Subgroups — #30860235

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22568856	C₂	1	2	yes	maximal, complemented
3	#22561098	C₂	1	1	yes	maximal, characteristic
4	#22560957	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560955	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560923	C₂	1	2	yes	maximal, complemented
7	#22560917	C₂	1	2	yes	maximal, complemented
8	#108×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560021	C₂	1	2	yes	maximal, complemented
10	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1974	C₂	1	1	yes	maximal, characteristic
12	#41×Dic₆₇	C₂²	1	2	yes	—
13	Q₁₆×Dic₆₇	C₄	1	2	yes	complemented
14	SD₁₆×Dic₆₇	C₄	1	4	yes	complemented
15	D₈×Dic₆₇	C₄	1	2	yes	complemented
16	#16863454	—	2	4	no	—
17	#16863450	—	2	4	no	—
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	#16863434	C₂²	1	1	yes	characteristic
20	#16863424	C₂²	1	2	yes	—
21	#16863422	C₂²	1	2	yes	—
22	C₂×#14523970	C₂²	1	2	yes	—
23	#16863392	C₂²	1	2	yes	—
24	#16863363	C₂²	1	2	yes	—
25	M₁₆×Dic₆₇	C₂²	1	2	yes	—
26	C₂×#14523954	C₂²	1	1	yes	characteristic
27	C₂×#14523953	C₂²	1	1	yes	characteristic
28	C₂×#14523953	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
31	#16863242	C₂²	1	1	yes	characteristic
32	#16863160	C₂²	1	2	yes	—
33	#16863157	C₂²	1	2	yes	—
34	#16863156	C₂²	1	2	yes	—
35	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#104×C₆₇	C₂²	1	2	yes	—
38	#408×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#11761612	C₂³	1	2	yes	—
41	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	Q₈×Dic₆₇	C₂×C₄	1	4	yes	—
44	D₄×Dic₆₇	—	2	4	no	—
45	D₄×Dic₆₇	C₂×C₄	1	4	yes	—
46	#14524002	—	2	2	no	—
47	#14524002	C₂³	1	2	yes	—
48	C₂×C₄×Dic₆₇	—	2	2	no	—
49	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
51	#14523979	—	2	2	no	—
52	#14523977	C₂×C₄	1	2	yes	—
53	#14523975	C₂×C₄	1	2	yes	—
54	Dic₆₇⋊C₈	—	2	4	no	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	#14523970	—	2	4	no	—
57	#14523967	C₂×C₄	1	4	yes	—
58	#14523964	C₂×C₄	1	4	yes	—
59	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
60	#14523954	—	2	2	no	—
61	#14523953	—	2	2	no	—
62	#14523953	—	2	2	no	—
63	#14523953	D₄	1	2	yes	—
64	#14523953	D₄	1	2	yes	—
65	C₈×Dic₆₇	—	2	2	no	—
66	C₈×Dic₆₇	—	2	2	no	—
67	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
68	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
69	C₆₇⋊#95	—	2	2	no	—
70	C₂×#41×C₆₇	C₂³	1	2	yes	—
71	#135×C₆₇	—	2	2	no	—
72	#135×C₆₇	—	2	2	no	—
73	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	#108×C₆₇	—	2	2	no	—
81	#104×C₆₇	—	2	4	no	—
82	C₂²⋊C₈×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	C₂²×Dic₆₇	—	2	1	no	—
85	C₂²×Dic₆₇	—	2	4	no	—
86	C₂²×Dic₆₇	#41	1	2	yes	—
87	#11761612	—	2	2	no	—
88	#11761612	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	1	no	—
93	C₂×C₆₇⋊C₈	—	2	2	no	—
94	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₄⋊Dic₆₇	—	2	4	no	—
97	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
98	C₂²⋊Dic₆₇	—	2	2	no	—
99	C₂²⋊Dic₆₇	—	2	4	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	1	no	—
102	C₄×Dic₆₇	—	2	4	no	—
103	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
