Subgroups — #30856201

Order 17152.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568636	C₂	1	1	yes	maximal, characteristic
2	#22560357	C₂	1	2	yes	maximal, complemented
3	#22560333	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560311	C₂	1	2	yes	maximal
5	#22560256	C₂	1	1	yes	maximal, characteristic
6	#22560252	C₂	1	1	yes	maximal, characteristic
7	#22560096	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560095	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560008	C₂	1	2	yes	maximal
10	#22559963	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	#16863864	C₂²	1	1	yes	characteristic
15	C₂×#14523979	C₂²	1	1	yes	characteristic
16	#16863434	C₂²	1	1	yes	characteristic
17	#16863378	C₂²	1	2	yes	—
18	#16863369	C₂²	1	2	yes	—
19	#16863354	C₂²	1	2	yes	—
20	#16863353	C₄	1	2	yes	complemented
21	#16863352	C₂²	1	2	yes	—
22	#16863352	C₂²	1	1	yes	characteristic
23	#16863350	C₄	1	2	yes	complemented
24	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
25	C₄×#11761611	C₄	1	2	yes	complemented
26	C₂×#14523953	C₂²	1	1	yes	characteristic
27	#97×D₆₇	C₄	1	2	yes	complemented
28	#16863167	C₂²	1	2	yes	—
29	#16863157	C₂²	1	2	yes	—
30	#16863155	C₂²	1	2	yes	—
31	C₆₇⋊#338	C₂²	1	2	yes	—
32	#16863152	C₂²	1	2	yes	—
33	#16863152	C₂²	1	2	yes	—
34	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
35	#427×C₆₇	C₂²	1	1	yes	characteristic
36	#405×C₆₇	C₂²	1	1	yes	characteristic
37	#344×C₆₇	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	—
39	C₂×#11761628	C₂³	1	1	yes	characteristic
40	#14524051	—	2	2	no	—
41	#14524051	—	2	2	no	—
42	#14524049	—	2	1	no	—
43	#14524049	—	2	1	no	—
44	#14524049	—	2	1	no	—
45	#14524049	—	2	1	no	—
46	C₂×#11761612	C₂³	1	2	yes	—
47	C₂×#11761612	C₂³	1	1	yes	characteristic
48	C₂×#11761611	C₂×C₄	1	2	yes	—
49	C₂×#11761611	C₂×C₄	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
53	#14524008	—	2	1	no	—
54	#14524005	—	2	1	no	—
55	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
56	#14524002	C₂³	1	2	yes	—
57	#14524000	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
59	C₄²×D₆₇	C₂×C₄	1	2	yes	—
60	#14523979	—	2	1	no	—
61	#14523979	—	2	1	no	—
62	#14523977	—	2	1	no	—
63	#14523975	—	2	1	no	—
64	#14523961	—	2	2	no	—
65	#14523959	—	2	2	no	—
66	#14523957	—	2	4	no	—
67	#14523954	—	2	2	no	—
68	#14523954	C₂×C₄	1	2	yes	—
69	#14523953	—	2	2	no	—
70	#14523953	C₂×C₄	1	1	yes	characteristic
71	#14523953	C₂×C₄	1	2	yes	—
72	#14523953	C₂×C₄	1	1	yes	characteristic
73	#14523953	C₂×C₄	1	1	yes	characteristic
74	#14523953	C₂×C₄	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	—	2	2	no	—
76	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
77	C₈×Dic₆₇	—	2	2	no	—
78	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#117×C₆₇	C₂³	1	2	yes	—
83	#117×C₆₇	C₂³	1	1	yes	characteristic
84	#108×C₆₇	—	2	1	no	—
85	#108×C₆₇	—	2	1	no	—
86	#107×C₆₇	—	2	1	no	—
87	#106×C₆₇	—	2	1	no	—
