Subgroups — #30859253

Order 17152.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22568772	C₂	1	2	yes	maximal
4	#22560962	C₂	1	4	yes	maximal, complemented
5	#22560957	C₂	1	4	yes	maximal, complemented
6	#22559919	C₂	1	2	yes	maximal
7	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	2	yes	—
9	C₂×#14524002	C₂²	1	2	yes	—
10	C₂×#14523979	—	2	4	no	—
11	C₂×#14523979	C₂²	1	2	yes	—
12	C₂×#14523979	C₂²	1	2	yes	—
13	#16863434	C₄	1	2	yes	complemented
14	#16863434	C₄	1	2	yes	complemented
15	#16863434	C₂²	1	2	yes	—
16	#16863434	C₂²	1	2	yes	—
17	C₂×#14523977	C₄	1	2	yes	complemented
18	C₂×#14523975	C₄	1	2	yes	complemented
19	#16863424	—	2	4	no	—
20	#16863422	C₂²	1	8	yes	—
21	#16863419	—	2	4	no	—
22	#16863156	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
26	#405×C₆₇	—	2	4	no	—
27	C₂×#97×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	4	no	—
31	C₂×#11761612	—	2	4	no	—
32	C₂×#11761612	C₂³	1	4	yes	—
33	C₂²×C₆₇⋊C₈	—	2	2	no	—
34	#14524002	—	2	2	no	—
35	#14524002	—	2	2	no	—
36	#14524002	C₂×C₄	1	4	yes	—
37	#14524002	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
40	C₂×C₄×Dic₆₇	—	2	2	no	—
41	#14523979	—	2	4	no	—
42	#14523979	—	2	4	no	—
43	#14523977	C₂×C₄	1	4	yes	—
44	#14523977	C₂×C₄	1	4	yes	—
45	#14523975	C₂×C₄	1	4	yes	—
46	#14523975	C₂×C₄	1	4	yes	—
47	Dic₆₇⋊C₈	—	2	16	no	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
52	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	D₄	1	4	yes	—
58	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	#117×C₆₇	—	2	2	no	—
63	#117×C₆₇	Q₈	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	#97×C₆₇	—	2	4	no	—
68	C₄×C₈×C₆₇	—	2	4	no	—
69	C₂²×Dic₆₇	—	2	4	no	—
70	#11761612	—	2	4	no	—
71	#11761612	—	2	4	no	—
72	#11761612	—	4	4	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	4	4	no	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
77	C₂²⋊Dic₆₇	—	2	4	no	—
78	C₂²⋊Dic₆₇	—	2	4	no	—
79	C₄×Dic₆₇	—	2	4	no	—
80	C₄×Dic₆₇	—	2	4	no	—
81	C₂⁴×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	#41	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	—	2	4	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
108	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₂²⋊C₄×C₆₇	—	2	4	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
117	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₆₇	#125	1	2	yes	—
119	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	4	4	no	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	#121	1	2	yes	—
129	C₂×C₄×C₆₇	#121	1	2	yes	—
130	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₆₇	#125	1	2	yes	—
132	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₆₇	—	2	8	no	—
141	C₂×Dic₆₇	—	8	4	no	—
142	C₈×C₆₇	—	2	4	no	—
143	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
144	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
145	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
146	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
147	C₆₇⋊C₈	—	4	4	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×C₆₇	#389	1	2	yes	—
154	C₂²×C₆₇	#389	1	2	yes	—
155	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
158	C₄×C₆₇	—	4	4	no	—
159	C₄×C₆₇	#389	1	2	yes	—
160	C₄×C₆₇	#389	1	2	yes	—
161	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
164	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	Dic₆₇	—	4	8	no	—
166	#21163	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	—	2	1	no	—
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	2	2	no	—
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
172	C₂×C₆₇	#2957	1	2	yes	—
173	C₂²×#108	—	67	1	no	—
174	C₂×#427	—	67	1	no	—
175	#2905	—	67	2	no	—
176	C₂×#405	D₆₇	1	1	yes	characteristic
177	#1946	—	67	4	no	—
178	#1945	—	67	4	no	—
179	#1733	—	67	2	no	—
180	C₆₇	#21163	1	1	yes	characteristic, complemented, Sylow
181	C₂²×M₁₆	—	67	2	no	—
182	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
183	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#117	—	67	2	no	—
185	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
186	C₂×#108	—	67	2	no	—
187	C₂×#108	—	67	2	no	—
188	C₂×#108	—	134	4	no	—
189	#427	—	67	2	no	—
190	#427	—	67	2	no	—
191	#427	—	67	2	no	—
192	#427	—	67	2	no	—
193	C₂×#107	—	67	2	no	—
194	C₂×#106	—	67	2	no	—
195	#424	—	67	8	no	—
196	#424	—	134	4	no	—
197	#423	—	134	4	no	—
198	#405	—	2	4	no	—
199	C₂×#97	C₂×D₆₇	1	2	yes	—
200	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
201	#339	—	67	8	no	—
202	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	67	4	no	—
