Subgroups — #30761643

Order 17152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562616	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863157	C₂	1	4	yes	maximal
4	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₆₇	—	2	2	no	—
7	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
8	#16863266	C₄	1	2	yes	complemented
9	#16863262	C₄	1	2	yes	complemented
10	#16863256	—	2	2	no	—
11	C₄×C₄⋊Dic₆₇	—	2	2	no	—
12	C₄²×Dic₆₇	—	2	2	no	—
13	C₄²×Dic₆₇	C₄	1	4	yes	complemented
14	#16863157	—	2	8	no	—
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
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	C₂³×Dic₆₇	—	2	2	no	—
21	#14524002	—	2	4	no	—
22	#14524002	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	4	no	—
24	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄×Dic₆₇	—	2	16	no	—
27	C₂×C₄×Dic₆₇	—	2	8	no	—
28	C₂×C₄×Dic₆₇	—	2	2	no	—
29	C₂×C₄×Dic₆₇	—	2	2	no	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	C₂×C₄×Dic₆₇	—	2	2	no	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₆₇	D₄	1	4	yes	—
34	C₂×C₄×Dic₆₇	Q₈	1	4	yes	—
35	C₆₇⋊#95	—	2	4	no	—
36	C₂³×C₄×C₆₇	D₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	4	no	—
40	C₂×M₁₆×C₆₇	—	2	4	no	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
50	C₄⋊C₈×C₆₇	—	2	8	no	—
51	C₂²×Dic₆₇	—	2	16	no	—
52	C₂²×Dic₆₇	—	2	4	no	—
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	2	8	no	—
57	C₄⋊Dic₆₇	—	4	8	no	—
58	C₂²⋊Dic₆₇	—	4	4	no	—
59	C₄×Dic₆₇	—	2	32	no	—
60	C₄×Dic₆₇	—	2	4	no	—
61	C₄×Dic₆₇	—	2	4	no	—
62	C₄×Dic₆₇	—	2	4	no	—
63	C₄×Dic₆₇	—	4	2	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	complemented
66	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	M₁₆×C₆₇	—	2	8	no	—
83	M₁₆×C₆₇	—	4	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	4	2	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	C₄²	1	2	yes	—
88	C₄²×C₆₇	C₄²	1	2	yes	—
89	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
92	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₂³×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₇	#121	1	2	yes	—
98	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	2	8	no	—
102	C₂×C₄×C₆₇	#95	1	4	yes	—
103	C₂×C₄×C₆₇	#95	1	4	yes	—
104	C₂×C₄×C₆₇	#104	1	8	yes	—
105	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	#121	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#125	1	2	yes	—
120	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
123	C₂×Dic₆₇	—	2	16	no	—
124	C₂×Dic₆₇	—	2	16	no	—
125	C₂×Dic₆₇	—	2	16	no	—
126	C₂×Dic₆₇	—	2	8	no	—
127	C₂×Dic₆₇	—	4	4	no	—
128	C₂×Dic₆₇	—	4	8	no	—
129	C₈×C₆₇	—	4	4	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
135	C₂²×C₆₇	#380	1	2	yes	—
136	C₂²×C₆₇	#389	1	2	yes	—
137	C₂²×C₆₇	#390	1	1	yes	characteristic
138	C₂²×C₆₇	#391	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×#104	1	4	yes	—
140	C₄×C₆₇	—	2	4	no	—
141	C₄×C₆₇	#339	1	8	yes	complemented
142	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
143	C₄×C₆₇	#382	1	1	yes	characteristic
144	C₄×C₆₇	#384	1	1	yes	characteristic
145	C₄×C₆₇	#389	1	2	yes	—
146	C₄×C₆₇	#390	1	1	yes	characteristic
147	C₄×C₆₇	#391	1	1	yes	characteristic
148	Dic₆₇	—	2	16	no	—
149	Dic₆₇	—	8	4	no	—
150	#10839	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#339	1	2	yes	—
155	C₂×C₆₇	#1962	1	2	yes	—
156	C₂×C₆₇	#1994	1	2	yes	—
157	C₂×#423	D₆₇	1	1	yes	characteristic
158	#2285	—	67	1	no	—
159	C₂×C₄³	—	67	1	no	—
160	C₂×#339	—	67	4	no	—
161	C₆₇	#10839	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
163	C₂×#117	—	67	2	no	—
164	C₂²×C₄²	—	67	2	no	—
165	C₂²×C₄²	—	134	2	no	—
166	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
167	#423	—	2	2	no	—
168	#423	Dic₆₇	1	4	yes	complemented
169	C₂×C₄⋊C₈	—	2	2	no	—
170	#384	—	67	2	no	—
171	#382	—	67	2	no	—
172	#380	—	134	2	no	—
173	C₄×C₄⋊C₄	—	134	2	no	—
174	C₄³	—	67	4	no	—
175	C₄³	—	134	2	no	—
176	#339	—	134	8	no	—
177	C₂³×C₄	—	134	2	no	—
178	C₂³×C₄	D₂₆₈	1	2	yes	—
179	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	#117	—	134	4	no	—