88	#97×C₆₇	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
90	#11761628	—	2	1	no	—
91	#11761628	—	2	1	no	—
92	#11761628	—	2	1	no	—
93	#11761628	—	2	1	no	—
94	C₂×Dic₁₃₄	#41	1	1	yes	characteristic
95	C₂×D₂₆₈	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₆₇	#41	1	2	yes	—
97	C₂²×Dic₆₇	—	2	1	no	—
98	C₂×C₄×D₆₇	—	2	2	no	—
99	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
100	#11761612	—	2	1	no	—
101	#11761612	—	2	1	no	—
102	#11761612	—	2	2	no	—
103	#11761612	—	2	2	no	—
104	#11761611	—	2	2	no	—
105	#11761611	—	2	2	no	—
106	#11761611	—	2	4	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	#41	1	2	yes	—
112	C₂×C₆₇⋊C₈	#41	1	2	yes	—
113	C₈×D₆₇	—	2	2	no	—
114	C₈×D₆₇	—	2	2	no	—
115	C₈×D₆₇	—	2	4	no	—
116	C₄⋊Dic₆₇	—	2	2	no	—
117	Dic₆₇⋊C₄	—	2	2	no	—
118	C₂²⋊Dic₆₇	—	2	2	no	—
119	(C₂×D₆₇)⋊C₄	—	2	2	no	—
120	C₄×Dic₆₇	—	2	1	no	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	—	2	2	no	—
123	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	M₁₆×C₆₇	—	2	1	no	—
127	M₁₆×C₆₇	—	2	1	no	—
128	M₁₆×C₆₇	—	2	2	no	—
129	M₁₆×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
135	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
136	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
137	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₂²⋊C₄×C₆₇	—	2	1	no	—
141	C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂²×D₆₇	#117	1	2	yes	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	C₂³×C₄	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×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₆₇	—	2	2	no	—
155	C₂²⋊D₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	2	1	no	—
157	C₂×Dic₆₇	—	2	1	no	—
158	C₂×Dic₆₇	—	4	2	no	—
159	C₂×Dic₆₇	—	8	1	no	—
160	C₂×Dic₆₇	#117	1	2	yes	—
161	D₂₆₈	—	2	1	no	—
162	D₂₆₈	—	2	1	no	—
163	C₄×D₆₇	—	2	2	no	—
164	C₄×D₆₇	—	2	2	no	—
165	C₄×D₆₇	—	8	2	no	—
166	Dic₁₃₄	—	2	1	no	—
167	Dic₁₃₄	—	2	1	no	—
168	C₈×C₆₇	—	2	2	no	—
169	C₈×C₆₇	—	4	2	no	—
170	C₈×C₆₇	#117	1	1	yes	characteristic
171	C₈×C₆₇	#117	1	1	yes	characteristic
172	C₈×C₆₇	#117	1	1	yes	characteristic
173	C₈×C₆₇	#117	1	1	yes	characteristic
174	C₆₇⋊C₈	—	2	2	no	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₆₇⋊C₈	#117	1	2	yes	—
177	C₆₇⋊C₈	#117	1	2	yes	—
178	C₂²×C₆₇	—	4	1	no	—
179	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
180	C₂²×C₆₇	#529	1	1	yes	characteristic
181	C₂²×C₆₇	#529	1	1	yes	characteristic
182	C₂×D₆₇	—	2	2	no	—
183	C₂×D₆₇	—	4	2	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	8	2	no	—
186	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
187	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
188	C₄×C₆₇	#529	1	1	yes	characteristic
189	C₄×C₆₇	#529	1	1	yes	characteristic
190	Dic₆₇	—	2	2	no	—
191	Dic₆₇	—	4	1	no	—
192	Dic₆₇	—	4	1	no	—
193	Dic₆₇	—	8	2	no	—
194	#20788	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	—	2	1	no	—
196	C₂×C₆₇	—	2	1	no	—
197	C₂×C₆₇	—	2	1	no	—