104	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄×Dic₆₇	#41	1	2	yes	—
107	#41×C₆₇	—	2	2	no	—
108	#41×C₆₇	—	2	2	no	—
109	#41×C₆₇	—	2	2	no	—
110	#41×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	Q₁₆×C₆₇	—	2	2	no	—
118	SD₁₆×C₆₇	—	2	4	no	—
119	D₈×C₆₇	—	2	2	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₆₇	#41	1	2	yes	—
129	C₂×C₈×C₆₇	#41	1	2	yes	—
130	C₄²×C₆₇	—	2	1	no	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	2	no	—
135	Q₈×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₂×Dic₆₇	—	2	4	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	4	2	no	—
154	C₂×Dic₆₇	—	4	2	no	—
155	C₂×Dic₆₇	—	8	4	no	—
156	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
157	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	#117	1	2	yes	—
161	C₈×C₆₇	#117	1	2	yes	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₆₇⋊C₈	—	2	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	—	4	4	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₆₇	#529	1	1	yes	characteristic
170	C₂²×C₆₇	#529	1	1	yes	characteristic
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
174	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
175	C₄×C₆₇	#532	1	1	yes	characteristic
176	C₄×C₆₇	#532	1	1	yes	characteristic
177	Dic₆₇	—	2	2	no	—
178	Dic₆₇	—	2	2	no	—
179	Dic₆₇	—	4	2	no	—
180	Dic₆₇	—	4	4	no	—
181	#21294	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	2	1	no	—
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	—	8	2	no	—
185	C₂×C₆₇	#2293	1	1	yes	characteristic
186	C₂×C₆₇	#2957	1	2	yes	—
187	C₄×#135	—	67	1	no	—
188	#2923	—	67	2	no	—
189	#1974	—	67	1	no	—
190	#1974	D₆₇	1	1	yes	characteristic
191	#1945	—	67	1	no	—
192	#1945	—	67	1	no	—
193	#1938	—	67	2	no	—
194	#1938	—	67	2	no	—
195	C₄×#108	—	67	1	no	—
196	#1759	—	67	2	no	—
197	#1726	—	67	1	no	—
198	C₆₇	#21294	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
200	C₄×#41	—	67	2	no	—
201	C₄×Q₁₆	—	67	2	no	—
202	C₄×SD₁₆	—	67	4	no	—
203	C₄×D₈	—	67	2	no	—
204	#433	—	134	4	no	—
205	#432	—	134	4	no	—
206	C₂×#108	—	67	1	no	—
207	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
208	#427	—	67	1	no	—
209	#424	—	67	2	no	—
210	#424	—	67	2	no	—
211	C₂×#104	—	67	2	no	—
212	C₂×#104	C₂×D₆₇	1	2	yes	—
213	#416	—	67	2	no	—
214	#408	—	67	2	no	—
215	#408	C₂×D₆₇	1	2	yes	—
216	C₄×M₁₆	—	67	2	no	—
217	C₂×#97	—	67	1	no	—
218	C₂×#97	—	67	1	no	—
219	C₂×#97	—	67	1	no	—
220	C₂×C₄×C₈	—	67	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
223	#376	—	67	1	no	—
224	#341	—	67	2	no	—
225	#339	—	67	2	no	—
226	#339	—	67	2	no	—
227	C₂×#41	C₂²×D₆₇	1	2	yes	—
228	#135	—	2	2	no	—
229	#135	—	2	2	no	—
230	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
231	C₂×SD₁₆	C₄×D₆₇	1	2	yes	—
232	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
233	C₂×M₁₆	—	67	2	no	—
234	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
235	C₂²×C₈	—	67	1	no	—
236	C₂²×C₈	—	67	1	no	—
237	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
239	C₄×Q₈	—	67	4	no	—
240	C₄×D₄	—	67	4	no	—
241	C₄×D₄	—	134	4	no	—
242	#117	—	67	2	no	—
243	#117	—	134	2	no	—
244	C₂×C₄²	—	67	1	no	—
245	C₂×C₄²	—	67	1	no	—
246	C₂×C₄²	—	134	2	no	—
247	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
