Subgroups — #31008480

Order 17152.

818 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863814	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863812	C₂	1	1	yes	maximal, characteristic
3	#22568631	C₂	1	1	yes	maximal, characteristic
4	#22561807	C₂	1	2	yes	maximal, complemented
5	#22561806	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561804	C₂	1	2	yes	maximal, complemented
7	M₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14523982	C₂	1	2	yes	maximal, complemented
9	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
10	#346×D₆₇	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	C₄	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	#16863864	C₄	1	1	yes	characteristic
15	#16863814	C₄	1	8	yes	complemented
16	#16863812	C₄	1	8	yes	complemented
17	C₂×#14523983	C₂²	1	1	yes	characteristic
18	C₂×#14523982	C₂²	1	2	yes	—
19	C₂×#14523982	C₂²	1	2	yes	—
20	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
22	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
23	#16863353	C₄	1	2	yes	—
24	#16863352	C₄	1	1	yes	characteristic
25	#16863350	C₂²	1	2	yes	—
26	#16863348	C₄	1	2	yes	—
27	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₆₇	C₂²	1	2	yes	—
30	#16863172	C₂²	1	2	yes	—
31	#16863171	C₂²	1	1	yes	characteristic
32	#16863171	C₂²	1	2	yes	—
33	#16863171	C₂²	1	1	yes	characteristic
34	#16863171	C₂²	1	1	yes	characteristic
35	#16863171	C₂²	1	1	yes	characteristic
36	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
37	C₁₆×Dic₆₇	C₂²	1	2	yes	—
38	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	#405×C₆₇	C₂²	1	1	yes	characteristic
41	#346×C₆₇	C₂²	1	2	yes	—
42	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
43	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
44	#14524051	C₈	1	8	yes	—
45	#14524049	C₂×C₄	1	4	yes	—
46	#14524049	C₂×C₄	1	4	yes	—
47	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
48	C₂×#11761611	C₂×C₄	1	2	yes	—
49	C₂×#11761611	C₂×C₄	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₆₇	C₂³	1	2	yes	—
53	#14524008	C₈	1	2	yes	—
54	#14524005	C₈	1	2	yes	—
55	(C₂²⋊D₆₇)⋊C₄	C₈	1	4	yes	—
56	#14524000	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₆₇	C₂×C₄	1	2	yes	—
59	#14523983	C₂×C₄	1	4	yes	complemented
60	#14523982	C₂×C₄	1	8	yes	complemented
61	#14523982	C₂×C₄	1	8	yes	complemented
62	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
63	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
65	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
66	C₁₆×D₆₇	C₂×C₄	1	8	yes	complemented
67	C₁₆×D₆₇	C₂×C₄	1	8	yes	complemented
68	#14523954	C₂×C₄	1	2	yes	—
69	#14523953	C₂×C₄	1	2	yes	—
70	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
71	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
72	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
75	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
79	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
80	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
81	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
84	#97×C₆₇	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	#11761628	C₄²	1	4	yes	—
87	#11761628	C₂×C₈	1	4	yes	—
88	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
89	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₆₇	C₂×C₈	1	2	yes	—
91	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₆₇	C₂×C₈	1	4	yes	—
93	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
94	#11761612	C₄²	1	4	yes	—
95	#11761611	C₄²	1	8	yes	—
96	#11761611	C₂×C₈	1	8	yes	—
97	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
98	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₆₇	C₂×C₈	1	8	yes	—
102	C₈×D₆₇	C₂²×C₄	1	4	yes	—
103	C₈×D₆₇	C₂²×C₄	1	4	yes	—
104	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
