Subgroups — M₁₆×C₄⋊Dic₆₇ #30853822

Order 17152.

1242 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	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22560081	C₂	1	4	yes	maximal, complemented
5	#22560076	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₆₇	C₂	1	2	yes	maximal, complemented
7	#22559902	C₂	1	2	yes	maximal
8	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761612	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₆₇	C₄	1	2	yes	—
11	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
12	#16863420	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	M₁₆×Dic₆₇	—	2	8	no	—
16	C₂×#14523953	C₂²	1	4	yes	—
17	C₂×C₈×Dic₆₇	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₆₇	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₆₇	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
23	C₂×C₆₇⋊#95	C₄	1	2	yes	—
24	#16863150	C₂²	1	4	yes	—
25	#16863150	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₄	1	8	yes	complemented
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
32	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
33	C₂×#11761612	—	2	4	no	—
34	C₂×#11761612	—	2	4	no	—
35	C₂×#11761612	C₂×C₄	1	8	yes	—
36	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₆₇	—	2	8	no	—
38	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₆₇	—	2	4	no	—
42	C₂×C₄×Dic₆₇	—	2	4	no	—
43	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
45	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
46	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	complemented
47	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	complemented
48	#14523953	—	2	8	no	—
49	C₈×Dic₆₇	—	2	8	no	—
50	C₆₇⋊#95	C₂×C₄	1	2	yes	—
51	C₆₇⋊#95	C₂×C₄	1	4	yes	—
52	C₆₇⋊#95	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	—	2	2	no	—
58	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
59	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
61	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	#97×C₆₇	C₂×C₄	1	8	yes	complemented
71	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
72	C₂²×Dic₆₇	—	2	8	no	—
73	C₂²×Dic₆₇	—	2	8	no	—
74	C₂²×Dic₆₇	—	4	4	no	—
75	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
76	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
77	#11761612	—	2	16	no	—
78	C₂×C₆₇⋊C₈	—	2	4	no	—
79	C₂×C₆₇⋊C₈	—	2	4	no	—
80	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
81	C₄⋊Dic₆₇	—	2	4	no	—
82	C₄⋊Dic₆₇	—	2	8	no	—
83	C₄⋊Dic₆₇	M₁₆	1	8	yes	direct product factor
84	C₄×Dic₆₇	—	2	8	no	—
85	C₄×Dic₆₇	—	2	2	no	—
86	C₄×Dic₆₇	—	2	2	no	—
87	C₄×Dic₆₇	—	2	2	no	—
88	C₄×Dic₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
93	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₆₇	#41	1	2	yes	—
104	C₂²×C₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₆₇	—	2	8	no	—
107	M₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
113	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
114	C₄²×C₆₇	C₄²	1	4	yes	—
115	C₄²×C₆₇	C₄²	1	4	yes	—
116	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
118	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
119	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
123	C₂³×C₆₇	C₄×D₄	1	2	yes	—
124	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
125	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
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	8	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₆₇	#117	1	4	yes	—
138	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
139	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
141	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×Q₈	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×Dic₆₇	—	2	4	no	—
150	C₂×Dic₆₇	—	2	4	no	—
151	C₂×Dic₆₇	—	2	4	no	—
152	C₂×Dic₆₇	—	2	4	no	—
153	C₂×Dic₆₇	—	2	4	no	—
154	C₂×Dic₆₇	—	4	16	no	—
155	C₂×Dic₆₇	C₂×M₁₆	1	8	yes	—
156	C₈×C₆₇	—	2	4	no	—
