Subgroups — #30866771

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562687	C₂	1	1	yes	maximal, characteristic
2	#22561150	C₂	1	1	yes	maximal, characteristic
3	#22560932	C₂	1	1	yes	maximal, characteristic
4	#22560930	C₂	1	1	yes	maximal, characteristic
5	#22560774	C₂	1	2	yes	maximal
6	#22560772	C₂	1	2	yes	maximal
7	#22560353	C₂	1	2	yes	maximal
8	#22560327	C₂	1	2	yes	maximal
9	#22560064	C₂	1	1	yes	maximal, characteristic
10	#22559957	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1985	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
14	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#14523954	C₂²	1	1	yes	characteristic
16	C₂×#14523953	C₂²	1	2	yes	—
17	#16863303	C₄	1	2	yes	—
18	#16863295	C₄	1	2	yes	—
19	#16863282	C₄	1	4	yes	—
20	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863242	C₂²	1	2	yes	—
23	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
24	#16863151	C₂²	1	2	yes	—
25	#16863151	C₂²	1	2	yes	—
26	#16863150	C₂²	1	1	yes	characteristic
27	#16863150	C₂²	1	2	yes	—
28	#16863150	C₂²	1	1	yes	characteristic
29	#16863150	C₂²	1	2	yes	—
30	#16863150	C₂²	1	2	yes	—
31	#16863150	C₂²	1	1	yes	characteristic
32	#16863150	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
48	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
49	Dic₆₇⋊C₈	—	4	2	no	—
50	Dic₆₇⋊C₈	—	4	2	no	—
51	#14523954	—	2	2	no	—
52	#14523953	—	4	2	no	—
53	C₆₇⋊#95	C₂×C₄	1	4	yes	—
54	C₆₇⋊#95	C₂×C₄	1	4	yes	—
55	C₆₇⋊#95	C₂×C₄	1	2	yes	—
56	C₆₇⋊#95	C₂×C₄	1	4	yes	—
57	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₆₇	—	2	2	no	—
68	#97×C₆₇	—	2	2	no	—
69	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
70	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
71	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	1	no	—
77	C₂×C₆₇⋊C₈	—	2	1	no	—
78	C₂×C₆₇⋊C₈	—	2	1	no	—
79	C₂×C₆₇⋊C₈	—	2	1	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₄⋊Dic₆₇	—	4	4	no	—
82	C₄⋊Dic₆₇	—	4	2	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₄×Dic₆₇	—	2	2	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₄×Dic₆₇	—	2	1	no	—
87	C₄×Dic₆₇	—	2	1	no	—
88	C₄×Dic₆₇	—	2	2	no	—
89	C₄×Dic₆₇	—	4	2	no	—
90	C₄×Dic₆₇	—	4	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	1	no	—
100	C₂×C₈×C₆₇	—	2	1	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	1	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	#41	1	2	yes	—
112	C₄²×C₆₇	#41	1	2	yes	—
113	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	Q₈×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	#117	1	2	yes	—
122	C₂×C₄×C₆₇	#117	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	2	2	no	—
131	C₂×Dic₆₇	—	2	4	no	—
132	C₂×Dic₆₇	—	2	2	no	—
133	C₂×Dic₆₇	—	2	2	no	—
134	C₂×Dic₆₇	—	2	2	no	—
135	C₂×Dic₆₇	—	2	2	no	—
136	C₂×Dic₆₇	—	2	4	no	—
137	C₂×Dic₆₇	—	2	4	no	—
138	C₈×C₆₇	—	4	1	no	—
139	C₈×C₆₇	—	4	2	no	—
140	C₈×C₆₇	—	4	1	no	—
141	C₆₇⋊C₈	—	4	1	no	—
142	C₆₇⋊C₈	—	4	1	no	—
143	C₆₇⋊C₈	—	4	2	no	—
144	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
145	C₂²×C₆₇	#519	1	1	yes	characteristic
146	C₂²×C₆₇	#520	1	1	yes	characteristic
147	C₂²×C₆₇	#568	1	2	yes	—
148	C₂²×C₆₇	#568	1	2	yes	—
149	C₄×C₆₇	—	2	2	no	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	#541	1	1	yes	characteristic
154	C₄×C₆₇	#541	1	1	yes	characteristic
155	C₄×C₆₇	#542	1	1	yes	characteristic
156	C₄×C₆₇	#542	1	1	yes	characteristic
157	Dic₆₇	—	4	2	no	—
158	Dic₆₇	—	4	2	no	—
159	Dic₆₇	—	4	4	no	—
160	#22025	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	#2320	1	1	yes	characteristic
162	C₂×C₆₇	#2921	1	1	yes	characteristic
163	C₂×C₆₇	#2923	1	1	yes	characteristic
164	C₂×C₆₇	#2987	1	2	yes	—
165	C₂×C₆₇	#2995	1	2	yes	—
166	#2295	—	67	1	no	—
167	#1985	—	67	1	no	—
168	#1985	D₆₇	1	1	yes	characteristic
169	#1940	—	67	1	no	—
170	#1940	—	67	1	no	—
171	#1917	—	67	2	no	—
172	#1917	—	67	2	no	—
173	#1840	—	67	2	no	—
174	#1832	—	67	2	no	—
175	#1767	—	67	1	no	—
176	#1745	—	67	1	no	—
177	C₆₇	#22025	1	1	yes	characteristic, complemented, Sylow
178	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
179	C₂×C₄⋊C₈	—	67	2	no	—
