Subgroups — #30866721

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561146	C₂	1	2	yes	maximal
2	#22560928	C₂	1	2	yes	maximal
3	#22560768	C₂	1	4	yes	maximal
4	#22560352	C₂	1	2	yes	maximal
5	#22560326	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#22559955	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
13	#16863151	C₂²	1	2	yes	—
14	#16863151	C₂²	1	4	yes	—
15	#16863151	C₂²	1	4	yes	—
16	#16863151	C₂²	1	4	yes	—
17	#16863151	C₂²	1	2	yes	—
18	#16863151	C₂²	1	1	yes	characteristic
19	#16863151	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄³×C₆₇	C₂²	1	1	yes	characteristic
30	C₄³×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
34	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
36	C₄×C₆₇⋊C₈	—	2	4	no	—
37	C₄×C₆₇⋊C₈	—	4	2	no	—
38	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₆₇	—	2	4	no	—
40	C₄×Q₈×C₆₇	—	2	4	no	—
41	C₄×Q₈×C₆₇	C₈	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#95×C₆₇	C₂×C₄	1	2	yes	—
62	#95×C₆₇	C₂×C₄	1	2	yes	—
63	#95×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	4	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
92	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	4	1	no	—
101	C₄²×C₆₇	—	4	1	no	—
102	C₄²×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	—	4	2	no	—
104	C₄²×C₆₇	C₂×C₈	1	8	yes	—
105	C₄²×C₆₇	C₂×C₈	1	4	yes	—
106	C₄²×C₆₇	M₁₆	1	4	yes	—
107	C₄²×C₆₇	#41	1	4	yes	—
108	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	8	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	#117	1	2	yes	—
131	C₂×C₄×C₆₇	#117	1	2	yes	—
132	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
134	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
144	C₆₇⋊C₈	—	4	2	no	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	—	4	4	no	—
147	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
148	C₂²×C₆₇	#519	1	1	yes	characteristic
149	C₂²×C₆₇	#520	1	1	yes	characteristic
150	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
151	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
152	C₂²×C₆₇	#568	1	1	yes	characteristic
153	C₂²×C₆₇	#568	1	1	yes	characteristic
154	C₄×C₆₇	—	2	4	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	2	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	4	no	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	4	2	no	—
162	C₄×C₆₇	—	4	2	no	—
163	C₄×C₆₇	#431	1	4	yes	—
164	C₄×C₆₇	#541	1	2	yes	—
165	C₄×C₆₇	#542	1	2	yes	—
166	#22018	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	#2320	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
169	C₂×C₆₇	#2923	1	1	yes	characteristic
170	C₂×C₆₇	#2982	1	1	yes	characteristic
171	C₂×C₆₇	#2983	1	1	yes	characteristic
172	C₂×C₆₇	#2988	1	1	yes	characteristic
173	C₂×C₆₇	#2989	1	1	yes	characteristic
174	C₄²×Q₈	D₆₇	1	1	yes	characteristic
175	#1984	—	67	2	no	—
176	#1939	—	67	2	no	—
177	#1916	—	67	4	no	—
178	#1839	—	67	2	no	—
179	#1831	—	67	2	no	—
180	C₄×C₄⋊C₈	—	67	1	no	—
181	#1744	—	67	1	no	—
182	C₆₇	#22018	1	1	yes	characteristic, complemented, Sylow
183	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
184	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
185	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	67	4	no	—
187	C₂×C₄⋊C₈	—	67	2	no	—
188	C₂×C₄×C₈	—	67	2	no	—
189	C₂×C₄×C₈	—	67	2	no	—
190	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
191	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
192	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
193	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
194	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
195	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
196	C₄³	C₂×D₆₇	1	2	yes	—
197	C₄³	C₂×D₆₇	1	1	yes	characteristic
198	#336	—	67	1	no	—
199	#336	—	67	2	no	—
200	#336	—	67	4	no	—
201	#336	—	67	1	no	—
202	#336	—	67	4	no	—
203	#336	—	67	4	no	—
204	#336	—	67	2	no	—
205	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂²×C₈	—	67	2	no	—
207	C₂²×C₈	—	67	2	no	—
208	C₂²×C₈	—	67	4	no	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	C₆₇⋊C₈	1	8	yes	complemented
212	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
214	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
219	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
220	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
221	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
224	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
227	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
230	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
231	C₄⋊C₈	—	268	4	no	—
232	C₄⋊C₈	—	134	4	no	—
233	C₄×C₈	—	268	2	no	—
234	C₄×C₈	—	134	4	no	—
235	#95	C₂×Dic₆₇	1	2	yes	—
236	#95	C₂×Dic₆₇	1	2	yes	—
237	#95	C₂×Dic₆₇	1	2	yes	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
252	C₂×C₈	—	134	2	no	—
253	C₂×C₈	—	134	2	no	—
254	C₂×C₈	—	134	4	no	—
255	C₂×C₈	—	134	4	no	—
256	C₂×C₈	—	134	2	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	2	no	—
260	C₂×C₈	—	134	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
266	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×C₆₇⋊C₈	1	8	yes	—
279	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
280	C₄²	#11761612	1	4	yes	—
281	C₄²	#11761628	1	4	yes	—
282	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
283	Q₈	—	2	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	#14524002	1	2	yes	—
305	C₂×C₄	#14524002	1	2	yes	—
306	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
307	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
308	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
309	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
310	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
311	C₂×C₄	C₂×#11761612	1	2	yes	—
312	C₂×C₄	C₂×#11761612	1	2	yes	—
313	C₂×C₄	C₂×#11761612	1	2	yes	—
314	C₂×C₄	C₂×#11761628	1	2	yes	—
315	C₂×C₄	#14524089	1	1	yes	characteristic
316	C₂×C₄	#14524090	1	1	yes	characteristic
317	C₂×C₄	#14524091	1	2	yes	—
318	C₈	—	268	2	no	—
319	C₈	—	268	2	no	—
320	C₈	—	268	4	no	—
321	C₂²	C₂×#14524002	1	1	yes	characteristic
322	C₂²	#16863874	1	1	yes	characteristic
323	C₂²	#16863879	1	1	yes	characteristic
324	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
325	C₂²	C₂²×#11761612	1	1	yes	characteristic
326	C₂²	#16864163	1	1	yes	characteristic
327	C₂²	#16864163	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	4	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	#16863449	1	4	yes	—
338	C₄	#16864004	1	2	yes	—
339	C₄	#16864018	1	2	yes	—
340	C₂	#22562860	1	1	yes	characteristic
341	C₂	C₂×#16863449	1	1	yes	characteristic
342	C₂	#22568860	1	1	yes	characteristic
343	C₂	#22569290	1	1	yes	characteristic
344	C₂	#22569294	1	1	yes	characteristic
345	C₂	#22569320	1	1	yes	characteristic
346	C₂	#22569326	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₄²×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	#22018
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
41. C₄×Q₈×C₆₇	C₈	semidirect
182. C₆₇	#22018	semidirect
211. C₄×Q₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22561146	2
#22560928	2
#22560768	2
#22560352	2
#22560326	2
C₄×C₄⋊(C₆₇⋊C₈)	2
#22559955	2
C₄²×Q₈×C₆₇	2
#22018	67