157	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	complemented
158	C₆₇⋊C₈	—	2	8	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	4	no	—
162	C₂²×C₆₇	—	2	4	no	—
163	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
164	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
165	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
168	C₂²×C₆₇	#568	1	2	yes	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
173	C₄×C₆₇	C₄×M₁₆	1	4	yes	complemented
174	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
175	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
176	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
177	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
178	Dic₆₇	—	2	8	no	—
179	Dic₆₇	—	4	4	no	—
180	C₄⋊C₄×M₁₆	—	67	1	no	maximal, Sylow
181	C₂×C₆₇	—	2	4	no	—
182	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
184	C₂×C₆₇	#2873	1	1	yes	characteristic
185	C₂×C₆₇	D₄×M₁₆	1	2	yes	—
186	C₂×C₆₇	Q₈×M₁₆	1	2	yes	—
187	C₂×#423	—	67	1	no	—
188	C₂×C₄×M₁₆	—	67	2	no	—
189	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
190	C₂×C₄×C₄⋊C₄	—	67	1	no	—
191	#1771	—	67	4	no	—
192	#1770	—	67	2	no	—
193	C₈×C₄⋊C₄	—	67	2	no	—
194	#1728	—	67	2	no	—
195	C₆₇	C₄⋊C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂²×M₁₆	—	67	2	no	—
197	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
198	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
199	C₂²×C₄⋊C₄	—	67	2	no	—
200	C₂²×C₄²	—	67	2	no	—
201	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
202	#423	—	67	8	no	—
203	C₂×C₄⋊C₈	—	67	2	no	—
204	C₂×C₄⋊C₈	—	67	2	no	—
205	C₄×M₁₆	—	134	8	no	—
206	C₄×M₁₆	Dic₆₇	1	8	yes	complemented
207	C₂×#97	—	67	4	no	—
208	C₂×#97	C₂×D₆₇	1	2	yes	—
209	C₂×C₄×C₈	—	67	4	no	—
210	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
211	C₄×C₄⋊C₄	—	67	2	no	—
212	C₄×C₄⋊C₄	—	67	2	no	—
213	C₄×C₄⋊C₄	—	67	1	no	—
214	C₄×C₄⋊C₄	—	67	1	no	—
215	C₄×C₄⋊C₄	—	67	2	no	—
216	C₂×#95	—	67	2	no	—
217	#336	—	67	4	no	—
218	#336	—	67	4	no	—
219	C₂³×C₄	—	67	4	no	—
220	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
221	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
222	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	67	8	no	—
226	C₂×M₁₆	—	134	4	no	—
227	C₂×M₁₆	—	134	4	no	—
228	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
229	C₂×M₁₆	D₂₆₈	1	4	yes	—
230	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
231	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
232	C₂²×C₈	—	67	4	no	—
233	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
234	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
235	C₂×C₄⋊C₄	—	67	2	no	—
236	C₂×C₄⋊C₄	—	67	4	no	—
237	C₂×C₄⋊C₄	—	67	2	no	—
238	C₂×C₄⋊C₄	—	134	8	no	—
239	C₂×C₄²	—	67	4	no	—
240	C₂×C₄²	—	67	2	no	—
241	C₂×C₄²	—	67	2	no	—
242	C₂×C₄²	—	134	4	no	—
243	C₂×C₄²	—	134	4	no	—
244	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₄²	C₄×D₆₇	1	4	yes	—
247	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
248	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
250	C₄⋊C₈	—	67	8	no	—
251	C₄⋊C₈	—	67	8	no	—
252	#97	—	134	8	no	—
253	#97	C₂×Dic₆₇	1	8	yes	complemented
254	C₄×C₈	—	134	8	no	—
255	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
256	#95	—	67	2	no	—
257	#95	—	67	4	no	—
258	#95	—	67	2	no	—
259	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	268	4	no	—
263	C₂²×C₄	—	67	4	no	—
264	C₂²×C₄	—	67	4	no	—
265	C₂²×C₄	—	134	8	no	—
266	C₂²×C₄	—	134	8	no	—
267	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
268	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
269	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
275	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
276	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
277	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
278	C₂²×C₄	#11761628	1	2	yes	—
279	C₂²×C₄	#11761628	1	2	yes	—
280	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
281	M₁₆	—	2	8	no	—
