Subgroups — #30859254

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	#22568773	C₂	1	2	yes	maximal
3	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
4	#22560963	C₂	1	4	yes	maximal, complemented
5	#22560958	C₂	1	4	yes	maximal, complemented
6	#22559918	C₂	1	2	yes	maximal
7	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	2	yes	—
9	C₂²×#11761612	C₂²	1	1	yes	characteristic
10	C₂×#14524002	C₂²	1	1	yes	characteristic
11	C₂×#14523979	—	2	4	no	—
12	C₂×#14523979	C₂²	1	2	yes	—
13	C₂×#14523979	C₂²	1	2	yes	—
14	#16863423	—	2	4	no	—
15	#16863423	C₂²	1	8	yes	—
16	#16863420	—	2	4	no	—
17	#16863351	—	2	4	no	—
18	C₂×#14523953	C₂²	1	2	yes	—
19	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
20	#16863155	C₂²	1	8	yes	—
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	2	yes	—
23	#427×C₆₇	C₄	1	2	yes	complemented
24	#427×C₆₇	C₄	1	2	yes	complemented
25	#427×C₆₇	C₂²	1	2	yes	—
26	#427×C₆₇	C₂²	1	2	yes	—
27	C₂×#107×C₆₇	C₄	1	2	yes	complemented
28	C₂×#106×C₆₇	C₄	1	2	yes	complemented
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	—	2	4	no	—
32	C₂×#11761612	—	2	2	no	—
33	C₂×#11761612	—	2	4	no	—
34	C₂×#11761612	D₄	1	2	yes	—
35	C₂×#11761612	D₄	1	2	yes	—
36	C₂×#11761612	C₂³	1	4	yes	—
37	C₂²×C₆₇⋊C₈	—	2	2	no	—
38	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
39	#14524002	—	2	2	no	—
40	#14524002	Q₈	1	4	yes	—
41	C₂×C₄⋊Dic₆₇	—	2	1	no	—
42	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
43	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
44	#14523979	—	2	4	no	—
45	#14523979	—	2	4	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
47	#14523953	—	2	4	no	—
48	C₈×Dic₆₇	—	2	4	no	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
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	D₄	1	4	yes	—
55	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	2	no	—
60	#117×C₆₇	C₂×C₄	1	4	yes	—
61	#117×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	#107×C₆₇	C₂×C₄	1	4	yes	—
66	#107×C₆₇	C₂×C₄	1	4	yes	—
67	#106×C₆₇	C₂×C₄	1	4	yes	—
68	#106×C₆₇	C₂×C₄	1	4	yes	—
69	C₂²×Dic₆₇	—	2	2	no	—
70	#11761612	—	2	4	no	—
71	#11761612	—	2	4	no	—
72	#11761612	—	2	4	no	—
73	#11761612	—	2	4	no	—
74	#11761612	—	4	4	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	4	4	no	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
80	C₄⋊Dic₆₇	—	2	4	no	—
81	C₂²⋊Dic₆₇	—	2	4	no	—
82	C₄×Dic₆₇	—	2	2	no	—
83	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
84	C₂⁴×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	#41	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
110	C₂²⋊C₄×C₆₇	—	2	4	no	—
111	C₂²⋊C₄×C₆₇	—	2	4	no	—
112	C₄²×C₆₇	—	2	4	no	—
113	C₄²×C₆₇	—	2	4	no	—
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	8	no	—
121	C₂×C₄×C₆₇	—	8	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	4	no	—
141	C₂×Dic₆₇	—	4	4	no	—
142	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
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	C₆₇⋊C₈	—	2	4	no	—
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	8	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	4	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	D₆₇	1	1	yes	characteristic
175	#2905	—	67	2	no	—
176	C₂×#405	—	67	1	no	—
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	1	no	—
182	C₂²×M₁₆	—	67	2	no	—
183	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#117	—	67	1	no	—
185	C₂×#117	C₂×D₆₇	1	2	yes	—
186	C₂×#108	—	67	2	no	—
187	C₂×#108	—	67	2	no	—
188	C₂×#108	—	134	4	no	—
189	#427	Dic₆₇	1	2	yes	complemented
190	#427	Dic₆₇	1	2	yes	complemented
191	#427	C₂×D₆₇	1	2	yes	—
192	#427	C₂×D₆₇	1	2	yes	—
193	C₂×#107	Dic₆₇	1	2	yes	complemented
194	C₂×#106	Dic₆₇	1	2	yes	complemented
195	#424	—	67	8	no	—
196	#424	—	134	4	no	—
197	#423	—	134	4	no	—
198	#405	—	134	4	no	—
199	C₂×#97	—	67	2	no	—
200	C₂×C₄×C₈	—	67	2	no	—
201	#339	—	67	8	no	—
202	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
203	C₂×M₁₆	—	67	2	no	—
204	C₂×M₁₆	—	67	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	—	134	2	no	—
210	C₂×M₁₆	—	134	2	no	—
211	C₂²×C₈	—	67	2	no	—
212	C₂²×C₈	—	134	2	no	—
213	C₂²×C₈	Dic₁₃₄	1	4	yes	—
214	C₂²×C₈	D₂₆₈	1	4	yes	—
215	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
