Subgroups — #30761638

Order 17152.

762 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₇⋊#338	C₂	1	4	yes	maximal
4	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₂²	1	2	yes	—
6	C₂²×#11761612	C₂²	1	2	yes	—
7	#16863420	—	2	1	no	—
8	#16863420	—	2	1	no	—
9	#16863420	—	2	2	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	—
11	C₄×#11761612	—	2	1	no	—
12	C₄×#11761612	—	2	1	no	—
13	C₄×#11761612	—	2	1	no	—
14	C₄×#11761612	—	2	1	no	—
15	C₂×#14523954	C₄	1	2	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
17	C₆₇⋊#338	—	2	8	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#423×C₆₇	—	2	1	no	—
21	#423×C₆₇	—	2	1	no	—
22	#423×C₆₇	—	2	2	no	—
23	C₂×C₄⋊C₈×C₆₇	C₄	1	4	yes	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	4	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	—	2	4	no	—
32	C₂×#11761612	D₄	1	4	yes	—
33	C₂×#11761612	Q₈	1	4	yes	—
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
38	#14523954	—	2	2	no	—
39	#14523954	—	2	2	no	—
40	C₄×C₆₇⋊C₈	—	2	2	no	—
41	C₄×C₆₇⋊C₈	—	2	2	no	—
42	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	4	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	2	yes	—
58	C₄⋊C₈×C₆₇	—	2	4	no	—
59	C₄⋊C₈×C₆₇	—	2	4	no	—
60	#11761612	—	2	4	no	—
61	#11761612	—	2	4	no	—
62	#11761612	—	2	4	no	—
63	#11761612	—	2	4	no	—
64	#11761612	—	4	4	no	—
65	#11761612	—	4	2	no	—
66	#11761612	—	4	2	no	—
67	C₂×C₆₇⋊C₈	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	8	no	—
71	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
72	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	4	4	no	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₆₇	#41	1	2	yes	—
89	C₂²×C₄×C₆₇	#41	1	2	yes	—
90	M₁₆×C₆₇	—	4	4	no	—
91	M₁₆×C₆₇	—	4	2	no	—
92	M₁₆×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	8	no	—
95	C₄²×C₆₇	C₄²	1	2	yes	—
96	C₄²×C₆₇	C₄²	1	2	yes	—
97	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
103	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
104	C₄²×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	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₆₇	#125	1	2	yes	—
110	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	#95	1	2	yes	—
118	C₂×C₄×C₆₇	#95	1	2	yes	—
119	C₂×C₄×C₆₇	#95	1	2	yes	—
120	C₂×C₄×C₆₇	#95	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₆₇	#117	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₆₇	#121	1	2	yes	—
132	C₂×C₄×C₆₇	#121	1	2	yes	—
133	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₆₇	—	4	4	no	—
140	C₆₇⋊C₈	—	2	8	no	—
141	C₆₇⋊C₈	—	4	4	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₆₇	#382	1	1	yes	characteristic
149	C₂²×C₆₇	#384	1	1	yes	characteristic
150	C₂²×C₆₇	#389	1	2	yes	—
151	C₂²×C₆₇	#389	1	2	yes	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	#338	1	4	yes	complemented
156	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
157	C₄×C₆₇	#380	1	2	yes	—
158	C₄×C₆₇	#390	1	1	yes	characteristic
159	C₄×C₆₇	#390	1	1	yes	characteristic
160	C₄×C₆₇	#391	1	1	yes	characteristic
161	C₄×C₆₇	#391	1	1	yes	characteristic
162	C₄⋊#338	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	—	2	2	no	—
167	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#338	1	2	yes	—
169	C₂×#423	—	67	1	no	—
170	C₂×#423	D₆₇	1	1	yes	characteristic
171	C₂×C₄×M₁₆	—	67	1	no	—
172	C₂×#338	—	67	4	no	—
173	C₆₇	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	67	2	no	—
175	C₂²×M₁₆	—	67	2	no	—
176	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
177	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
178	#423	—	2	1	no	—
179	#423	—	2	1	no	—
180	#423	—	2	2	no	—
181	#423	—	134	1	no	—
182	#423	—	134	1	no	—
183	#423	—	134	2	no	—
184	C₂×C₄⋊C₈	—	67	4	no	—
185	C₂×C₄⋊C₈	Dic₆₇	1	4	yes	—
186	C₄×M₁₆	—	134	1	no	—
187	C₄×M₁₆	—	134	1	no	—
188	C₄×M₁₆	—	134	1	no	—
189	C₄×M₁₆	—	134	1	no	—
190	C₂×#97	—	67	2	no	—
191	C₂×C₄×C₈	—	67	2	no	—
192	#338	—	134	8	no	—
193	C₂³×C₄	Dic₁₃₄	1	2	yes	—
194	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	67	4	no	—