206	C₂×M₁₆	—	134	4	no	—
207	C₂×M₁₆	—	134	4	no	—
208	C₂×M₁₆	—	134	2	no	—
209	C₂×M₁₆	D₂₆₈	1	2	yes	—
210	C₂×M₁₆	D₂₆₈	1	2	yes	—
211	C₂²×C₈	—	134	2	no	—
212	C₂²×C₈	Dic₁₃₄	1	4	yes	—
213	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
214	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
218	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
219	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	67	4	no	—
223	#117	—	67	4	no	—
224	#117	—	134	2	no	—
225	#117	—	134	2	no	—
226	#117	Dic₁₃₄	1	4	yes	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	67	4	no	—
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	134	2	no	—
231	C₂×C₄²	—	134	2	no	—
232	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
233	#108	—	134	4	no	—
234	#108	—	134	4	no	—
235	#107	—	67	4	no	—
236	#107	—	67	4	no	—
237	#106	—	67	4	no	—
238	#106	—	67	4	no	—
239	C₄⋊C₈	—	134	16	no	—
240	#97	—	2	4	no	—
241	C₄×C₈	—	2	4	no	—
242	C₂⁴	#11761628	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	134	4	no	—
245	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
248	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
251	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
252	C₂²×C₄	#11761628	1	1	yes	characteristic
253	C₂²×C₄	#11761628	1	2	yes	—
254	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	268	4	no	—
258	M₁₆	—	134	4	no	—
259	M₁₆	—	134	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	268	4	no	—
263	C₂×C₈	—	134	4	no	—
264	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
265	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
266	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
267	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
268	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
274	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
275	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
276	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	67	8	no	—
279	C₄⋊C₄	—	134	4	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	134	4	no	—
282	C₂²⋊C₄	—	134	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	134	4	no	—
285	C₄²	—	134	4	no	—
286	C₄²	C₂×Dic₁₃₄	1	4	yes	—
287	C₂³	—	4	2	no	—
288	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
289	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
290	C₂³	#14524022	1	2	yes	—
291	C₂³	#14524035	1	2	yes	—
292	C₂³	C₂×#11761628	1	1	yes	characteristic
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	536	4	no	—
296	C₂×C₄	—	134	8	no	—
297	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
298	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
299	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
301	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
302	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
303	C₂×C₄	#14524017	1	2	yes	—
304	C₂×C₄	#14524017	1	2	yes	—
305	C₂×C₄	#14524022	1	2	yes	—
306	C₂×C₄	#14524035	1	2	yes	—
307	C₂×C₄	#14524043	1	2	yes	—
308	C₂×C₄	#14524043	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
312	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
314	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	268	4	no	—
317	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
318	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
319	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
320	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	#16863294	1	2	yes	—
327	C₂²	#16863294	1	2	yes	—
328	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
329	C₂²	C₂×#14524022	1	1	yes	characteristic
330	C₂²	C₂×#14524035	1	1	yes	characteristic
331	C₄	—	4	4	no	—
332	C₄	—	268	8	no	—
333	C₄	#16863294	1	2	yes	—
334	C₄	#16863294	1	2	yes	—
335	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
336	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
337	C₄	C₂×#14524017	1	1	yes	characteristic
338	C₄	C₂×#14524043	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	C₂×#16863294	1	1	yes	characteristic
344	C₂	#22569127	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#21163
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #22560962	C₂	semidirect
5. #22560957	C₂	semidirect
13. #16863434	C₄	semidirect
14. #16863434	C₄	semidirect
17. C₂×#14523977	C₄	semidirect
18. C₂×#14523975	C₄	semidirect
180. C₆₇	#21163	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523979	2
C₂×#16863434	2
#22568772	2
#22560962	2
#22560957	2
#22559919	2
C₂×#405×C₆₇	2
#21163	67