Subgroups — #30859385

Order 17152.

874 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568799	C₂	1	2	yes	maximal
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
4	#22560995	C₂	1	4	yes	maximal, complemented
5	#22560990	C₂	1	4	yes	maximal, complemented
6	#22559947	C₂	1	2	yes	maximal
7	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	1	yes	characteristic
10	C₂×#14524002	C₂²	1	2	yes	—
11	#16863438	—	2	4	no	—
12	#16863438	C₂²	1	8	yes	—
13	#16863434	—	2	4	no	—
14	#16863434	—	2	1	no	—
15	#16863434	—	2	1	no	—
16	#16863434	—	2	1	no	—
17	#16863434	—	2	1	no	—
18	C₂×#14523977	C₂²	1	2	yes	—
19	C₂×#14523975	C₂²	1	2	yes	—
20	#16863351	—	2	1	no	—
21	#16863351	—	2	1	no	—
22	#16863351	C₂²	1	2	yes	—
23	#16863351	C₂²	1	2	yes	—
24	C₂×#14523953	—	2	1	no	—
25	C₂×C₈×Dic₆₇	—	2	1	no	—
26	#16863164	C₂²	1	8	yes	—
27	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
28	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#108×C₆₇	C₄	1	8	yes	complemented
30	C₂×#108×C₆₇	C₂²	1	2	yes	—
31	C₂×#108×C₆₇	C₂²	1	2	yes	—
32	C₂×#11761612	—	2	1	no	—
33	C₂×#11761612	—	2	2	no	—
34	C₂×#11761612	Q₈	1	4	yes	—
35	C₂×#11761612	C₂³	1	2	yes	—
36	C₂²×C₆₇⋊C₈	—	2	1	no	—
37	#14524002	—	2	1	no	—
38	#14524002	—	2	2	no	—
39	#14524002	—	2	2	no	—
40	#14524002	—	2	2	no	—
41	#14524002	—	2	2	no	—
42	#14524002	—	2	1	no	—
43	#14524002	D₄	1	2	yes	—
44	#14524002	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
46	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
48	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
49	C₂×C₄×Dic₆₇	—	2	1	no	—
50	C₂×C₄×Dic₆₇	—	2	2	no	—
51	#14523977	—	2	8	no	—
52	#14523977	—	2	2	no	—
53	#14523977	—	2	2	no	—
54	#14523975	—	2	8	no	—
55	#14523975	—	2	2	no	—
56	#14523975	—	2	2	no	—
57	#14523953	—	2	2	no	—
58	#14523953	—	2	2	no	—
59	C₈×Dic₆₇	—	2	2	no	—
60	C₈×Dic₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂²×C₈×C₆₇	—	2	2	no	—
67	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#108×C₆₇	C₂×C₄	1	8	yes	complemented
76	#108×C₆₇	C₂×C₄	1	8	yes	complemented
77	C₂²×Dic₆₇	—	2	2	no	—
78	C₂²×Dic₆₇	—	2	4	no	—
79	#11761612	—	2	4	no	—
80	#11761612	—	2	2	no	—
81	#11761612	C₂×Q₈	1	4	yes	—
82	C₂×C₆₇⋊C₈	—	2	4	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₄⋊Dic₆₇	—	2	2	no	—
85	C₄⋊Dic₆₇	—	2	4	no	—
86	C₄⋊Dic₆₇	—	2	4	no	—
87	C₄⋊Dic₆₇	C₂×D₄	1	4	yes	—
88	C₂²⋊Dic₆₇	—	2	2	no	—
89	C₂²⋊Dic₆₇	—	2	2	no	—
90	C₂²⋊Dic₆₇	—	4	2	no	—
91	C₂²⋊Dic₆₇	—	4	2	no	—
92	C₄×Dic₆₇	—	2	2	no	—
93	C₄×Dic₆₇	—	2	2	no	—
94	C₄×Dic₆₇	—	4	2	no	—
95	C₄×Dic₆₇	—	4	2	no	—
96	C₂⁴×C₆₇	#41	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₆₇	—	2	8	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
110	C₂×C₈×C₆₇	—	2	8	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
113	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
114	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
115	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
116	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
123	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
124	C₂³×C₆₇	—	4	2	no	—
125	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
126	C₂³×C₆₇	#121	1	1	yes	characteristic
127	C₂³×C₆₇	#121	1	1	yes	characteristic
128	C₂³×C₆₇	#125	1	2	yes	—
129	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#125	1	2	yes	—
140	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₆₇	—	2	4	no	—
149	C₂×Dic₆₇	—	4	4	no	—
150	C₂×Dic₆₇	—	4	4	no	—
151	C₂×Dic₆₇	—	8	4	no	—
152	C₈×C₆₇	—	2	8	no	—
153	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
154	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