248	#108	—	2	2	no	—
249	#108	—	134	2	no	—
250	#107	—	67	2	no	—
251	#106	—	67	2	no	—
252	C₄⋊C₈	—	134	4	no	—
253	C₄⋊C₈	—	134	4	no	—
254	#104	—	2	4	no	—
255	#104	—	134	4	no	—
256	#103	—	67	4	no	—
257	#102	—	67	4	no	—
258	C₂²⋊C₈	—	2	4	no	—
259	C₂²⋊C₈	—	134	4	no	—
260	#97	—	67	2	no	—
261	#97	—	67	2	no	—
262	#97	—	134	2	no	—
263	#97	—	134	2	no	—
264	#97	—	134	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	67	2	no	—
267	C₄×C₈	—	67	2	no	—
268	C₄×C₈	—	134	2	no	—
269	C₄×C₈	—	134	2	no	—
270	#95	—	134	2	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	67	2	no	—
281	C₂²×C₄	—	134	1	no	—
282	C₂²×C₄	—	134	4	no	—
283	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
284	Q₁₆	—	2	2	no	—
285	SD₁₆	—	2	4	no	—
286	D₈	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	134	2	no	—
290	M₁₆	—	134	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	67	2	no	—
295	C₂×C₈	—	67	2	no	—
296	C₂×C₈	—	134	2	no	—
297	C₂×C₈	—	134	1	no	—
298	C₂×C₈	—	134	1	no	—
299	C₂×C₈	—	134	1	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂×C₈	D₄×D₆₇	1	2	yes	—
304	C₂×C₈	#11761627	1	2	yes	—
305	C₂×C₈	#11761627	1	2	yes	—
306	C₄⋊C₄	—	67	4	no	—
307	C₄⋊C₄	—	134	4	no	—
308	C₂²⋊C₄	—	134	4	no	—
309	C₂²⋊C₄	—	134	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	67	2	no	—
312	C₄²	—	67	1	no	—
313	C₄²	—	67	1	no	—
314	C₄²	—	67	2	no	—
315	C₄²	—	134	4	no	—
316	C₄²	—	134	2	no	—
317	C₄²	—	134	1	no	—
318	C₄²	#11761627	1	2	yes	—
319	C₂³	—	2	2	no	—
320	C₂³	#41×D₆₇	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	268	2	no	—
333	C₂×C₄	—	268	2	no	—
334	C₂×C₄	—	536	4	no	—
335	C₂×C₄	—	67	2	no	—
336	C₂×C₄	—	67	2	no	—
337	C₂×C₄	—	134	4	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
343	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	268	2	no	—
348	C₈	—	134	2	no	—
349	C₈	—	134	2	no	—
350	C₈	#14523999	1	2	yes	—
351	C₈	#14523999	1	2	yes	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
356	C₂²	#16863925	1	1	yes	characteristic
357	C₂²	#16863927	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	268	2	no	—
361	C₄	—	268	4	no	—
362	C₄	—	134	2	no	—
363	C₄	—	134	2	no	—
364	C₄	C₂×#14523999	1	1	yes	characteristic
365	C₄	C₄×#11761627	1	1	yes	characteristic
366	C₄	#16863941	1	1	yes	characteristic
367	C₄	#16863944	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	#22562671	1	1	yes	characteristic
372	C₂	#22569121	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1974	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1974	D₆₇
hypocenter	C₆₇	#21294
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1974

Complements

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

N	H ≅ G/N	type
2. #22568856	C₂	semidirect
4. #22560957	C₂	semidirect
5. #22560955	C₂	semidirect
6. #22560923	C₂	semidirect
7. #22560917	C₂	semidirect
8. #108×Dic₆₇	C₂	semidirect
9. #22560021	C₂	semidirect
10. #22559895	C₂	semidirect
13. Q₁₆×Dic₆₇	C₄	semidirect
14. SD₁₆×Dic₆₇	C₄	semidirect
15. D₈×Dic₆₇	C₄	semidirect
198. C₆₇	#21294	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₆₇	2
#22568856	2
#22561098	2
#22560957	2
#22560955	2
#22560923	2
#22560917	2
#108×Dic₆₇	2
#22560021	2
#22559895	2
C₆₇×#1974	2
#21294	67