Subgroups — #30763939

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568799	C₂	1	1	yes	maximal, characteristic
2	#22559947	C₂	1	2	yes	maximal
3	C₂×#16863156	C₂	1	2	yes	maximal
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	#16863438	—	2	4	no	—
9	#16863434	—	2	2	no	—
10	#16863164	—	2	4	no	—
11	#16863156	—	2	2	no	—
12	#16863156	—	2	2	no	—
13	C₆₇⋊#338	—	2	2	no	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#423×C₆₇	—	2	2	no	—
18	#423×C₆₇	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	Q₈	1	4	yes	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	D₄	1	4	yes	—
32	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	—	2	2	no	—
34	#14523977	—	2	4	no	—
35	#14523975	—	2	4	no	—
36	C₂³×C₄×C₆₇	—	2	1	no	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	1	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
55	C₄⋊C₈×C₆₇	—	2	8	no	—
56	C₂²×Dic₆₇	—	2	4	no	—
57	#11761612	—	2	4	no	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	2	no	—
60	#11761612	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	—	2	4	no	—
66	C₂²⋊Dic₆₇	—	4	2	no	—
67	C₂²⋊Dic₆₇	—	4	2	no	—
68	C₄×Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	C₂⁴×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	M₁₆×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	4	1	no	—
94	C₂×C₈×C₆₇	—	4	1	no	—
95	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	#117	1	1	yes	characteristic
104	C₂³×C₆₇	#121	1	1	yes	characteristic
105	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₆₇	#123	1	2	yes	—
107	C₂³×C₆₇	#125	1	2	yes	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#121	1	2	yes	—
120	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#121	1	2	yes	—
122	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#125	1	2	yes	—
129	C₂×C₄×C₆₇	#126	1	2	yes	—
130	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
132	C₂×Dic₆₇	—	4	4	no	—
133	C₂×Dic₆₇	—	8	4	no	—
134	C₈×C₆₇	—	2	4	no	—
135	C₈×C₆₇	—	4	2	no	—
136	C₆₇⋊C₈	—	4	4	no	—
137	C₂²×C₆₇	—	2	4	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	4	1	no	—
140	C₂²×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	#380	1	1	yes	characteristic
143	C₂²×C₆₇	#387	1	2	yes	—
144	C₂²×C₆₇	#394	1	1	yes	characteristic
145	C₂²×C₆₇	#398	1	1	yes	characteristic
146	C₂²×C₆₇	#398	1	2	yes	—
147	C₄×C₆₇	—	2	4	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	#386	1	1	yes	characteristic
150	C₄×C₆₇	#389	1	2	yes	—
151	C₄×C₆₇	#391	1	1	yes	characteristic
152	C₄×C₆₇	#395	1	1	yes	characteristic
153	C₄×C₆₇	#396	1	2	yes	—
154	C₄×C₆₇	#400	1	1	yes	characteristic
155	Dic₆₇	—	8	4	no	—
156	#11243	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	—	2	2	no	—
161	C₂×C₆₇	#1441	1	1	yes	characteristic
162	C₂×C₆₇	#2050	1	2	yes	—
163	#2911	—	67	1	no	—
164	C₂×#423	D₆₇	1	1	yes	characteristic
165	#1742	—	67	2	no	—
166	C₂×#339	—	67	2	no	—
167	C₂×#338	—	67	1	no	—
168	C₆₇	#11243	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	67	2	no	—
170	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
171	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#117	—	67	2	no	—
173	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
174	#428	—	134	4	no	—
175	#427	—	134	2	no	—
176	#423	—	2	2	no	—
177	#423	Dic₆₇	1	4	yes	complemented
178	C₂×C₄⋊C₈	—	2	2	no	—
179	#343	—	134	4	no	—
180	#339	—	134	2	no	—
181	#339	—	134	2	no	—
182	#338	—	134	2	no	—
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	67	4	no	—
191	C₂×M₁₆	—	134	2	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	—	134	2	no	—
196	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