186	#117	—	134	4	no	—
187	C₂×C₄⋊C₄	—	67	4	no	—
188	C₂×C₄⋊C₄	—	134	4	no	—
189	C₂×C₂²⋊C₄	—	134	2	no	—
190	C₂×C₄²	—	67	4	no	—
191	C₂×C₄²	—	67	4	no	—
192	C₂×C₄²	—	67	4	no	—
193	C₂×C₄²	—	134	2	no	—
194	C₂×C₄²	—	134	16	no	—
195	C₂×C₄²	—	134	8	no	—
196	C₂×C₄²	—	134	2	no	—
197	C₂×C₄²	—	134	2	no	—
198	C₂×C₄²	—	134	2	no	—
199	C₂×C₄²	Dic₁₃₄	1	2	yes	—
200	C₂×C₄²	C₄×D₆₇	1	2	yes	—
201	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
204	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
206	C₄⋊C₈	—	2	8	no	—
207	#95	—	134	4	no	—
208	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	134	4	no	—
211	C₂²×C₄	—	134	16	no	—
212	C₂²×C₄	—	134	4	no	—
213	C₂²×C₄	—	134	4	no	—
214	C₂²×C₄	—	134	4	no	—
215	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
216	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
217	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
218	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
219	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₆₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
227	C₂²×C₄	#11761627	1	2	yes	—
228	C₂²×C₄	#11761629	1	2	yes	—
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	268	8	no	—
235	C₄⋊C₄	—	134	8	no	—
236	C₂²⋊C₄	—	268	4	no	—
237	C₄²	—	268	2	no	—
238	C₄²	—	268	2	no	—
239	C₄²	—	67	8	no	—
240	C₄²	—	134	32	no	—
241	C₄²	—	134	4	no	—
242	C₄²	—	134	4	no	—
243	C₄²	—	134	4	no	—
244	C₄²	C₄×Dic₆₇	1	2	yes	—
245	C₄²	C₄×Dic₆₇	1	2	yes	—
246	C₄²	C₂²⋊Dic₆₇	1	4	yes	—
247	C₄²	Dic₆₇⋊C₄	1	2	yes	—
248	C₄²	Dic₆₇⋊C₄	1	4	yes	—
249	C₄²	Dic₆₇⋊C₄	1	2	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
252	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
253	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
254	C₂³	#14524018	1	2	yes	—
255	C₂³	#14524023	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	268	4	no	—
260	C₂×C₄	—	268	8	no	—
261	C₂×C₄	—	134	16	no	—
262	C₂×C₄	—	134	16	no	—
263	C₂×C₄	—	134	16	no	—
264	C₂×C₄	—	134	8	no	—
265	C₂×C₄	C₆₇⋊#95	1	4	yes	—
266	C₂×C₄	C₆₇⋊#95	1	4	yes	—
267	C₂×C₄	#14523969	1	8	yes	—
268	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
271	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
273	C₂×C₄	#14524001	1	2	yes	—
274	C₂×C₄	#14524005	1	2	yes	—
275	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
276	C₂×C₄	#14524008	1	2	yes	—
277	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
278	C₂×C₄	#14524014	1	2	yes	—
279	C₂×C₄	#14524019	1	2	yes	—
280	C₂×C₄	#14524030	1	1	yes	characteristic
281	C₂×C₄	#14524032	1	1	yes	characteristic
282	C₂×C₄	#14524034	1	2	yes	—
283	C₂×C₄	#14524039	1	1	yes	characteristic
284	C₂×C₄	#14524041	1	1	yes	characteristic
285	C₂×C₄	#14524042	1	2	yes	—
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
292	C₂²	#16863253	1	2	yes	—
293	C₂²	#16863295	1	2	yes	—
294	C₂²	#16863297	1	1	yes	characteristic
295	C₂²	#16863301	1	1	yes	characteristic
296	C₂²	C₂×#14523969	1	4	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	536	4	no	—
299	C₄	—	134	16	no	—
300	C₄	#16863157	1	8	yes	complemented
301	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
302	C₄	#16863261	1	1	yes	characteristic
303	C₄	#16863265	1	1	yes	characteristic
304	C₄	#16863292	1	2	yes	—
305	C₄	#16863299	1	1	yes	characteristic
306	C₄	#16863303	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#22558438	1	1	yes	characteristic
310	C₂	C₂×#16863157	1	2	yes	—
311	C₂	#22561036	1	2	yes	—
312	C₂	#22561185	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	#10839
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
8. #16863266	C₄	semidirect
9. #16863262	C₄	semidirect
13. C₄²×Dic₆₇	C₄	semidirect
18. #423×C₆₇	C₄	semidirect
65. C₄×Dic₆₇	C₄⋊C₄	semidirect
141. C₄×C₆₇	#339	semidirect
161. C₆₇	#10839	semidirect
168. #423	Dic₆₇	semidirect
300. C₄	#16863157	semidirect

Maximal subgroups

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

subgroup	index
#22562616	2
C₂×C₄²×Dic₆₇	2
C₂×#16863157	2
C₂×#423×C₆₇	2
#10839	67