Subgroups — #30761645

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863242	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863155	C₂	1	4	yes	maximal
4	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
7	#16863420	—	2	2	no	—
8	#16863420	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
10	#16863242	—	2	2	no	—
11	#16863242	C₄	1	4	yes	complemented
12	C₂×C₆₇⋊#95	—	2	2	no	—
13	#16863155	—	2	8	no	—
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#384×C₆₇	C₄	1	2	yes	complemented
17	#382×C₆₇	C₄	1	2	yes	complemented
18	#380×C₆₇	—	2	2	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
20	C₂³×Dic₆₇	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	4	no	—
23	C₂×#11761612	—	2	4	no	—
24	C₂×#11761612	C₂×C₄	1	4	yes	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂×C₄×Dic₆₇	—	2	2	no	—
27	C₂×C₄×Dic₆₇	—	2	2	no	—
28	C₂×C₄×Dic₆₇	—	2	2	no	—
29	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₆₇	D₄	1	4	yes	—
31	C₂×C₄×Dic₆₇	Q₈	1	4	yes	—
32	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
33	C₆₇⋊#95	—	2	8	no	—
34	C₂³×C₄×C₆₇	D₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	#117×C₆₇	—	2	4	no	—
37	#117×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
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	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
49	#95×C₆₇	—	2	4	no	—
50	C₂²×Dic₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	—	2	4	no	—
52	C₂²×Dic₆₇	—	2	4	no	—
53	C₂²×Dic₆₇	—	4	8	no	—
54	#11761612	—	2	8	no	—
55	#11761612	—	4	4	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₄×Dic₆₇	—	2	4	no	—
60	C₄×Dic₆₇	—	2	4	no	—
61	C₄×Dic₆₇	—	2	4	no	—
62	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	complemented
63	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	4	4	no	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	4	8	no	—
82	C₂²⋊C₄×C₆₇	—	4	4	no	—
83	C₄²×C₆₇	—	4	2	no	—
84	C₄²×C₆₇	—	4	2	no	—
85	C₄²×C₆₇	C₄²	1	2	yes	—
86	C₄²×C₆₇	C₄²	1	2	yes	—
87	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
88	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
89	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₇	#121	1	2	yes	—
96	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	8	no	—
103	C₂×C₄×C₆₇	#95	1	4	yes	—
104	C₂×C₄×C₆₇	#95	1	4	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	8	no	—
124	C₂×Dic₆₇	—	4	8	no	—
125	C₂×Dic₆₇	—	4	8	no	—
126	C₂×Dic₆₇	—	4	8	no	—
127	C₆₇⋊C₈	—	4	4	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
133	C₂²×C₆₇	#380	1	2	yes	—
134	C₂²×C₆₇	#389	1	2	yes	—
135	C₂²×C₆₇	#390	1	1	yes	characteristic
136	C₂²×C₆₇	#391	1	1	yes	characteristic
137	C₂²×C₆₇	#421	1	4	yes	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	—	8	4	no	—
141	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
142	C₄×C₆₇	#382	1	1	yes	characteristic
143	C₄×C₆₇	#384	1	1	yes	characteristic
144	C₄×C₆₇	#389	1	2	yes	—
145	C₄×C₆₇	#390	1	1	yes	characteristic
146	C₄×C₆₇	#391	1	1	yes	characteristic
147	Dic₆₇	—	4	8	no	—
148	#10840	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
152	C₂×C₆₇	#1733	1	2	yes	—
153	C₂×C₆₇	#1963	1	2	yes	—
154	C₂×C₆₇	#1995	1	2	yes	—
155	C₂×#423	—	67	1	no	—
156	#2285	D₆₇	1	1	yes	characteristic
157	C₂×#376	—	67	1	no	—
158	C₂×#339	—	67	4	no	—
159	C₆₇	#10840	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	67	2	no	—
161	C₂×#117	C₂×D₆₇	1	2	yes	—
162	C₂²×C₄²	—	67	2	no	—
163	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
164	#423	—	67	4	no	—
165	#423	—	134	2	no	—
166	C₂×C₄⋊C₈	—	134	2	no	—
167	#384	Dic₆₇	1	2	yes	complemented
168	#382	Dic₆₇	1	2	yes	complemented
169	#380	—	2	2	no	—
170	C₄×C₄⋊C₄	—	2	2	no	—
171	#376	—	67	4	no	—
172	#376	—	134	2	no	—
173	C₂×#95	—	134	2	no	—
174	#339	—	134	8	no	—
175	C₂³×C₄	—	134	2	no	—
176	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	67	4	no	—
