Subgroups — C₂×#22569119 #31720144

Order 17152.

2674 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864163	C₂	1	2	yes	maximal, complemented
2	#22569119	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863518	C₂	1	4	yes	maximal, complemented
4	C₂×#16863513	C₂	1	4	yes	maximal, complemented
5	C₂×#16863445	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#14523970	C₂	1	2	yes	maximal, complemented
7	C₂×M₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#575×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16864163	—	2	8	no	—
10	#16864163	C₂²	1	16	yes	complemented
11	C₂×#14524050	—	2	4	no	—
12	C₂×#14524050	C₂²	1	8	yes	complemented
13	C₂²×#11761612	—	2	4	no	—
14	C₂²×#11761612	C₂²	1	2	yes	—
15	C₂²×#11761612	C₂²	1	1	yes	characteristic
16	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863518	C₂²	1	32	yes	complemented
18	#16863513	C₂²	1	32	yes	complemented
19	#16863445	C₂²	1	8	yes	complemented
20	C₂×#14523979	C₂²	1	4	yes	complemented
21	C₂×#14523970	C₂²	1	4	yes	complemented
22	C₂×#14523970	C₂²	1	4	yes	complemented
23	C₂×#14523970	C₂²	1	16	yes	complemented
24	M₁₆×Dic₆₇	C₂²	1	8	yes	complemented
25	C₂×#14523953	C₂²	1	2	yes	complemented
26	C₂×C₈×Dic₆₇	C₂²	1	2	yes	complemented
27	C₂²×#41×C₆₇	C₂²	1	2	yes	—
28	#575×C₆₇	C₄	1	8	yes	complemented
29	#575×C₆₇	C₂²	1	8	yes	—
30	C₂×#137×C₆₇	C₄	1	4	yes	complemented
31	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	4	yes	complemented
32	C₂×#135×C₆₇	C₂²	1	4	yes	—
33	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₂³×Dic₆₇	—	2	1	no	—
35	#14524050	—	2	32	no	—
36	#14524050	—	2	16	no	—
37	#14524050	C₂³	1	32	yes	complemented
38	C₂×#11761612	—	2	32	no	—
39	C₂×#11761612	—	2	8	no	—
40	C₂×#11761612	—	2	2	no	—
41	C₂×#11761612	—	2	4	no	—
42	C₂×#11761612	D₄	1	4	yes	—
43	C₂×#11761612	D₄	1	4	yes	—
44	C₂×#11761612	C₂³	1	4	yes	—
45	C₂×#11761612	C₂³	1	8	yes	—
46	C₂×#11761612	C₂³	1	4	yes	—
47	C₂²×C₆₇⋊C₈	—	2	8	no	—
48	C₂²×C₆₇⋊C₈	—	2	2	no	—
49	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₂×C₄×Dic₆₇	—	2	1	no	—
51	C₂×C₄×Dic₆₇	—	2	2	no	—
52	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
55	#14523979	C₂³	1	16	yes	complemented
56	#14523970	C₂³	1	16	yes	complemented
57	#14523970	C₂³	1	16	yes	complemented
58	#14523953	C₂³	1	8	yes	complemented
59	C₈×Dic₆₇	C₂³	1	8	yes	complemented
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	4	no	—
62	C₂×#41×C₆₇	—	2	2	no	—
63	C₂×#41×C₆₇	—	2	4	no	—
64	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
66	C₂×#41×C₆₇	C₂³	1	4	yes	—
67	C₂×#41×C₆₇	C₂³	1	8	yes	—
68	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
69	C₂²×D₄×C₆₇	—	2	2	no	—
70	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂³×C₄×C₆₇	—	2	2	no	—
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	#137×C₆₇	C₂×C₄	1	32	yes	complemented
74	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	32	yes	complemented
75	#135×C₆₇	C₂×C₄	1	16	yes	complemented
76	#135×C₆₇	C₂³	1	16	yes	—
77	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
78	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
