Subgroups — C₂×#22558330 #30770786

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863370	C₂	1	2	yes	maximal
2	#22558330	C₂	1	4	yes	maximal, direct product factor
3	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524002	—	2	2	no	—
5	#16863370	—	2	2	no	—
6	#16863370	—	2	2	no	—
7	#16863370	C₄	1	4	yes	complemented
8	#16863370	C₂²	1	4	yes	—
9	C₂×#14523958	—	2	2	no	—
10	C₂×#14523958	C₄	1	4	yes	complemented
11	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic, complemented
14	#408×C₆₇	—	2	2	no	—
15	#408×C₆₇	C₄	1	2	yes	complemented
16	#408×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
18	#14524002	—	2	4	no	—
19	#14524002	—	2	4	no	—
20	#14524002	—	4	2	no	—
21	#14524002	—	4	2	no	—
22	C₂×C₄⋊Dic₆₇	—	4	2	no	—
23	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
24	C₂×C₄×Dic₆₇	—	4	2	no	—
25	#14523958	—	2	4	no	—
26	#14523958	—	2	4	no	—
27	#14523958	—	2	4	no	—
28	#14523958	C₂×C₄	1	8	yes	complemented
29	C₂×#41×C₆₇	—	4	1	no	—
30	C₂×#41×C₆₇	—	4	1	no	—
31	C₂×#41×C₆₇	—	4	1	no	—
32	C₂×#41×C₆₇	—	4	1	no	—
33	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
36	C₂×#41×C₆₇	D₄	1	2	yes	—
37	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	2	yes	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	4	1	no	—
46	C₂×M₁₆×C₆₇	—	4	1	no	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
48	C₂²×C₈×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₂²×C₈×C₆₇	—	4	1	no	—
52	C₂²×C₈×C₆₇	—	4	1	no	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	complemented
54	C₂²⋊C₈×C₆₇	—	2	4	no	—
55	C₂²⋊C₈×C₆₇	—	2	4	no	—
56	C₂²×Dic₆₇	—	4	4	no	—
57	C₄⋊Dic₆₇	—	4	4	no	—
58	C₄⋊Dic₆₇	—	4	4	no	—
59	C₂²⋊Dic₆₇	—	2	8	no	—
60	C₂²⋊Dic₆₇	—	4	4	no	—
61	C₄×Dic₆₇	—	4	4	no	—
62	C₄×Dic₆₇	—	4	4	no	—
63	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
64	C₂⁴×C₆₇	C₂²⋊C₄	1	2	yes	—
65	#41×C₆₇	—	4	2	no	—
66	#41×C₆₇	—	4	2	no	—
67	#41×C₆₇	—	4	1	no	—
68	#41×C₆₇	—	4	1	no	—
69	#41×C₆₇	—	4	1	no	—
70	#41×C₆₇	—	4	1	no	—
71	#41×C₆₇	—	4	2	no	—
72	#41×C₆₇	—	4	2	no	—
73	#41×C₆₇	—	4	1	no	—
74	#41×C₆₇	—	4	1	no	—
75	#41×C₆₇	—	4	1	no	—
76	#41×C₆₇	—	4	1	no	—
77	C₂×Q₈×C₆₇	—	4	1	no	—
78	C₂×Q₈×C₆₇	—	4	1	no	—
79	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂×Q₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₂×D₄×C₆₇	—	4	2	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂×D₄×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂×D₄×C₆₇	C₄²	1	2	yes	—
87	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	M₁₆×C₆₇	—	4	2	no	—
113	M₁₆×C₆₇	—	4	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	4	2	no	—
117	C₂×C₈×C₆₇	—	4	1	no	—
118	C₂×C₈×C₆₇	—	4	1	no	—
119	C₂×C₈×C₆₇	—	4	1	no	—
120	C₂×C₈×C₆₇	—	4	1	no	—
121	C₂×C₈×C₆₇	—	4	2	no	—
122	C₂³×C₆₇	—	2	1	no	—
123	C₂³×C₆₇	—	2	1	no	—
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	—	4	4	no	—
128	C₂³×C₆₇	#95	1	1	yes	characteristic
129	C₂³×C₆₇	#95	1	2	yes	—
130	C₂³×C₆₇	#95	1	1	yes	characteristic
131	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
132	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	4	1	no	—
135	Q₈×C₆₇	—	4	1	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	4	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	1	no	—
141	D₄×C₆₇	—	4	2	no	—
142	D₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
164	C₂×C₄×C₆₇	#95	1	2	yes	—
165	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
166	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
167	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
168	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