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	—	2	2	no	—
223	#117	—	67	4	no	—
224	#117	—	134	2	no	—
225	#117	C₂×Dic₆₇	1	4	yes	—
226	#117	C₂²×D₆₇	1	4	yes	—
227	C₂×C₄⋊C₄	—	134	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	134	1	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	67	2	no	—
233	#108	—	134	4	no	—
234	#108	—	134	4	no	—
235	#107	C₂×Dic₆₇	1	4	yes	—
236	#107	C₂×Dic₆₇	1	4	yes	—
237	#106	C₂×Dic₆₇	1	4	yes	—
238	#106	C₂×Dic₆₇	1	4	yes	—
239	C₄⋊C₈	—	134	16	no	—
240	#97	—	134	4	no	—
241	C₄×C₈	—	134	4	no	—
242	C₂⁴	#11761627	1	1	yes	characteristic
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	134	2	no	—
245	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
250	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
251	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
252	C₂²×C₄	#11761627	1	1	yes	characteristic
253	C₂²×C₄	#11761629	1	2	yes	—
254	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
255	M₁₆	—	268	4	no	—
256	M₁₆	—	134	4	no	—
257	M₁₆	—	134	4	no	—
258	M₁₆	—	134	4	no	—
259	M₁₆	—	134	4	no	—
260	C₂×C₈	—	268	4	no	—
261	C₂×C₈	—	67	4	no	—
262	C₂×C₈	—	134	4	no	—
263	C₂×C₈	—	134	2	no	—
264	C₂×C₈	—	134	2	no	—
265	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
266	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
267	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
268	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
269	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
273	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
274	C₂×C₈	C₂×D₂₆₈	1	2	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	—	134	4	no	—
279	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	134	4	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	67	4	no	—
286	C₄²	—	134	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
289	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
290	C₂³	#14524023	1	2	yes	—
291	C₂³	#14524034	1	2	yes	—
292	C₂³	C₂×#11761627	1	1	yes	characteristic
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	268	4	no	—
296	C₂×C₄	—	134	4	no	—
297	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
299	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
301	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
302	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
303	C₂×C₄	#14524018	1	2	yes	—
304	C₂×C₄	#14524018	1	2	yes	—
305	C₂×C₄	#14524023	1	2	yes	—
306	C₂×C₄	#14524034	1	2	yes	—
307	C₂×C₄	#14524042	1	2	yes	—
308	C₂×C₄	#14524042	1	2	yes	—
309	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
312	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
314	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
315	C₈	—	268	4	no	—
316	C₈	—	134	4	no	—
317	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
318	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
319	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
320	C₈	C₂×C₄⋊Dic₆₇	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	#16863295	1	2	yes	—
327	C₂²	#16863295	1	2	yes	—
328	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
329	C₂²	C₂×#14524023	1	1	yes	characteristic
330	C₂²	C₂×#14524034	1	1	yes	characteristic
331	C₄	—	4	8	no	—
332	C₄	—	268	4	no	—
333	C₄	#16863295	1	2	yes	—
334	C₄	#16863295	1	2	yes	—
335	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
336	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
337	C₄	C₂×#14524018	1	1	yes	characteristic
338	C₄	C₂×#14524042	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×#16863295	1	1	yes	characteristic
344	C₂	#22569126	1	2	yes	—

	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	#21163
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
4. #22560963	C₂	semidirect
5. #22560958	C₂	semidirect
23. #427×C₆₇	C₄	semidirect
24. #427×C₆₇	C₄	semidirect
27. C₂×#107×C₆₇	C₄	semidirect
28. C₂×#106×C₆₇	C₄	semidirect
180. C₆₇	#21163	semidirect
189. #427	Dic₆₇	semidirect
190. #427	Dic₆₇	semidirect
193. C₂×#107	Dic₆₇	semidirect
194. 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
C₂²×#14523979	2
#22568773	2
C₂×#16863351	2
#22560963	2
#22560958	2
#22559918	2
C₂×#427×C₆₇	2
#21163	67