79	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
80	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
81	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
82	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
83	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×Dic₆₇	—	2	1	no	—
85	C₂²×Dic₆₇	—	2	4	no	—
86	C₂²×Dic₆₇	—	2	1	no	—
87	C₂²×Dic₆₇	—	4	4	no	—
88	#11761612	—	2	8	no	—
89	#11761612	—	2	4	no	—
90	#11761612	—	2	32	no	—
91	#11761612	C₂×D₄	1	16	yes	—
92	#11761612	C₂⁴	1	8	yes	—
93	C₂×C₆₇⋊C₈	—	2	4	no	—
94	C₂×C₆₇⋊C₈	—	2	8	no	—
95	C₂×C₆₇⋊C₈	—	2	32	no	—
96	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
97	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
98	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
99	C₄×Dic₆₇	—	2	1	no	—
100	C₄×Dic₆₇	—	2	4	no	—
101	C₄×Dic₆₇	—	2	1	no	—
102	C₄×Dic₆₇	C₂⁴	1	2	yes	—
103	C₄×Dic₆₇	C₂⁴	1	2	yes	—
104	C₂⁴×C₆₇	—	2	2	no	—
105	C₂⁴×C₆₇	#41	1	1	yes	characteristic
106	#41×C₆₇	—	2	8	no	—
107	#41×C₆₇	—	2	8	no	—
108	#41×C₆₇	—	2	8	no	—
109	#41×C₆₇	—	2	8	no	—
110	#41×C₆₇	—	2	4	no	—
111	#41×C₆₇	—	2	4	no	—
112	#41×C₆₇	—	2	8	no	—
113	#41×C₆₇	C₂²×C₄	1	8	yes	—
114	#41×C₆₇	C₂²×C₄	1	16	yes	—
115	#41×C₆₇	C₂⁴	1	8	yes	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	4	no	—
118	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	8	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
127	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	#41	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
141	Q₁₆×C₆₇	C₂²×C₄	1	16	yes	complemented
142	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	complemented
143	D₈×C₆₇	C₂²×C₄	1	16	yes	complemented
144	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
145	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
146	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
147	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
148	C₂³×C₆₇	—	2	4	no	—
149	C₂³×C₆₇	—	2	8	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	8	4	no	—
152	C₂³×C₆₇	C₄×D₄	1	2	yes	—
153	C₂³×C₆₇	C₄×D₄	1	2	yes	—
154	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₂³×C₆₇	C₂×#41	1	2	yes	—
156	Q₈×C₆₇	—	2	8	no	—
157	Q₈×C₆₇	—	2	4	no	—
158	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
159	D₄×C₆₇	—	2	16	no	—
160	D₄×C₆₇	—	2	8	no	—
161	D₄×C₆₇	—	2	8	no	—
162	D₄×C₆₇	—	2	8	no	—
163	D₄×C₆₇	—	2	4	no	—
164	D₄×C₆₇	C₂³×C₄	1	8	yes	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	16	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
172	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
173	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
182	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
183	C₂×Dic₆₇	—	2	4	no	—
184	C₂×Dic₆₇	—	2	2	no	—
185	C₂×Dic₆₇	—	2	2	no	—
186	C₂×Dic₆₇	—	4	8	no	—
187	C₂×Dic₆₇	—	4	2	no	—
188	C₈×C₆₇	C₂³×C₄	1	8	yes	—
189	C₆₇⋊C₈	—	2	8	no	—
190	C₆₇⋊C₈	C₂²×D₄	1	8	yes	—
191	C₂²×C₆₇	—	2	2	no	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	2	8	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	8	no	—
197	C₂²×C₆₇	—	8	8	no	—
198	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