172	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×Dic₆₇	—	4	8	no	—
174	C₂×Dic₆₇	—	8	8	no	—
175	C₈×C₆₇	—	4	2	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	8	2	no	—
190	C₂²×C₆₇	#342	1	1	yes	characteristic
191	C₂²×C₆₇	#342	1	1	yes	characteristic
192	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	2	2	no	—
195	C₄×C₆₇	—	2	2	no	—
196	C₄×C₆₇	—	2	1	no	—
197	C₄×C₆₇	#342	1	1	yes	characteristic
198	C₄×C₆₇	#342	1	1	yes	characteristic
199	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
200	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
201	Dic₆₇	—	8	8	no	—
202	C₂×#1391	—	67	1	no	maximal, Sylow
203	C₂×C₆₇	—	2	1	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	4	2	no	—
206	C₂×C₆₇	—	4	1	no	—
207	C₂×C₆₇	—	4	1	no	—
208	C₂×C₆₇	—	4	2	no	—
209	C₂×C₆₇	#1391	1	2	yes	complemented
210	C₂×C₆₇	C₂×#342	1	1	yes	characteristic
211	C₂×#410	—	67	2	no	—
212	C₂×#408	D₆₇	1	1	yes	characteristic
213	#1391	—	67	4	no	—
214	C₆₇	C₂×#1391	1	1	yes	characteristic, complemented, Sylow
215	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
216	C₂²×M₁₆	Dic₆₇	1	1	yes	characteristic, complemented
217	C₂³×C₈	Dic₆₇	1	1	yes	characteristic, complemented
218	C₂×#117	—	134	2	no	—
219	#410	—	67	4	no	—
220	#410	—	67	4	no	—
221	#410	—	134	2	no	—
222	#410	—	134	2	no	—
223	C₂×#99	—	67	4	no	—
224	C₂×#99	—	134	2	no	—
225	#408	—	2	2	no	—
226	#408	Dic₆₇	1	2	yes	complemented
227	#408	C₂×D₆₇	1	2	yes	—
228	C₂×C₂²⋊C₈	—	2	2	no	—
229	C₂×#41	—	4	1	no	—
230	C₂×#41	—	4	1	no	—
231	C₂×#41	—	4	1	no	—
232	C₂×#41	—	4	1	no	—
233	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
234	C₂×#41	C₄×D₆₇	1	2	yes	—
235	C₂×#41	D₂₆₈	1	1	yes	characteristic
236	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
237	C₂×#41	C₂²⋊D₆₇	1	2	yes	—
238	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
239	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
240	C₂²×D₄	C₄×D₆₇	1	2	yes	—
241	C₂²×D₄	D₂₆₈	1	1	yes	characteristic
242	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
243	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
244	C₂×M₁₆	—	2	1	no	—
245	C₂×M₁₆	—	4	1	no	—
246	C₂×M₁₆	—	4	1	no	—
247	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	complemented
248	C₂²×C₈	—	2	1	no	—
249	C₂²×C₈	—	2	1	no	—
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	4	1	no	—
252	C₂²×C₈	—	4	1	no	—
253	C₂²×C₈	C₂×Dic₆₇	1	2	yes	complemented
254	#117	—	268	2	no	—
255	#117	—	268	2	no	—
256	#117	—	134	4	no	—
257	#117	—	134	4	no	—
258	C₂×C₄⋊C₄	—	268	2	no	—
259	C₂×C₂²⋊C₄	—	134	4	no	—
260	C₂×C₄²	—	268	2	no	—
261	#99	—	67	8	no	—
262	#99	—	134	4	no	—
263	#99	—	134	4	no	—
264	#99	—	134	4	no	—
265	C₂²⋊C₈	—	2	4	no	—
266	C₂²⋊C₈	—	2	4	no	—
267	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
268	C₂⁴	(C₂×D₆₇)⋊C₄	1	2	yes	—
269	#41	—	4	1	no	—
270	#41	—	4	1	no	—
271	#41	—	4	2	no	—
272	#41	—	4	2	no	—
273	#41	—	4	1	no	—
274	#41	—	4	1	no	—
275	#41	—	4	1	no	—
276	#41	—	4	1	no	—
277	#41	—	4	2	no	—
278	#41	—	4	2	no	—
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×Q₈	Dic₆₇⋊C₄	1	2	yes	—
284	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
285	C₂×Q₈	C₂×Dic₁₃₄	1	1	yes	characteristic
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	1	no	—
290	C₂×D₄	C₄×Dic₆₇	1	2	yes	—
291	C₂×D₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
292	C₂×D₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
293	C₂×D₄	Dic₆₇⋊C₄	1	2	yes	—
294	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
295	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