155	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
156	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
157	C₆₇⋊C₈	—	2	4	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	4	2	no	—
162	C₂²×C₆₇	—	4	2	no	—
163	C₂²×C₆₇	#389	1	2	yes	—
164	C₂²×C₆₇	#389	1	2	yes	—
165	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
167	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
168	C₄×C₆₇	#389	1	2	yes	—
169	C₄×C₆₇	#389	1	2	yes	—
170	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
171	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	Dic₆₇	—	4	4	no	—
175	Dic₆₇	—	8	4	no	—
176	#21181	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	—	2	1	no	—
179	C₂×C₆₇	—	2	2	no	—
180	C₂×C₆₇	—	2	2	no	—
181	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
182	C₂×C₆₇	#3067	1	2	yes	—
183	C₂²×#108	D₆₇	1	1	yes	characteristic
184	#2911	—	67	2	no	—
185	C₂×#427	—	67	1	no	—
186	C₂×#405	—	67	1	no	—
187	#1952	—	67	4	no	—
188	#1951	—	67	4	no	—
189	#1742	—	67	2	no	—
190	C₆₇	#21181	1	1	yes	characteristic, complemented, Sylow
191	C₂²×M₁₆	—	67	1	no	—
192	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
193	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₂×#117	—	67	1	no	—
195	C₂×#117	—	67	2	no	—
196	C₂×#108	Dic₆₇	1	8	yes	complemented
197	C₂×#108	C₂×D₆₇	1	2	yes	—
198	C₂×#108	C₂×D₆₇	1	2	yes	—
199	#428	—	67	8	no	—
200	#428	—	134	4	no	—
201	#427	—	134	4	no	—
202	#427	—	134	1	no	—
203	#427	—	134	1	no	—
204	#427	—	134	1	no	—
205	#427	—	134	1	no	—
206	C₂×#107	—	67	2	no	—
207	C₂×#106	—	67	2	no	—
208	#405	—	67	2	no	—
209	#405	—	67	2	no	—
210	#405	—	134	1	no	—
211	#405	—	134	1	no	—
212	C₂×#97	—	134	1	no	—
213	C₂×C₄×C₈	—	134	1	no	—
214	#343	—	67	8	no	—
215	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	—	67	4	no	—
219	C₂×M₁₆	—	67	2	no	—
220	C₂×M₁₆	—	134	1	no	—
221	C₂×M₁₆	—	134	2	no	—
222	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
223	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	134	1	no	—
226	C₂²×C₈	Dic₁₃₄	1	4	yes	—
227	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
228	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
229	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
230	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
231	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
232	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	#117	—	67	2	no	—
235	#117	—	67	2	no	—
236	#117	—	134	2	no	—
237	#117	—	134	2	no	—
238	#117	—	134	2	no	—
239	#117	—	134	2	no	—
240	#117	—	134	1	no	—
241	#117	—	134	1	no	—
242	C₂×C₄⋊C₄	—	67	2	no	—
243	C₂×C₄⋊C₄	—	67	4	no	—
244	C₂×C₂²⋊C₄	—	134	2	no	—
245	C₂×C₂²⋊C₄	—	134	1	no	—
246	C₂×C₄²	—	134	2	no	—
247	C₂×C₄²	—	134	1	no	—
248	#108	C₂×Dic₆₇	1	8	yes	complemented
249	#108	C₂×Dic₆₇	1	8	yes	complemented
250	#107	—	134	8	no	—
251	#107	—	134	2	no	—
252	#107	—	134	2	no	—
253	#106	—	134	8	no	—
254	#106	—	134	2	no	—
255	#106	—	134	2	no	—
256	#97	—	134	2	no	—
257	#97	—	134	2	no	—
258	C₄×C₈	—	134	2	no	—
259	C₄×C₈	—	134	2	no	—
260	C₂⁴	#11761629	1	1	yes	characteristic
261	C₂²×C₄	—	134	2	no	—
262	C₂²×C₄	—	134	4	no	—
263	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
264	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
268	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
269	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
270	C₂²×C₄	#11761627	1	2	yes	—
271	C₂²×C₄	#11761629	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
273	M₁₆	—	2	8	no	—
274	M₁₆	—	2	4	no	—
275	M₁₆	—	67	4	no	—
276	M₁₆	—	134	4	no	—
277	M₁₆	—	134	2	no	—
278	M₁₆	C₂²×Dic₆₇	1	8	yes	—