199	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
202	C₄×C₆₇	—	2	8	no	—
203	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
205	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
207	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
208	Dic₆₇	—	2	4	no	—
209	Dic₆₇	—	4	2	no	—
210	C₂×#2956	—	67	1	no	maximal, Sylow
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	2	4	no	—
213	C₂×C₆₇	—	2	1	no	—
214	C₂×C₆₇	—	4	8	no	—
215	C₂×C₆₇	#2956	1	2	yes	complemented
216	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
217	C₂×#575	D₆₇	1	1	yes	characteristic
218	C₂×#568	—	67	2	no	—
219	#2956	—	67	16	no	—
220	C₂×#444	—	67	4	no	—
221	C₂×#443	—	67	4	no	—
222	C₂×#430	—	67	1	no	—
223	C₂²×#104	—	67	2	no	—
224	C₂×C₄×M₁₆	—	67	1	no	—
225	C₆₇	C₂×#2956	1	1	yes	characteristic, complemented, Sylow
226	C₂²×#41	C₂×D₆₇	1	2	yes	—
227	#575	Dic₆₇	1	8	yes	complemented
228	#575	C₂×D₆₇	1	8	yes	—
229	C₂×#137	Dic₆₇	1	4	yes	complemented
230	C₂×AGL(1,ℤ/8)	Dic₆₇	1	4	yes	complemented
231	C₂×#135	C₂×D₆₇	1	4	yes	—
232	#568	—	67	16	no	—
233	#568	—	134	8	no	—
234	C₂×#131	—	67	8	no	—
235	C₂×#131	—	134	4	no	—
236	C₂²×M₁₆	—	67	1	no	—
237	C₂²×M₁₆	—	67	2	no	—
238	C₂²×M₁₆	—	134	4	no	—
239	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
240	C₂²×C₄²	—	67	1	no	—
241	#444	—	67	32	no	—
242	#443	—	67	32	no	—
243	#430	—	67	8	no	—
244	C₂×#108	—	67	4	no	—
245	C₂×#104	—	67	4	no	—
246	C₂×#104	—	67	4	no	—
247	C₂×#104	—	67	16	no	—
248	C₄×M₁₆	—	67	8	no	—
249	C₂×#97	—	67	2	no	—
250	C₂×C₄×C₈	—	67	2	no	—
251	C₂×#41	—	2	2	no	—
252	C₂×#41	—	2	4	no	—
253	C₂×#41	—	2	2	no	—
254	C₂×#41	—	2	4	no	—
255	C₂×#41	C₂×Dic₆₇	1	4	yes	—
256	C₂×#41	C₂×Dic₆₇	1	8	yes	—
257	C₂×#41	C₂²×D₆₇	1	4	yes	—
258	C₂×#41	C₂²×D₆₇	1	8	yes	—
259	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
260	C₂²×D₄	—	2	2	no	—
261	C₂²×D₄	C₂×Dic₆₇	1	2	yes	—
262	C₂³×C₄	—	2	2	no	—
263	C₂³×C₄	—	134	1	no	—
264	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
265	#137	C₂×Dic₆₇	1	32	yes	complemented
266	AGL(1,ℤ/8)	C₂×Dic₆₇	1	32	yes	complemented
267	#135	C₂×Dic₆₇	1	16	yes	complemented
268	#135	C₂²×D₆₇	1	16	yes	—
269	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	complemented
270	C₂×SD₁₆	C₂×Dic₆₇	1	8	yes	complemented
271	C₂×D₈	C₂×Dic₆₇	1	4	yes	complemented
272	#131	—	67	32	no	—
273	#131	—	134	16	no	—
274	#131	—	134	32	no	—
275	C₂×M₁₆	—	67	4	no	—
276	C₂×M₁₆	—	67	4	no	—
277	C₂×M₁₆	—	67	4	no	—
278	C₂×M₁₆	—	67	4	no	—
279	C₂×M₁₆	—	67	8	no	—
280	C₂×M₁₆	—	134	8	no	—
281	C₂×M₁₆	—	134	32	no	—
282	C₂×M₁₆	—	134	2	no	—
283	C₂×M₁₆	—	134	4	no	—
284	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
285	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
286	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
287	C₂²×C₈	—	67	2	no	—
288	C₂²×C₈	—	134	8	no	—
289	C₂²×C₈	—	134	2	no	—
290	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
291	C₂×C₄²	—	67	1	no	—
292	C₂×C₄²	—	67	1	no	—
293	C₂×C₄²	—	67	4	no	—
294	C₂×C₄²	—	134	2	no	—
295	C₂×C₄²	—	134	1	no	—
296	#108	—	67	16	no	—
297	#104	—	67	16	no	—
298	#104	—	67	16	no	—