296	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	268	4	no	—
306	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
307	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
308	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
309	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
311	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
312	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
313	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
317	M₁₆	—	4	2	no	—
318	M₁₆	—	4	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	4	1	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	—	4	2	no	—
327	C₄⋊C₄	—	268	4	no	—
328	C₄⋊C₄	—	268	4	no	—
329	C₂²⋊C₄	—	268	4	no	—
330	C₂²⋊C₄	—	134	8	no	—
331	C₄²	—	268	4	no	—
332	C₄²	—	268	4	no	—
333	C₂³	—	2	2	no	—
334	C₂³	—	2	1	no	—
335	C₂³	—	2	1	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	4	no	—
339	C₂³	C₆₇⋊#95	1	1	yes	characteristic
340	C₂³	C₆₇⋊#95	1	2	yes	—
341	C₂³	C₆₇⋊#95	1	1	yes	characteristic
342	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
343	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
344	Q₈	—	4	2	no	—
345	Q₈	—	4	1	no	—
346	Q₈	—	4	1	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	2	no	—
351	D₄	—	4	1	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	268	8	no	—
375	C₂×C₄	—	536	8	no	—
376	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
377	C₂×C₄	C₆₇⋊#95	1	2	yes	—
378	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
379	C₂×C₄	#14523958	1	1	yes	characteristic
380	C₂×C₄	#14523958	1	1	yes	characteristic
381	C₂×C₄	#14523958	1	1	yes	characteristic
382	C₂×C₄	#14523958	1	1	yes	characteristic
383	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
385	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	4	2	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	#16863163	1	1	yes	characteristic
402	C₂²	#16863163	1	1	yes	characteristic
403	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	2	no	—
407	C₄	—	2	1	no	—
408	C₄	—	536	8	no	—
409	C₄	#16863163	1	1	yes	characteristic
410	C₄	#16863163	1	1	yes	characteristic
411	C₄	C₂×#14523958	1	1	yes	characteristic
412	C₄	C₂×#14523958	1	1	yes	characteristic
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	—	4	2	no	—
416	C₂	—	4	1	no	—
417	C₂	—	4	1	no	—
418	C₂	—	4	2	no	—
419	C₂	#22558330	1	2	yes	direct product factor
420	C₂	C₂×#16863163	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523958
commutator	C₂³×C₆₇	C₂×C₄²
Frattini	C₂×#41	C₂²×D₆₇
Fitting	C₂×#408×C₆₇	C₂
socle	C₂²×C₆₇	#342
prime radical	C₂²×#41×C₆₇	C₂²
cosocle	C₂×#41×C₆₇	C₂³
hypercenter	C₂×#408	D₆₇
hypocenter	C₆₇	C₂×#1391
LCS	G ▷ C₂³×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#408

Complements

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

N	H ≅ G/N	type
2. #22558330	C₂	direct (N × H)
7. #16863370	C₄	semidirect
10. C₂×#14523958	C₄	semidirect
12. C₂²×M₁₆×C₆₇	C₄	semidirect
13. C₂³×C₈×C₆₇	C₄	semidirect
15. #408×C₆₇	C₄	semidirect
28. #14523958	C₂×C₄	semidirect
47. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
53. C₂²×C₈×C₆₇	C₂×C₄	semidirect
209. C₂×C₆₇	#1391	semidirect
214. C₆₇	C₂×#1391	semidirect
216. C₂²×M₁₆	Dic₆₇	semidirect
217. C₂³×C₈	Dic₆₇	semidirect
226. #408	Dic₆₇	semidirect
247. C₂×M₁₆	C₂×Dic₆₇	semidirect
253. C₂²×C₈	C₂×Dic₆₇	semidirect
419. C₂	#22558330	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×#16863370	2
#22558330	2
C₂×#408×C₆₇	2
C₂×#1391	67

Subgroups — C2×#22558330 #30770786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558330 #30770786