279	C₂×C₈	—	2	8	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	134	4	no	—
282	C₂×C₈	—	134	2	no	—
283	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
284	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
285	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
286	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
287	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
288	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
289	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
290	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
291	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
292	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
293	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
294	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
295	C₄⋊C₄	—	67	4	no	—
296	C₄⋊C₄	—	134	4	no	—
297	C₄⋊C₄	—	134	4	no	—
298	C₄⋊C₄	—	134	2	no	—
299	C₂²⋊C₄	—	268	2	no	—
300	C₂²⋊C₄	—	268	2	no	—
301	C₂²⋊C₄	—	134	2	no	—
302	C₂²⋊C₄	—	134	2	no	—
303	C₄²	—	268	2	no	—
304	C₄²	—	268	2	no	—
305	C₄²	—	134	2	no	—
306	C₄²	—	134	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	Q₈×Dic₆₇	1	2	yes	—
309	C₂³	#14524018	1	1	yes	characteristic
310	C₂³	#14524018	1	1	yes	characteristic
311	C₂³	#14524034	1	2	yes	—
312	C₂³	C₂×#11761629	1	1	yes	characteristic
313	C₂×C₄	—	268	4	no	—
314	C₂×C₄	—	268	4	no	—
315	C₂×C₄	—	536	4	no	—
316	C₂×C₄	—	134	4	no	—
317	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
318	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
319	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
320	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
321	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
322	C₂×C₄	#14524018	1	1	yes	characteristic
323	C₂×C₄	#14524018	1	1	yes	characteristic
324	C₂×C₄	#14524023	1	2	yes	—
325	C₂×C₄	#14524023	1	2	yes	—
326	C₂×C₄	#14524034	1	2	yes	—
327	C₂×C₄	#14524042	1	2	yes	—
328	C₂×C₄	#14524042	1	2	yes	—
329	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
332	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
334	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
335	C₈	—	2	8	no	—
336	C₈	—	134	4	no	—
337	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
338	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
339	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
340	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#16863295	1	2	yes	—
347	C₂²	#16863295	1	2	yes	—
348	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
349	C₂²	C₂×#14524018	1	1	yes	characteristic
350	C₂²	C₂×#14524034	1	1	yes	characteristic
351	C₄	—	268	4	no	—
352	C₄	—	536	4	no	—
353	C₄	#16863295	1	2	yes	—
354	C₄	#16863295	1	2	yes	—
355	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
356	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
357	C₄	C₂×#14524023	1	1	yes	characteristic
358	C₄	C₂×#14524042	1	1	yes	characteristic
359	C₂	—	2	2	no	—
360	C₂	—	2	2	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	C₂×#16863295	1	1	yes	characteristic
364	C₂	#22569930	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524018
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	#21181
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #22560995	C₂	semidirect
5. #22560990	C₂	semidirect
29. C₂×#108×C₆₇	C₄	semidirect
75. #108×C₆₇	C₂×C₄	semidirect
76. #108×C₆₇	C₂×C₄	semidirect
190. C₆₇	#21181	semidirect
196. C₂×#108	Dic₆₇	semidirect
248. #108	C₂×Dic₆₇	semidirect
249. #108	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568799	2
C₂×#16863434	2
C₂×#16863351	2
#22560995	2
#22560990	2
#22559947	2
C₂²×#108×C₆₇	2
#21181	67