299	#97	—	67	8	no	—
300	C₄×C₈	—	67	8	no	—
301	C₂⁴	—	2	2	no	—
302	C₂⁴	#11761627	1	1	yes	characteristic
303	#41	—	2	8	no	—
304	#41	—	2	8	no	—
305	#41	—	2	8	no	—
306	#41	—	2	4	no	—
307	#41	—	2	8	no	—
308	#41	—	2	4	no	—
309	#41	—	2	8	no	—
310	#41	C₂²×Dic₆₇	1	8	yes	—
311	#41	C₂²×Dic₆₇	1	16	yes	—
312	#41	C₂³×D₆₇	1	8	yes	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
316	C₂×Q₈	C₂²×Dic₆₇	1	8	yes	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	8	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	8	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
324	C₂×D₄	C₂²×Dic₆₇	1	8	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	268	4	no	—
331	C₂²×C₄	—	134	1	no	—
332	C₂²×C₄	—	134	4	no	—
333	C₂²×C₄	—	134	1	no	—
334	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
335	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
336	C₂²×C₄	D₄×D₆₇	1	2	yes	—
337	C₂²×C₄	D₄×D₆₇	1	2	yes	—
338	C₂²×C₄	#11761627	1	1	yes	characteristic
339	C₂²×C₄	#11761627	1	2	yes	—
340	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
341	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
342	Q₁₆	C₂²×Dic₆₇	1	16	yes	complemented
343	SD₁₆	C₂²×Dic₆₇	1	32	yes	complemented
344	D₈	C₂²×Dic₆₇	1	16	yes	complemented
345	M₁₆	—	67	16	no	—
346	M₁₆	—	67	8	no	—
347	M₁₆	—	134	32	no	—
348	M₁₆	—	134	8	no	—
349	M₁₆	—	134	4	no	—
350	M₁₆	C₂²×Dic₆₇	1	16	yes	—
351	C₂×C₈	—	67	4	no	—
352	C₂×C₈	—	67	4	no	—
353	C₂×C₈	—	67	4	no	—
354	C₂×C₈	—	134	32	no	—
355	C₂×C₈	—	134	8	no	—
356	C₂×C₈	—	134	4	no	—
357	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
358	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
359	C₂×C₈	C₂³×D₆₇	1	4	yes	—
360	C₄²	—	67	2	no	—
361	C₄²	—	67	2	no	—
362	C₄²	—	134	1	no	—
363	C₄²	—	134	4	no	—
364	C₄²	—	134	1	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	2	8	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	8	4	no	—
369	C₂³	D₄×Dic₆₇	1	2	yes	—
370	C₂³	D₄×Dic₆₇	1	2	yes	—
371	C₂³	C₂×#11761627	1	1	yes	characteristic
372	C₂³	C₂×#11761627	1	2	yes	—
373	Q₈	—	2	8	no	—
374	Q₈	—	2	4	no	—
375	Q₈	C₂³×Dic₆₇	1	8	yes	—
376	D₄	—	2	16	no	—
377	D₄	—	2	8	no	—
378	D₄	—	2	8	no	—
379	D₄	—	2	8	no	—
380	D₄	—	2	4	no	—
381	D₄	C₂³×Dic₆₇	1	8	yes	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	16	no	—
386	C₂×C₄	—	2	8	no	—
387	C₂×C₄	—	268	2	no	—
388	C₂×C₄	—	268	8	no	—
389	C₂×C₄	—	134	2	no	—
390	C₂×C₄	—	134	4	no	—
391	C₂×C₄	—	134	2	no	—
392	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
393	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
394	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
395	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
396	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
397	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
399	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
400	C₂×C₄	C₂×D₄×D₆₇	1	4	yes	—
401	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
403	C₂×C₄	C₂×#11761627	1	2	yes	—
404	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
405	C₈	—	67	8	no	—
406	C₈	—	134	8	no	—
