Subgroups — C₂×C₆₇×#2143 #30712635

Order 17152.

1154 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2143	C₂	1	8	yes	maximal, direct product factor
5	C₂×#359×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂²×#41×C₆₇	—	2	1	no	—
7	C₂×#135×C₆₇	—	2	2	no	—
8	C₂×#135×C₆₇	—	2	1	no	—
9	C₂×#135×C₆₇	—	2	1	no	—
10	C₂×#135×C₆₇	—	2	2	no	—
11	C₂×#135×C₆₇	C₄	1	2	yes	—
12	C₂×#135×C₆₇	C₄	1	4	yes	—
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	C₂×#135×C₆₇	C₂²	1	4	yes	—
15	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
16	C₂²×SD₁₆×C₆₇	—	2	1	no	—
17	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	—	2	1	no	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₆₇	—	2	1	no	—
21	C₂×M₃₂×C₆₇	—	2	1	no	—
22	C₂×M₃₂×C₆₇	—	2	2	no	—
23	C₂×M₃₂×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#108×C₆₇	—	2	2	no	—
26	C₂×#108×C₆₇	—	2	1	no	—
27	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
29	#359×C₆₇	C₂²	1	16	yes	complemented
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	—	2	4	no	—
32	C₂×#41×C₆₇	—	2	4	no	—
33	C₂×#41×C₆₇	—	2	2	no	—
34	C₂×#41×C₆₇	—	4	2	no	—
35	C₂×#41×C₆₇	—	4	1	no	—
36	C₂×#41×C₆₇	—	4	1	no	—
37	C₂×#41×C₆₇	—	4	2	no	—
38	C₂²×Q₈×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	—	4	1	no	—
41	C₂³×C₄×C₆₇	—	4	1	no	—
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	#135×C₆₇	—	2	4	no	—
44	#135×C₆₇	—	2	4	no	—
45	#135×C₆₇	—	2	4	no	—
46	#135×C₆₇	—	2	2	no	—
47	#135×C₆₇	—	2	4	no	—
48	#135×C₆₇	—	2	2	no	—
49	#135×C₆₇	—	2	4	no	—
50	#135×C₆₇	C₂×C₄	1	4	yes	—
51	#135×C₆₇	C₂×C₄	1	8	yes	—
52	#135×C₆₇	C₂³	1	4	yes	—
53	C₂×Q₁₆×C₆₇	—	2	2	no	—
54	C₂×Q₁₆×C₆₇	—	2	1	no	—
55	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×SD₁₆×C₆₇	—	2	4	no	—
58	C₂×SD₁₆×C₆₇	—	2	2	no	—
59	C₂×SD₁₆×C₆₇	—	2	2	no	—
60	C₂×SD₁₆×C₆₇	—	2	4	no	—
61	C₂×D₈×C₆₇	—	2	2	no	—
62	C₂×D₈×C₆₇	—	2	1	no	—
63	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	—	2	2	no	—
67	C₂×M₁₆×C₆₇	—	4	2	no	—
68	C₂×M₁₆×C₆₇	—	4	1	no	—
69	C₂²×C₈×C₆₇	—	4	1	no	—
70	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
73	C₂²×C₈×C₆₇	D₄	1	2	yes	—
74	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	D₄	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	M₃₂×C₆₇	—	2	4	no	—
81	M₃₂×C₆₇	—	2	4	no	—
82	C₂×C₁₆×C₆₇	—	2	2	no	—
83	C₂×C₁₆×C₆₇	—	2	2	no	—
84	C₂×C₁₆×C₆₇	C₂³	1	4	yes	complemented
85	#108×C₆₇	—	2	2	no	—
86	#108×C₆₇	—	2	4	no	—
87	#108×C₆₇	C₂³	1	4	yes	complemented
88	C₂⁴×C₆₇	—	4	1	no	—
89	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
90	#41×C₆₇	—	2	8	no	—
91	#41×C₆₇	—	2	4	no	—
92	#41×C₆₇	—	2	4	no	—
93	#41×C₆₇	—	4	4	no	—
94	#41×C₆₇	—	4	4	no	—
95	#41×C₆₇	—	4	4	no	—
96	#41×C₆₇	—	4	4	no	—
97	#41×C₆₇	—	4	2	no	—
98	#41×C₆₇	—	4	4	no	—
99	#41×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	—	2	4	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	4	2	no	—
103	C₂×Q₈×C₆₇	—	4	1	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	4	2	no	—
107	C₂×D₄×C₆₇	—	4	4	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂×D₄×C₆₇	—	4	4	no	—
110	C₂×D₄×C₆₇	—	4	2	no	—
