	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₂×#117	C₂×D₆₇	1	2	yes	—
195	#433	—	134	4	no	—
196	#432	—	134	2	no	—
197	#431	—	134	2	no	—
198	C₂×#108	—	67	2	no	—
199	#427	Dic₆₇	1	1	yes	characteristic, complemented
200	#427	Dic₆₇	1	1	yes	characteristic, complemented
201	#427	Dic₆₇	1	2	yes	complemented
202	#427	C₂×D₆₇	1	2	yes	—
203	#427	C₂×D₆₇	1	1	yes	characteristic
204	#427	C₂×D₆₇	1	1	yes	characteristic
205	C₂×#107	Dic₆₇	1	1	yes	characteristic, complemented
206	C₂×#107	Dic₆₇	1	1	yes	characteristic, complemented
207	C₂×#106	Dic₆₇	1	2	yes	complemented
208	#424	—	67	4	no	—
209	#407	—	67	2	no	—
210	C₂×C₂²⋊C₈	—	67	1	no	—
211	#405	—	67	4	no	—
212	C₂×#97	—	67	2	no	—
213	C₄×C₂²⋊C₄	—	67	1	no	—
214	#341	—	67	2	no	—
215	#339	—	67	4	no	—
216	#339	—	67	4	no	—
217	#336	—	67	2	no	—
218	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
219	C₂×M₁₆	—	67	4	no	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	67	2	no	—
222	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
223	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
225	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	C₂×Dic₆₇	1	2	yes	—
231	#117	C₂×Dic₆₇	1	2	yes	—
232	#117	C₂²×D₆₇	1	2	yes	—
233	#117	C₂²×D₆₇	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	134	1	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	—	67	2	no	—
240	#108	—	134	4	no	—
241	#107	C₂×Dic₆₇	1	2	yes	—
242	#107	C₂×Dic₆₇	1	2	yes	—
243	#107	C₂×Dic₆₇	1	1	yes	characteristic
244	#107	C₂×Dic₆₇	1	1	yes	characteristic
245	#107	C₂×Dic₆₇	1	1	yes	characteristic
246	#107	C₂×Dic₆₇	1	1	yes	characteristic
247	#106	C₂×Dic₆₇	1	2	yes	—
248	#106	C₂×Dic₆₇	1	2	yes	—
249	#106	C₂×Dic₆₇	1	2	yes	—
250	#106	C₂×Dic₆₇	1	2	yes	—
251	C₄⋊C₈	—	134	4	no	—
252	C₄⋊C₈	—	134	4	no	—
253	C₂²⋊C₈	—	67	2	no	—
254	C₂²⋊C₈	—	67	2	no	—
255	C₂²⋊C₈	—	134	2	no	—
256	C₂²⋊C₈	—	134	2	no	—
257	#97	—	134	4	no	—
258	#97	—	134	4	no	—
259	C₄×C₈	—	134	4	no	—
260	#95	—	134	1	no	—
261	C₂⁴	#11761627	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	134	2	no	—
267	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
271	C₂²×C₄	#11761627	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
273	M₁₆	—	134	4	no	—
274	M₁₆	—	134	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	67	4	no	—
281	C₂×C₈	—	134	4	no	—
282	C₂×C₈	—	134	2	no	—
283	C₂×C₈	—	134	2	no	—
284	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
285	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
286	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
287	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
288	C₂×C₈	#11761628	1	4	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
292	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
293	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
294	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	268	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	134	2	no	—
305	C₄²	—	134	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
310	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
311	C₂³	C₂×#11761627	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	268	2	no	—
327	C₂×C₄	—	268	2	no	—
328	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×#11761628	1	2	yes	—
333	C₂×C₄	#14524091	1	2	yes	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	268	4	no	—
337	C₈	—	134	4	no	—
338	C₈	#14524002	1	2	yes	—
339	C₈	#14524002	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#16863518	1	2	yes	—
349	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
350	C₂²	#16864081	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	536	2	no	—
358	C₄	C₂×#14524002	1	1	yes	characteristic
359	C₄	#16863874	1	1	yes	characteristic
360	C₄	#16864047	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#22563078	1	1	yes	characteristic
365	C₂	C₂×#16863518	1	1	yes	characteristic
366	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#22607
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #22560963	C₂	semidirect
3. #22560810	C₂	semidirect
6. #22560095	C₂	semidirect
25. #427×C₆₇	C₄	semidirect
26. #427×C₆₇	C₄	semidirect
27. #427×C₆₇	C₄	semidirect
31. C₂×#107×C₆₇	C₄	semidirect
32. C₂×#107×C₆₇	C₄	semidirect
33. C₂×#106×C₆₇	C₄	semidirect
192. C₆₇	#22607	semidirect
199. #427	Dic₆₇	semidirect
200. #427	Dic₆₇	semidirect
201. #427	Dic₆₇	semidirect
205. C₂×#107	Dic₆₇	semidirect
206. C₂×#107	Dic₆₇	semidirect
207. C₂×#106	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568867	2
#22560963	2
#22560810	2
#22560308	2
#22560247	2
#22560095	2
#22559985	2
C₂×#427×C₆₇	2
#22607	67