105	(C₂×D₆₇)⋊C₄	C₂×C₈	1	4	yes	—
106	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
107	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
108	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
111	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	—
112	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
115	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
121	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
122	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
123	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
124	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
125	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
126	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₆₇	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₆₇	C₄×C₈	1	8	yes	—
134	C₂×Dic₆₇	—	4	2	no	—
135	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
136	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
137	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
138	D₂₆₈	C₄×C₈	1	4	yes	—
139	C₄×D₆₇	—	4	4	no	—
140	C₄×D₆₇	C₂×C₄²	1	4	yes	—
141	C₄×D₆₇	C₂²×C₈	1	4	yes	—
142	Dic₁₃₄	C₄×C₈	1	4	yes	—
143	C₈×C₆₇	C₂×C₄²	1	2	yes	—
144	C₈×C₆₇	C₂²×C₈	1	4	yes	—
145	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
147	C₆₇⋊C₈	C₂×C₄²	1	4	yes	—
148	C₆₇⋊C₈	C₂²×C₈	1	2	yes	—
149	C₆₇⋊C₈	C₂²×C₈	1	2	yes	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
152	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₆₇	—	2	4	no	—
154	C₂×D₆₇	C₂×C₄×C₈	1	4	yes	—
155	C₄×C₆₇	—	2	4	no	—
156	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
157	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
159	Dic₆₇	—	2	2	no	—
160	Dic₆₇	—	2	2	no	—
161	Dic₆₇	C₂×C₄×C₈	1	4	yes	—
162	#33326	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
165	C₂×C₆₇	#3407	1	2	yes	—
166	D₆₇	—	2	4	no	—
167	C₂×#504	—	67	1	no	—
168	C₂×#504	—	67	1	no	—
169	C₄×#131	—	67	1	no	—
170	#2108	—	67	1	no	—
171	#2108	—	67	2	no	—
172	#2108	—	67	2	no	—
173	#2108	D₆₇	1	1	yes	characteristic, complemented
174	C₄×M₃₂	—	67	1	no	—
175	C₄×M₃₂	—	67	2	no	—
176	C₂×#346	—	67	1	no	—
177	C₂×#346	—	67	2	no	—
178	C₆₇	#33326	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	—	67	1	no	—
180	C₂×#131	—	67	1	no	—
181	C₄×#41	—	67	1	no	—
182	#504	—	67	8	no	—
183	#504	—	67	8	no	—
184	C₂×M₃₂	—	67	1	no	—
185	C₂×M₃₂	—	67	2	no	—
186	C₂×M₃₂	—	67	2	no	—
187	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
188	C₂²×C₁₆	—	67	1	no	—
189	C₂²×C₁₆	—	67	2	no	—
190	C₂²×C₁₆	—	67	2	no	—
191	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
192	#405	—	67	1	no	—
193	#405	—	67	2	no	—
194	#405	—	67	2	no	—
195	#405	C₂×D₆₇	1	1	yes	characteristic
196	C₄×M₁₆	—	67	1	no	—
197	C₄×M₁₆	—	67	2	no	—
198	C₂×C₄×C₈	—	67	1	no	—
199	C₂×C₄×C₈	—	67	2	no	—
200	#346	—	67	1	no	—
201	#346	—	67	2	no	—
202	#346	—	67	1	no	—
203	#346	—	67	1	no	—
204	#346	—	67	1	no	—
205	#346	—	67	2	no	—
206	#346	C₂×D₆₇	1	2	yes	—
207	C₄×C₁₆	—	67	2	no	—
208	C₄×C₁₆	—	67	2	no	—
209	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
210	C₂×#41	—	67	1	no	—
211	#131	—	67	8	no	—
212	#131	—	67	4	no	—
213	#131	—	67	4	no	—
214	C₂×M₁₆	—	67	1	no	—
215	C₂×M₁₆	—	67	2	no	—
216	C₂×M₁₆	—	67	2	no	—
217	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
218	C₂²×C₈	—	67	1	no	—
219	C₂²×C₈	—	67	2	no	—
220	C₂²×C₈	—	67	2	no	—
221	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
222	C₄×Q₈	—	67	2	no	—
223	C₄×D₄	—	67	2	no	—
224	C₄×D₄	—	67	4	no	—
225	#117	—	67	2	no	—
226	#117	C₄×D₆₇	1	1	yes	characteristic
227	C₂×C₄²	—	67	1	no	—
228	C₂×C₄²	—	67	2	no	—
229	M₃₂	—	67	4	no	—
230	M₃₂	—	67	8	no	—
231	M₃₂	—	67	8	no	—
232	M₃₂	C₄×D₆₇	1	4	yes	complemented
233	C₂×C₁₆	—	67	4	no	—
234	C₂×C₁₆	—	67	2	no	—
235	C₂×C₁₆	—	67	1	no	—
236	C₂×C₁₆	—	67	1	no	—
237	C₂×C₁₆	—	67	8	no	—