282	M₁₆	—	134	16	no	—
283	M₁₆	C₄⋊Dic₆₇	1	16	yes	direct product factor
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	67	8	no	—
287	C₂×C₈	—	134	4	no	—
288	C₂×C₈	—	134	4	no	—
289	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
290	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
291	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
292	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
293	C₄⋊C₄	—	67	8	no	—
294	C₄⋊C₄	—	134	8	no	—
295	C₄⋊C₄	—	134	4	no	—
296	C₄²	—	134	8	no	—
297	C₄²	—	134	2	no	—
298	C₄²	—	134	2	no	—
299	C₄²	—	134	2	no	—
300	C₄²	—	134	2	no	—
301	C₄²	C₄×Dic₆₇	1	4	yes	—
302	C₄²	C₄×Dic₆₇	1	4	yes	—
303	C₄²	C₂×C₄×D₆₇	1	2	yes	—
304	C₄²	C₂×C₄×D₆₇	1	2	yes	—
305	C₄²	C₂²×Dic₆₇	1	2	yes	—
306	C₄²	C₂²×Dic₆₇	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
310	C₂³	C₄×D₂₆₈	1	2	yes	—
311	C₂³	C₄×Dic₁₃₄	1	2	yes	—
312	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	268	16	no	—
319	C₂×C₄	—	67	8	no	—
320	C₂×C₄	—	134	4	no	—
321	C₂×C₄	—	134	4	no	—
322	C₂×C₄	—	134	4	no	—
323	C₂×C₄	—	134	4	no	—
324	C₂×C₄	—	134	4	no	—
325	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
326	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
327	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
328	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
329	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
330	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
331	C₂×C₄	#14524002	1	4	yes	—
332	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
333	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
334	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
335	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
336	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
337	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
340	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
341	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
343	C₈	—	2	4	no	—
344	C₈	—	134	8	no	—
345	C₈	C₂×C₄⋊Dic₆₇	1	8	yes	complemented
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	2	4	no	—
350	C₂²	C₄×C₄⋊Dic₆₇	1	4	yes	—
351	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
352	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
353	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
354	C₂²	C₂×M₁₆×D₆₇	1	2	yes	—
355	C₂²	#16864162	1	2	yes	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	268	4	no	—
360	C₄	—	134	8	no	—
361	C₄	C₄×C₄⋊Dic₆₇	1	4	yes	—
362	C₄	M₁₆×Dic₆₇	1	4	yes	complemented
363	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
364	C₄	C₂×#14524002	1	1	yes	characteristic
365	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
366	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
369	C₂	C₂×M₁₆×Dic₆₇	1	1	yes	characteristic
370	C₂	#22568627	1	1	yes	characteristic
371	C₂	M₁₆×D₂₆₈	1	2	yes	—
372	C₂	M₁₆×Dic₁₃₄	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₄⋊C₄×M₁₆
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #22560081	C₂	semidirect
5. #22560076	C₂	semidirect
6. C₈×C₄⋊Dic₆₇	C₂	semidirect
12. #16863420	C₄	semidirect
29. C₄×M₁₆×C₆₇	C₄	semidirect
46. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
47. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
70. #97×C₆₇	C₂×C₄	semidirect
71. C₄×C₈×C₆₇	C₂×C₄	semidirect
83. C₄⋊Dic₆₇	M₁₆	direct (N × H)
107. M₁₆×C₆₇	C₄⋊C₄	semidirect
157. C₈×C₆₇	C₂×C₄⋊C₄	semidirect
173. C₄×C₆₇	C₄×M₁₆	semidirect
195. C₆₇	C₄⋊C₄×M₁₆	semidirect
206. C₄×M₁₆	Dic₆₇	semidirect
253. #97	C₂×Dic₆₇	semidirect
255. C₄×C₈	C₂×Dic₆₇	semidirect
283. M₁₆	C₄⋊Dic₆₇	direct (N × H)
345. C₈	C₂×C₄⋊Dic₆₇	semidirect
362. C₄	M₁₆×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×M₁₆×Dic₆₇	2
C₂×C₄×C₄⋊Dic₆₇	2
#22560081	2
#22560076	2
C₈×C₄⋊Dic₆₇	2
#22559902	2
C₂×C₄×M₁₆×C₆₇	2
C₄⋊C₄×M₁₆	67

Subgroups — M16×C4⋊Dic67 #30853822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×C₄⋊Dic₆₇ #30853822