Subgroups — #30853952

Order 17152.

994 conjugacy classes of subgroups.

The subgroup lattice has 320 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	#22568627	C₂	1	2	yes	maximal
3	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
4	#22560117	C₂	1	4	yes	maximal, complemented
5	#108×Dic₆₇	C₂	1	4	yes	maximal, complemented
6	#22559935	C₂	1	2	yes	maximal
7	C₂²×#108×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	C₄	1	8	yes	complemented
12	C₂×#14523979	C₂²	1	2	yes	—
13	C₂×#14523979	C₂²	1	2	yes	—
14	#16863354	—	2	4	no	—
15	#16863351	—	2	4	no	—
16	#16863351	C₄	1	4	yes	—
17	#16863351	C₄	1	4	yes	—
18	M₁₆×Dic₆₇	C₂²	1	8	yes	—
19	C₂×#14523953	C₂²	1	2	yes	—
20	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
21	#16863160	C₂²	1	8	yes	—
22	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#108×C₆₇	C₄	1	8	yes	complemented
25	C₂×#108×C₆₇	C₂²	1	2	yes	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	4	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	C₂×C₄	1	8	yes	—
32	C₂×#11761612	C₂×C₄	1	2	yes	—
33	C₂×#11761612	C₂×C₄	1	2	yes	—
34	C₂×#11761612	C₂³	1	4	yes	—
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
37	#14524002	C₂×C₄	1	2	yes	—
38	#14524002	C₂×C₄	1	2	yes	—
39	#14524002	D₄	1	2	yes	—
40	#14524002	D₄	1	2	yes	—
41	C₂×C₄⋊Dic₆₇	Q₈	1	2	yes	—
42	C₂×C₂²⋊Dic₆₇	Q₈	1	2	yes	—
43	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
44	#14523979	C₂×C₄	1	8	yes	—
45	#14523979	C₂×C₄	1	8	yes	—
46	#14523953	—	2	8	no	—
47	#14523953	C₂×C₄	1	8	yes	—
48	C₈×Dic₆₇	—	2	8	no	—
49	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
54	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	#108×C₆₇	C₂×C₄	1	8	yes	—
63	#108×C₆₇	C₂×C₄	1	8	yes	—
64	C₂²×Dic₆₇	C₂×Q₈	1	4	yes	—
65	#11761612	—	2	4	no	—
66	#11761612	—	2	4	no	—
67	#11761612	—	2	8	no	—
68	#11761612	—	2	4	no	—
69	#11761612	C₂²×C₄	1	8	yes	—
70	C₂×C₆₇⋊C₈	—	2	8	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
75	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
76	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
77	C₂²⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
78	C₂²⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
80	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
81	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	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	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	2	yes	—
90	C₂²×C₄×C₆₇	#41	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₆₇	—	2	8	no	—
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂×C₈×C₆₇	—	2	8	no	—
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	C₄²	1	8	yes	—
98	C₂×C₈×C₆₇	C₄²	1	8	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂³×C₆₇	—	4	2	no	—
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
109	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
110	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#117	1	2	yes	—
116	C₂×C₄×C₆₇	#117	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₆₇	—	4	4	no	—
130	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
131	C₈×C₆₇	—	2	8	no	—
132	C₈×C₆₇	C₂×C₄²	1	4	yes	—
133	C₈×C₆₇	C₂×C₄²	1	4	yes	—
134	C₆₇⋊C₈	—	2	4	no	—
135	C₆₇⋊C₈	—	2	8	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	4	2	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
142	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
148	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
150	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	Dic₆₇	—	2	8	no	—
153	#20460	—	67	1	no	maximal, Sylow
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	2	2	no	—
158	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₆₇	#2916	1	2	yes	—
160	C₂²×#108	—	67	1	no	—
161	C₂²×#108	D₆₇	1	1	yes	characteristic
162	#2873	—	67	2	no	—
163	C₂×#405	—	67	1	no	—
164	#1778	—	67	4	no	—
165	C₄×#108	—	67	4	no	—
166	#1738	—	67	2	no	—
167	C₆₇	#20460	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	67	1	no	—
169	C₂²×M₁₆	—	67	2	no	—
170	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
171	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#117	—	67	1	no	—
173	C₂×#108	—	67	8	no	—
174	C₂×#108	—	67	2	no	—
175	C₂×#108	—	67	2	no	—
176	C₂×#108	Dic₆₇	1	8	yes	complemented
177	C₂×#108	C₂×D₆₇	1	2	yes	—
178	C₂×#108	C₂×D₆₇	1	2	yes	—
179	#405	—	67	4	no	—