198	C₂×C₆₇	#2283	1	1	yes	characteristic
199	D₆₇	—	4	2	no	—
200	#2875	—	67	1	no	—
201	#1841	—	67	2	no	—
202	#1833	—	67	1	no	—
203	#1833	D₆₇	1	1	yes	characteristic, complemented
204	#1827	—	67	2	no	—
205	#1813	—	67	1	no	—
206	#1812	—	67	1	no	—
207	#1773	—	67	1	no	—
208	#1773	—	67	1	no	—
209	#1757	—	67	2	no	—
210	#1747	—	67	2	no	—
211	C₆₇	#20788	1	1	yes	characteristic, complemented, Sylow
212	C₂×#131	—	67	1	no	—
213	C₂×#131	—	67	1	no	—
214	C₄×#41	—	67	1	no	—
215	C₂×#108	—	67	1	no	—
216	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
217	#427	—	67	1	no	—
218	#427	C₂×D₆₇	1	1	yes	characteristic
219	#413	—	67	2	no	—
220	#410	—	67	2	no	—
221	#405	—	67	1	no	—
222	#405	—	67	2	no	—
223	#405	—	67	2	no	—
224	#405	—	67	2	no	—
225	#405	—	67	2	no	—
226	#405	C₂×D₆₇	1	1	yes	characteristic
227	C₄×M₁₆	—	67	1	no	—
228	C₄×M₁₆	—	67	2	no	—
229	C₂×#97	—	67	1	no	—
230	C₂×#97	—	67	2	no	—
231	#344	—	67	2	no	—
232	#344	C₂×D₆₇	1	2	yes	—
233	#339	—	67	2	no	—
234	#339	—	67	2	no	—
235	#338	—	67	2	no	—
236	H(3,ℤ/4)	—	67	2	no	—
237	H(3,ℤ/4)	—	67	2	no	—
238	H(3,ℤ/4)	C₂×D₆₇	1	2	yes	—
239	C₂×#41	—	67	1	no	—
240	#131	—	134	2	no	—
241	#131	—	134	2	no	—
242	#131	—	134	1	no	—
243	#131	—	134	1	no	—
244	#131	—	134	1	no	—
245	#131	—	134	1	no	—
246	C₂×M₁₆	—	67	1	no	—
247	C₂×M₁₆	—	67	2	no	—
248	C₂×M₁₆	—	67	2	no	—
249	C₂×M₁₆	—	67	2	no	—
250	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
251	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
252	C₂²×C₈	—	67	1	no	—
253	C₂²×C₈	—	67	2	no	—
254	C₂²×C₈	—	67	2	no	—
255	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
256	C₄×Q₈	—	134	1	no	—
257	C₄×D₄	—	134	2	no	—
258	C₄×D₄	—	134	1	no	—
259	#117	—	67	2	no	—
260	#117	—	67	2	no	—
261	#117	C₂²×D₆₇	1	2	yes	—
262	#117	C₂²×D₆₇	1	1	yes	characteristic
263	C₂×C₄²	—	67	1	no	—
264	C₂×C₄²	—	67	2	no	—
265	#108	—	2	1	no	—
266	#108	—	2	1	no	—
267	#108	—	134	1	no	—
268	#108	—	134	1	no	—
269	#107	—	2	1	no	—
270	#107	—	134	1	no	—
271	#106	—	2	1	no	—
272	#106	—	134	1	no	—
273	#101	—	134	2	no	—
274	#100	—	134	2	no	—
275	#99	—	134	4	no	—
276	#97	—	67	1	no	—
277	#97	—	67	1	no	—
278	#97	—	67	2	no	—
279	#97	—	67	1	no	—
280	#97	—	67	1	no	—
281	#97	—	67	2	no	—
282	#97	—	134	2	no	—
283	#97	—	134	2	no	—
284	#97	C₄×D₆₇	1	2	yes	—
285	C₄×C₈	—	67	2	no	—
286	C₄×C₈	—	67	2	no	—
287	C₄×C₈	—	134	2	no	—
288	C₄×C₈	—	134	2	no	—
289	C₄×C₈	C₄×D₆₇	1	2	yes	—
290	#41	—	134	1	no	—
291	#41	—	134	1	no	—
292	#41	—	134	1	no	—
293	#41	—	134	1	no	—
294	C₂×Q₈	—	67	1	no	—
295	C₂×D₄	—	67	2	no	—
296	C₂×D₄	—	67	1	no	—
297	C₂²×C₄	—	67	2	no	—
298	C₂²×C₄	—	134	2	no	—
299	C₂²×C₄	—	134	1	no	—
300	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	134	4	no	—
306	M₁₆	—	134	2	no	—
307	M₁₆	—	134	2	no	—
308	M₁₆	—	134	1	no	—
309	M₁₆	—	134	1	no	—
310	M₁₆	—	134	2	no	—
311	M₁₆	—	134	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	67	1	no	—
314	C₂×C₈	—	67	1	no	—
315	C₂×C₈	—	67	2	no	—