179	C₂×M₁₆	—	134	4	no	—
180	C₂×M₁₆	—	134	4	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂²×C₈	—	134	2	no	—
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	67	4	no	—
190	C₂×C₄²	—	67	4	no	—
191	C₂×C₄²	—	67	4	no	—
192	C₂×C₄²	—	134	2	no	—
193	C₂×C₄²	—	134	2	no	—
194	C₂×C₄²	—	134	2	no	—
195	C₂×C₄²	Dic₁₃₄	1	2	yes	—
196	C₂×C₄²	C₄×D₆₇	1	2	yes	—
197	C₂×C₄²	C₄×D₆₇	1	2	yes	—
198	C₂×C₄²	D₂₆₈	1	2	yes	—
199	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
202	C₄⋊C₈	—	134	8	no	—
203	#95	—	2	4	no	—
204	#95	—	134	8	no	—
205	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	268	8	no	—
209	C₂²×C₄	—	134	4	no	—
210	C₂²×C₄	—	134	4	no	—
211	C₂²×C₄	—	134	4	no	—
212	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
213	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
215	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
216	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₆₇	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
222	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
223	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
224	C₂²×C₄	#11761627	1	2	yes	—
225	C₂²×C₄	#11761629	1	2	yes	—
226	M₁₆	—	268	4	no	—
227	M₁₆	—	134	8	no	—
228	C₂×C₈	—	268	2	no	—
229	C₂×C₈	—	268	2	no	—
230	C₂×C₈	—	134	4	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	4	8	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	67	8	no	—
237	C₄²	—	134	4	no	—
238	C₄²	—	134	4	no	—
239	C₄²	—	134	4	no	—
240	C₄²	C₄×Dic₆₇	1	2	yes	—
241	C₄²	C₄×Dic₆₇	1	2	yes	—
242	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
243	C₄²	Dic₆₇⋊C₄	1	4	yes	—
244	C₄²	C₄⋊Dic₆₇	1	2	yes	—
245	C₄²	C₄⋊Dic₆₇	1	2	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
250	C₂³	#14524014	1	2	yes	—
251	C₂³	#14524019	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	268	8	no	—
259	C₂×C₄	—	268	8	no	—
260	C₂×C₄	—	268	8	no	—
261	C₂×C₄	—	134	8	no	—
262	C₂×C₄	C₆₇⋊#95	1	4	yes	—
263	C₂×C₄	C₆₇⋊#95	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
267	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
269	C₂×C₄	#14524001	1	2	yes	—
270	C₂×C₄	#14524005	1	2	yes	—
271	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
272	C₂×C₄	#14524008	1	2	yes	—
273	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
274	C₂×C₄	#14524018	1	2	yes	—
275	C₂×C₄	#14524023	1	2	yes	—
276	C₂×C₄	#14524030	1	1	yes	characteristic
277	C₂×C₄	#14524032	1	1	yes	characteristic
278	C₂×C₄	#14524034	1	2	yes	—
279	C₂×C₄	#14524039	1	1	yes	characteristic
280	C₂×C₄	#14524041	1	1	yes	characteristic
281	C₂×C₄	#14524042	1	2	yes	—
282	C₈	—	268	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
288	C₂²	#16863253	1	2	yes	—
289	C₂²	#16863292	1	2	yes	—
290	C₂²	#16863299	1	1	yes	characteristic
291	C₂²	#16863303	1	1	yes	characteristic
292	C₂²	#16863413	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	8	4	no	—
296	C₄	—	268	8	no	—
297	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
298	C₄	#16863261	1	1	yes	characteristic
299	C₄	#16863265	1	1	yes	characteristic
300	C₄	#16863295	1	2	yes	—
301	C₄	#16863297	1	1	yes	characteristic
302	C₄	#16863301	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#22558438	1	1	yes	characteristic
306	C₂	#22559918	1	2	yes	—
307	C₂	#22561042	1	2	yes	—
308	C₂	#22561191	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#2285	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#2285	D₆₇
hypocenter	C₆₇	#10840
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
8. #16863420	C₄	semidirect
11. #16863242	C₄	semidirect
16. #384×C₆₇	C₄	semidirect
17. #382×C₆₇	C₄	semidirect
62. C₄×Dic₆₇	C₄⋊C₄	semidirect
159. C₆₇	#10840	semidirect
167. #384	Dic₆₇	semidirect
168. #382	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
C₂×#16863242	2
C₂×#16863155	2
C₆₇×#2285	2
#10840	67