238	C₂×C₁₆	—	67	8	no	—
239	C₂×C₁₆	C₄×D₆₇	1	4	yes	complemented
240	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
241	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
242	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
243	#97	—	67	2	no	—
244	#97	—	67	2	no	—
245	#97	C₄×D₆₇	1	2	yes	—
246	C₄×C₈	—	67	1	no	—
247	C₄×C₈	—	67	2	no	—
248	C₄×C₈	—	67	1	no	—
249	C₄×C₈	—	67	1	no	—
250	C₄×C₈	—	67	1	no	—
251	C₄×C₈	—	67	2	no	—
252	C₄×C₈	C₂²×D₆₇	1	2	yes	—
253	#41	—	67	4	no	—
254	#41	—	67	4	no	—
255	C₂×Q₈	—	67	1	no	—
256	C₂×D₄	—	67	2	no	—
257	C₂×D₄	—	67	1	no	—
258	C₂²×C₄	—	67	2	no	—
259	C₂²×C₄	—	67	4	no	—
260	C₂²×C₄	—	67	2	no	—
261	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
262	M₁₆	—	67	4	no	—
263	M₁₆	—	67	8	no	—
264	M₁₆	—	67	8	no	—
265	M₁₆	C₈×D₆₇	1	4	yes	—
266	C₂×C₈	—	67	4	no	—
267	C₂×C₈	—	67	1	no	—
268	C₂×C₈	—	67	2	no	—
269	C₂×C₈	—	67	1	no	—
270	C₂×C₈	—	67	8	no	—
271	C₂×C₈	—	67	4	no	—
272	C₂×C₈	—	67	4	no	—
273	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
275	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
276	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
277	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
278	C₄⋊C₄	—	67	4	no	—
279	C₄⋊C₄	C₈×D₆₇	1	2	yes	—
280	C₂²⋊C₄	—	67	4	no	—
281	C₂²⋊C₄	C₈×D₆₇	1	2	yes	—
282	C₄²	—	67	4	no	—
283	C₄²	—	67	2	no	—
284	C₄²	—	67	1	no	—
285	C₄²	—	67	1	no	—
286	C₄²	C₂×C₄×D₆₇	1	2	yes	—
287	C₁₆	—	67	2	no	—
288	C₁₆	—	67	2	no	—
289	C₁₆	—	67	4	no	—
290	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
291	C₁₆	C₂×C₄×D₆₇	1	4	yes	—
292	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
293	C₂³	—	67	2	no	—
294	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
295	Q₈	—	67	4	no	—
296	D₄	—	67	8	no	—
297	D₄	—	67	4	no	—
298	C₂×C₄	—	268	4	no	—
299	C₂×C₄	—	268	2	no	—
300	C₂×C₄	—	67	2	no	—
301	C₂×C₄	—	67	4	no	—
302	C₂×C₄	—	67	4	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	C₄²×D₆₇	1	2	yes	—
306	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
307	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
310	C₈	—	67	4	no	—
311	C₈	—	67	2	no	—
312	C₈	—	67	2	no	—
313	C₈	C₄²×D₆₇	1	2	yes	—
314	C₈	C₂×C₈×D₆₇	1	4	yes	—
315	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
316	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	67	4	no	—
319	C₂²	—	134	4	no	—
320	C₂²	C₄×C₈×D₆₇	1	2	yes	—
321	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	67	4	no	—
324	C₄	—	134	2	no	—
325	C₄	—	134	2	no	—
326	C₄	C₄×C₈×D₆₇	1	2	yes	—
327	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
328	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	134	4	no	—
331	C₂	C₂×C₄×C₈×D₆₇	1	1	yes	characteristic
332	C₂	#22573310	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator	C₂×C₆₇	C₂²×C₄×C₈
Fitting	C₆₇×#2108	C₂
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#33326
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
4. #22561807	C₂	semidirect
5. #22561806	C₂	semidirect
6. #22561804	C₂	semidirect
7. M₃₂×Dic₆₇	C₂	semidirect
8. C₄×#14523982	C₂	semidirect
9. C₂×#16863171	C₂	semidirect
10. #346×D₆₇	C₂	semidirect
11. C₆₇×#2108	C₂	semidirect
15. #16863814	C₄	semidirect
16. #16863812	C₄	semidirect
59. #14523983	C₂×C₄	semidirect
60. #14523982	C₂×C₄	semidirect
61. #14523982	C₂×C₄	semidirect
62. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
66. C₁₆×D₆₇	C₂×C₄	semidirect
67. C₁₆×D₆₇	C₂×C₄	semidirect
79. M₃₂×C₆₇	C₂×C₄	semidirect
80. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
173. #2108	D₆₇	semidirect
178. C₆₇	#33326	semidirect
232. M₃₂	C₄×D₆₇	semidirect
239. C₂×C₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863814	2
C₂×#16863812	2
#22568631	2
#22561807	2
#22561806	2
#22561804	2
M₃₂×Dic₆₇	2
C₄×#14523982	2
C₂×#16863171	2
#346×D₆₇	2
C₆₇×#2108	2
#33326	67