316	C₂×C₈	—	67	2	no	—
317	C₂×C₈	—	67	2	no	—
318	C₂×C₈	—	134	4	no	—
319	C₂×C₈	—	134	2	no	—
320	C₂×C₈	—	134	2	no	—
321	C₂×C₈	—	134	2	no	—
322	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
325	C₂×C₈	#11761627	1	1	yes	characteristic
326	C₂×C₈	#11761627	1	1	yes	characteristic
327	C₂×C₈	#11761629	1	1	yes	characteristic
328	C₂×C₈	#11761629	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	134	2	no	—
332	C₄⋊C₄	—	134	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	134	2	no	—
336	C₂²⋊C₄	—	134	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	67	1	no	—
339	C₄²	—	67	1	no	—
340	C₄²	—	134	2	no	—
341	C₄²	—	134	2	no	—
342	C₄²	—	134	1	no	—
343	C₄²	C₂×C₄×D₆₇	1	2	yes	—
344	C₂³	—	67	2	no	—
345	C₂³	#41×D₆₇	1	1	yes	characteristic
346	Q₈	—	134	1	no	—
347	Q₈	—	134	1	no	—
348	D₄	—	134	2	no	—
349	D₄	—	134	2	no	—
350	D₄	—	134	1	no	—
351	D₄	—	134	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	268	2	no	—
355	C₂×C₄	—	536	2	no	—
356	C₂×C₄	—	536	1	no	—
357	C₂×C₄	—	67	2	no	—
358	C₂×C₄	—	134	1	no	—
359	C₂×C₄	—	134	1	no	—
360	C₂×C₄	—	134	2	no	—
361	C₂×C₄	—	134	2	no	—
362	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
363	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
364	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
365	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
367	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	268	2	no	—
371	C₈	—	67	2	no	—
372	C₈	—	67	2	no	—
373	C₈	—	134	2	no	—
374	C₈	#14524001	1	1	yes	characteristic
375	C₈	#14524001	1	1	yes	characteristic
376	C₈	#14524001	1	1	yes	characteristic
377	C₈	#14524001	1	1	yes	characteristic
378	C₂²	—	4	1	no	—
379	C₂²	—	268	2	no	—
380	C₂²	—	134	2	no	—
381	C₂²	#117×D₆₇	1	1	yes	characteristic
382	C₂²	#16863925	1	1	yes	characteristic
383	C₂²	#16863931	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	268	1	no	—
387	C₄	—	268	1	no	—
388	C₄	—	536	2	no	—
389	C₄	—	134	2	no	—
390	C₄	#117×D₆₇	1	1	yes	characteristic
391	C₄	C₂×#14524001	1	1	yes	characteristic
392	C₄	#16863925	1	1	yes	characteristic
393	C₄	#16863931	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	268	2	no	—
398	C₂	#22562602	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1833	C₂
socle	C₂×C₆₇	#2283
prime radical	C₂×#11761628	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1833	D₆₇
hypocenter	C₆₇	#20788
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #22560357	C₂	semidirect
3. #22560333	C₂	semidirect
7. #22560096	C₂	semidirect
8. #22560095	C₂	semidirect
10. #22559963	C₂	semidirect
11. C₆₇×#1833	C₂	semidirect
20. #16863353	C₄	semidirect
23. #16863350	C₄	semidirect
25. C₄×#11761611	C₄	semidirect
27. #97×D₆₇	C₄	semidirect
203. #1833	D₆₇	semidirect
211. C₆₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#22568636	2
#22560357	2
#22560333	2
#22560311	2
#22560256	2
#22560252	2
#22560096	2
#22560095	2
#22560008	2
#22559963	2
C₆₇×#1833	2
#20788	67