407	C₈	C₂³×Dic₆₇	1	8	yes	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	8	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	8	no	—
414	C₂²	—	8	8	no	—
415	C₂²	C₂×D₄×Dic₆₇	1	4	yes	—
416	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
417	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
418	C₂²	C₂²×#11761627	1	1	yes	characteristic
419	C₄	—	2	8	no	—
420	C₄	—	268	2	no	—
421	C₄	—	134	4	no	—
422	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
423	C₄	C₂×D₄×Dic₆₇	1	4	yes	—
424	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
425	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
426	C₄	C₂²×D₄×D₆₇	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	4	no	—
429	C₂	—	2	1	no	—
430	C₂	—	4	8	no	—
431	C₂	#22569119	1	2	yes	direct product factor
432	C₂	C₂²×D₄×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting, prime radical	C₂×#575×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×#575	D₆₇
hypocenter	C₆₇	C₂×#2956
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#575

Complements

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

N	H ≅ G/N	type
1. C₂×#16864163	C₂	semidirect
2. #22569119	C₂	direct (N × H)
3. C₂×#16863518	C₂	semidirect
4. C₂×#16863513	C₂	semidirect
5. C₂×#16863445	C₂	semidirect
6. C₂²×#14523970	C₂	semidirect
7. C₂×M₁₆×Dic₆₇	C₂	semidirect
10. #16864163	C₂²	semidirect
12. C₂×#14524050	C₂²	semidirect
17. #16863518	C₂²	semidirect
18. #16863513	C₂²	semidirect
19. #16863445	C₂²	semidirect
20. C₂×#14523979	C₂²	semidirect
21. C₂×#14523970	C₂²	semidirect
22. C₂×#14523970	C₂²	semidirect
23. C₂×#14523970	C₂²	semidirect
24. M₁₆×Dic₆₇	C₂²	semidirect
25. C₂×#14523953	C₂²	semidirect
26. C₂×C₈×Dic₆₇	C₂²	semidirect
28. #575×C₆₇	C₄	semidirect
30. C₂×#137×C₆₇	C₄	semidirect
31. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
37. #14524050	C₂³	semidirect
55. #14523979	C₂³	semidirect
56. #14523970	C₂³	semidirect
57. #14523970	C₂³	semidirect
58. #14523953	C₂³	semidirect
59. C₈×Dic₆₇	C₂³	semidirect
73. #137×C₆₇	C₂×C₄	semidirect
74. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
75. #135×C₆₇	C₂×C₄	semidirect
77. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
78. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
79. C₂×D₈×C₆₇	C₂×C₄	semidirect
141. Q₁₆×C₆₇	C₂²×C₄	semidirect
142. SD₁₆×C₆₇	C₂²×C₄	semidirect
143. D₈×C₆₇	C₂²×C₄	semidirect
215. C₂×C₆₇	#2956	semidirect
225. C₆₇	C₂×#2956	semidirect
227. #575	Dic₆₇	semidirect
229. C₂×#137	Dic₆₇	semidirect
230. C₂×AGL(1,ℤ/8)	Dic₆₇	semidirect
265. #137	C₂×Dic₆₇	semidirect
266. AGL(1,ℤ/8)	C₂×Dic₆₇	semidirect
267. #135	C₂×Dic₆₇	semidirect
269. C₂×Q₁₆	C₂×Dic₆₇	semidirect
270. C₂×SD₁₆	C₂×Dic₆₇	semidirect
271. C₂×D₈	C₂×Dic₆₇	semidirect
342. Q₁₆	C₂²×Dic₆₇	semidirect
343. SD₁₆	C₂²×Dic₆₇	semidirect
344. D₈	C₂²×Dic₆₇	semidirect
431. C₂	#22569119	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16864163	2
#22569119	2
C₂×#16863518	2
C₂×#16863513	2
C₂×#16863445	2
C₂²×#14523970	2
C₂×M₁₆×Dic₆₇	2
C₂×#575×C₆₇	2
C₂×#2956	67

Subgroups — C2×#22569119 #31720144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569119 #31720144