180	#405	—	67	4	no	—
181	#405	—	134	4	no	—
182	#405	—	134	4	no	—
183	C₄×M₁₆	—	67	8	no	—
184	C₂×#97	—	67	2	no	—
185	C₂×C₄×C₈	—	67	2	no	—
186	#341	—	67	8	no	—
187	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	67	2	no	—
191	C₂×M₁₆	—	67	2	no	—
192	C₂×M₁₆	—	67	8	no	—
193	C₂×M₁₆	—	67	4	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	—	134	2	no	—
196	C₂×M₁₆	—	134	4	no	—
197	C₂×M₁₆	—	134	2	no	—
198	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
199	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	67	2	no	—
202	C₂²×C₈	—	134	2	no	—
203	C₂²×C₈	C₄×D₆₇	1	2	yes	—
204	C₂²×C₈	C₄×D₆₇	1	2	yes	—
205	C₂²×C₈	C₄×D₆₇	1	4	yes	—
206	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
207	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
209	#117	—	67	2	no	—
210	#117	—	67	2	no	—
211	#117	—	67	2	no	—
212	#117	—	67	2	no	—
213	C₂×C₄⋊C₄	—	67	2	no	—
214	C₂×C₂²⋊C₄	—	67	2	no	—
215	C₂×C₄²	—	67	2	no	—
216	#108	—	67	8	no	—
217	#108	—	67	8	no	—
218	#108	C₂×Dic₆₇	1	8	yes	—
219	#108	C₂×Dic₆₇	1	8	yes	—
220	#97	—	67	8	no	—
221	#97	—	134	8	no	—
222	C₄×C₈	—	67	8	no	—
223	C₄×C₈	—	134	8	no	—
224	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	—	67	4	no	—
226	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
230	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	#11761627	1	2	yes	—
234	C₂²×C₄	#11761629	1	2	yes	—
235	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
236	M₁₆	—	2	8	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	67	8	no	—
239	M₁₆	—	134	4	no	—
240	M₁₆	—	134	4	no	—
241	M₁₆	—	134	8	no	—
242	M₁₆	—	134	4	no	—
243	M₁₆	C₂²×Dic₆₇	1	8	yes	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	67	4	no	—
247	C₂×C₈	—	134	8	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	—	134	2	no	—
250	C₂×C₈	—	134	2	no	—
251	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
252	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
253	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
254	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
257	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
258	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
259	C₄⋊C₄	—	67	4	no	—
260	C₄⋊C₄	—	67	4	no	—
261	C₂²⋊C₄	—	67	4	no	—
262	C₂²⋊C₄	—	67	4	no	—
263	C₄²	—	67	4	no	—
264	C₄²	—	67	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
267	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
268	C₂³	#14524008	1	2	yes	—
269	C₂³	Q₈×Dic₆₇	1	2	yes	—
270	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
271	C₂×C₄	—	268	4	no	—
272	C₂×C₄	—	67	8	no	—
273	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
274	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	#14524001	1	2	yes	—
278	C₂×C₄	#14524001	1	2	yes	—
279	C₂×C₄	#14524005	1	2	yes	—
280	C₂×C₄	#14524005	1	2	yes	—
281	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
282	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
283	C₂×C₄	#14524008	1	2	yes	—
284	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
285	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
290	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
291	C₈	—	2	8	no	—
292	C₈	—	134	4	no	—
293	C₈	—	134	8	no	—
294	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
295	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
302	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
303	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
304	C₂²	C₂×#14524008	1	1	yes	characteristic
305	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
306	C₄	—	134	8	no	—
307	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
308	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
309	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
310	C₄	C₂×#14524001	1	1	yes	characteristic
311	C₄	C₂×#14524005	1	1	yes	characteristic
312	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
318	C₂	#22568817	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	#20460
LCS	G ▷ 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. #22560117	C₂	semidirect
5. #108×Dic₆₇	C₂	semidirect
11. C₂×#14523979	C₄	semidirect
24. C₂×#108×C₆₇	C₄	semidirect
167. C₆₇	#20460	semidirect
176. C₂×#108	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
#22568627	2
C₂×#16863351	2
#22560117	2
#108×Dic₆₇	2
#22559935	2
C₂²×#108×C₆₇	2
#20460	67