197	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
198	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	134	2	no	—
201	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
202	#117	—	134	2	no	—
203	#117	—	134	2	no	—
204	#117	—	134	2	no	—
205	#117	—	134	2	no	—
206	C₂×C₄⋊C₄	—	67	4	no	—
207	C₂×C₂²⋊C₄	—	134	2	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	134	2	no	—
212	C₂×C₄²	Dic₁₃₄	1	2	yes	—
213	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
214	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
216	#107	—	134	4	no	—
217	#106	—	134	4	no	—
218	C₄⋊C₈	—	2	8	no	—
219	C₂⁴	#11761628	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	134	4	no	—
226	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₆₇	1	2	yes	—
228	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
231	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
232	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
233	C₂²×C₄	#11761627	1	2	yes	—
234	C₂²×C₄	#11761628	1	1	yes	characteristic
235	C₂²×C₄	#11761629	1	2	yes	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	268	2	no	—
241	M₁₆	—	268	2	no	—
242	M₁₆	—	134	4	no	—
243	M₁₆	—	134	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	268	2	no	—
250	C₂×C₈	—	268	2	no	—
251	C₂×C₈	—	134	4	no	—
252	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
253	C₄⋊C₄	—	134	4	no	—
254	C₄⋊C₄	—	134	4	no	—
255	C₂²⋊C₄	—	268	2	no	—
256	C₂²⋊C₄	—	268	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	268	2	no	—
262	C₄²	—	268	2	no	—
263	C₄²	C₄⋊Dic₆₇	1	2	yes	—
264	C₄²	C₄⋊Dic₆₇	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	#14524002	1	1	yes	characteristic
267	C₂³	#14524017	1	1	yes	characteristic
268	C₂³	#14524022	1	1	yes	characteristic
269	C₂³	#14524028	1	2	yes	—
270	C₂³	#14524033	1	2	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	268	4	no	—
278	C₂×C₄	—	536	4	no	—
279	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
281	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
282	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
283	C₂×C₄	#14524011	1	1	yes	characteristic
284	C₂×C₄	#14524015	1	2	yes	—
285	C₂×C₄	#14524017	1	1	yes	characteristic
286	C₂×C₄	#14524018	1	2	yes	—
287	C₂×C₄	#14524019	1	2	yes	—
288	C₂×C₄	#14524021	1	2	yes	—
289	C₂×C₄	#14524022	1	1	yes	characteristic
290	C₂×C₄	#14524023	1	2	yes	—
291	C₂×C₄	#14524024	1	1	yes	characteristic
292	C₂×C₄	#14524032	1	1	yes	characteristic
293	C₂×C₄	#14524034	1	2	yes	—
294	C₂×C₄	#14524038	1	2	yes	—
295	C₂×C₄	#14524041	1	1	yes	characteristic
296	C₂×C₄	#14524042	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	268	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#16863256	1	1	yes	characteristic
306	C₂²	#16863282	1	2	yes	—
307	C₂²	#16863313	1	1	yes	characteristic
308	C₂²	#16863326	1	1	yes	characteristic
309	C₂²	#16863329	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	536	4	no	—
313	C₄	#16863276	1	1	yes	characteristic
314	C₄	#16863295	1	2	yes	—
315	C₄	#16863303	1	1	yes	characteristic
316	C₄	#16863316	1	1	yes	characteristic
317	C₄	#16863318	1	2	yes	—
318	C₄	#16863335	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#22558462	1	1	yes	characteristic
324	C₂	#22561496	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524032
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂²×C₆₇	#394
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	#11243
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
18. #423×C₆₇	C₄	semidirect
168. C₆₇	#11243	semidirect
177. #423	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
#22559947	2
C₂×#16863156	2
C₂×C₆₇⋊#338	2
C₂×#423×C₆₇	2
#11243	67