200	C₂×M₁₆	—	67	4	no	—
201	C₂×M₁₆	—	134	4	no	—
202	C₂×M₁₆	—	134	2	no	—
203	C₂×M₁₆	—	134	2	no	—
204	C₂×M₁₆	—	134	4	no	—
205	C₂×M₁₆	—	134	2	no	—
206	C₂×M₁₆	—	134	2	no	—
207	C₂×M₁₆	—	134	2	no	—
208	C₂×M₁₆	—	134	2	no	—
209	C₂²×C₈	—	67	4	no	—
210	C₂²×C₈	—	67	4	no	—
211	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
212	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₆₇	1	2	yes	—
215	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
216	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
218	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	134	4	no	—
224	C₄⋊C₈	—	134	4	no	—
225	#97	—	134	2	no	—
226	#97	—	134	2	no	—
227	C₄×C₈	—	134	2	no	—
228	C₄×C₈	—	134	2	no	—
229	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
232	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
233	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
234	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₆₇	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
245	C₂²×C₄	#11761627	1	2	yes	—
246	C₂²×C₄	#11761629	1	2	yes	—
247	M₁₆	—	4	4	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	268	4	no	—
251	M₁₆	—	268	2	no	—
252	M₁₆	—	268	2	no	—
253	M₁₆	—	134	4	no	—
254	M₁₆	—	134	4	no	—
255	M₁₆	—	134	4	no	—
256	M₁₆	—	134	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	67	8	no	—
260	C₂×C₈	—	134	4	no	—
261	C₂×C₈	—	134	8	no	—
262	C₂×C₈	—	134	4	no	—
263	C₂×C₈	—	134	4	no	—
264	C₄²	C₄×Dic₆₇	1	2	yes	—
265	C₄²	C₄×Dic₆₇	1	2	yes	—
266	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
267	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
268	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
269	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
270	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
271	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
272	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
273	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
274	C₂³	—	4	2	no	—
275	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
276	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
277	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
278	C₂³	#14524034	1	2	yes	—
279	C₂³	#14524042	1	2	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₆₇⋊#95	1	2	yes	—
287	C₂×C₄	C₆₇⋊#95	1	2	yes	—
288	C₂×C₄	C₆₇⋊#95	1	2	yes	—
289	C₂×C₄	C₆₇⋊#95	1	2	yes	—
290	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
294	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
295	C₂×C₄	#14524001	1	2	yes	—
296	C₂×C₄	#14524005	1	2	yes	—
297	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
298	C₂×C₄	#14524008	1	2	yes	—
299	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
300	C₂×C₄	#14524014	1	2	yes	—
301	C₂×C₄	#14524018	1	2	yes	—
302	C₂×C₄	#14524019	1	2	yes	—
303	C₂×C₄	#14524023	1	2	yes	—
304	C₂×C₄	#14524030	1	1	yes	characteristic
305	C₂×C₄	#14524032	1	1	yes	characteristic
306	C₂×C₄	#14524039	1	1	yes	characteristic
307	C₂×C₄	#14524041	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	268	4	no	—
310	C₈	—	134	8	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
317	C₂²	#16863261	1	1	yes	characteristic
318	C₂²	#16863265	1	1	yes	characteristic
319	C₂²	#16863292	1	2	yes	—
320	C₂²	#16863295	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	C₆₇⋊#338	1	4	yes	complemented
325	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
326	C₄	#16863253	1	2	yes	—
327	C₄	#16863297	1	1	yes	characteristic
328	C₄	#16863299	1	1	yes	characteristic
329	C₄	#16863301	1	1	yes	characteristic
330	C₄	#16863303	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	#22558438	1	1	yes	characteristic
336	C₂	C₂×C₆₇⋊#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	C₄⋊#338
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
155. C₄×C₆₇	#338	semidirect
173. C₆₇	C₄⋊#338	semidirect
324. C₄	C₆₇⋊#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
C₂×C₄×#11761612	2
C₂×C₆₇⋊#338	2
C₂×#423×C₆₇	2
C₄⋊#338	67