180	C₂×C₄⋊C₈	—	67	2	no	—
181	C₂×C₄⋊C₈	—	67	1	no	—
182	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
183	C₂×#97	—	67	2	no	—
184	C₂×#97	—	67	1	no	—
185	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
186	#391	—	67	2	no	—
187	#389	—	67	2	no	—
188	#387	—	67	4	no	—
189	C₄×C₄⋊C₄	—	67	2	no	—
190	C₄×C₄⋊C₄	—	67	1	no	—
191	#376	—	67	2	no	—
192	C₄³	—	67	1	no	—
193	#336	—	67	1	no	—
194	#336	—	67	2	no	—
195	#336	—	67	1	no	—
196	#336	—	67	2	no	—
197	#336	—	67	2	no	—
198	#336	—	67	1	no	—
199	#336	—	67	1	no	—
200	#336	—	67	2	no	—
201	#336	—	67	2	no	—
202	#336	C₂×D₆₇	1	2	yes	—
203	#336	C₂×D₆₇	1	2	yes	—
204	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₈	—	67	1	no	—
206	C₂²×C₈	—	67	1	no	—
207	C₂²×C₈	—	67	2	no	—
208	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
210	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₂×C₄⋊C₄	—	67	4	no	—
214	C₂×C₄⋊C₄	—	67	2	no	—
215	C₂×C₄⋊C₄	C₄×D₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
217	C₂×C₄²	—	67	2	no	—
218	C₂×C₄²	—	67	2	no	—
219	C₂×C₄²	—	67	1	no	—
220	C₂×C₄²	—	67	1	no	—
221	C₂×C₄²	—	67	2	no	—
222	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	268	2	no	—
226	C₄⋊C₈	—	268	2	no	—
227	C₄⋊C₈	—	134	2	no	—
228	#97	—	2	2	no	—
229	#97	—	268	2	no	—
230	#97	—	134	2	no	—
231	#95	—	67	4	no	—
232	#95	—	67	4	no	—
233	#95	—	67	2	no	—
234	#95	—	67	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂²×C₄	—	67	2	no	—
238	C₂²×C₄	—	67	4	no	—
239	C₂²×C₄	—	67	2	no	—
240	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	134	1	no	—
255	C₂×C₈	—	134	1	no	—
256	C₂×C₈	—	134	2	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	1	no	—
259	C₂×C₈	—	134	1	no	—
260	C₂×C₈	—	134	1	no	—
261	C₂×C₈	—	134	1	no	—
262	C₂×C₈	—	134	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	268	2	no	—
266	C₄⋊C₄	—	268	4	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	268	2	no	—
270	C₄²	—	268	2	no	—
271	C₄²	—	134	1	no	—
272	C₄²	—	134	2	no	—
273	C₄²	—	134	1	no	—
274	C₄²	—	134	1	no	—
275	C₄²	—	134	1	no	—
276	C₄²	—	134	2	no	—
277	C₄²	#11761627	1	2	yes	—
278	C₄²	#11761627	1	2	yes	—
279	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
280	Q₈	—	4	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	134	2	no	—
288	C₂×C₄	—	134	2	no	—
289	C₂×C₄	—	134	4	no	—
290	C₂×C₄	—	134	2	no	—
291	C₂×C₄	—	134	2	no	—
292	C₂×C₄	—	134	4	no	—
293	C₂×C₄	—	134	2	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	—	134	4	no	—
296	C₂×C₄	#14523999	1	2	yes	—
297	C₂×C₄	#14523999	1	2	yes	—
298	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
299	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
300	C₂×C₄	#14524089	1	1	yes	characteristic
301	C₂×C₄	#14524090	1	1	yes	characteristic
302	C₂×C₄	#14524091	1	1	yes	characteristic
303	C₂×C₄	#14524091	1	1	yes	characteristic
304	C₈	—	4	1	no	—
305	C₈	—	4	2	no	—
306	C₈	—	4	1	no	—
307	C₈	—	268	1	no	—
308	C₈	—	268	1	no	—
309	C₈	—	268	2	no	—
310	C₂²	C₂×#14523999	1	1	yes	characteristic
311	C₂²	#16863872	1	1	yes	characteristic
312	C₂²	#16863878	1	1	yes	characteristic
313	C₂²	#16864161	1	2	yes	—
314	C₂²	#16864162	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	268	2	no	—
320	C₄	—	268	2	no	—
321	C₄	—	268	4	no	—
322	C₄	#16864006	1	1	yes	characteristic
323	C₄	#16864007	1	1	yes	characteristic
324	C₄	#16864015	1	1	yes	characteristic
325	C₄	#16864019	1	1	yes	characteristic
326	C₂	#22562854	1	1	yes	characteristic
327	C₂	#22568844	1	1	yes	characteristic
328	C₂	#22568856	1	1	yes	characteristic
329	C₂	#22569312	1	2	yes	—
330	C₂	#22569378	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1985	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1985	D₆₇
hypocenter	C₆₇	#22025
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1985

Complements

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

N	H ≅ G/N	type
177. C₆₇	#22025	semidirect

Maximal subgroups

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

subgroup	index
#22562687	2
#22561150	2
#22560932	2
#22560930	2
#22560774	2
#22560772	2
#22560353	2
#22560327	2
#22560064	2
#22559957	2
C₆₇×#1985	2
#22025	67