111	C₂×D₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	—	4	1	no	—
113	C₂²×C₄×C₆₇	—	4	4	no	—
114	C₂²×C₄×C₆₇	—	4	2	no	—
115	C₂²×C₄×C₆₇	—	4	4	no	—
116	C₂²×C₄×C₆₇	—	4	2	no	—
117	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
120	C₂²×C₄×C₆₇	D₈	1	2	yes	—
121	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	Q₁₆×C₆₇	—	2	2	no	—
128	Q₁₆×C₆₇	—	2	4	no	—
129	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
130	SD₁₆×C₆₇	—	2	4	no	—
131	SD₁₆×C₆₇	—	2	8	no	—
132	SD₁₆×C₆₇	—	2	4	no	—
133	D₈×C₆₇	—	2	2	no	—
134	D₈×C₆₇	—	2	4	no	—
135	D₈×C₆₇	C₂²×C₄	1	4	yes	—
136	M₁₆×C₆₇	—	2	4	no	—
137	M₁₆×C₆₇	—	4	4	no	—
138	M₁₆×C₆₇	—	4	2	no	—
139	C₂×C₈×C₆₇	—	4	4	no	—
140	C₂×C₈×C₆₇	—	4	2	no	—
141	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
146	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
147	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
148	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
157	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₁₆×C₆₇	—	2	4	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	4	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	8	4	no	—
164	C₂³×C₆₇	#102	1	1	yes	characteristic
165	C₂³×C₆₇	#102	1	1	yes	characteristic
166	C₂³×C₆₇	#102	1	2	yes	—
167	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
168	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
169	Q₈×C₆₇	—	2	4	no	—
170	Q₈×C₆₇	—	4	4	no	—
171	Q₈×C₆₇	—	4	2	no	—
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	4	4	no	—
174	D₄×C₆₇	—	4	8	no	—
175	D₄×C₆₇	—	4	4	no	—
176	D₄×C₆₇	—	4	4	no	—
177	D₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	4	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	4	no	—
181	C₂×C₄×C₆₇	—	4	8	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
184	C₂×C₄×C₆₇	#102	1	2	yes	—
185	C₂×C₄×C₆₇	#102	1	2	yes	—
186	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
187	C₂×C₄×C₆₇	#102	1	2	yes	—
188	C₂×C₄×C₆₇	#102	1	2	yes	—
189	C₂×C₄×C₆₇	#102	1	2	yes	—
190	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
196	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	C₈×C₆₇	—	4	4	no	—
202	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
204	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
206	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
207	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	4	4	no	—
214	C₂²×C₆₇	—	8	8	no	—
215	C₂²×C₆₇	—	8	4	no	—
216	C₂²×C₆₇	C₂×#102	1	2	yes	—
217	C₂²×C₆₇	C₂×#102	1	2	yes	—
218	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
220	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
221	C₄×C₆₇	—	4	4	no	—
222	C₄×C₆₇	C₂×#102	1	2	yes	—
223	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
224	C₄×C₆₇	C₂×#102	1	2	yes	—
225	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
226	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
227	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
228	C₂×#2143	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	2	2	no	—
230	C₂×C₆₇	—	2	1	no	—
231	C₂×C₆₇	—	2	2	no	—
232	C₂×C₆₇	—	2	1	no	—
233	C₂×C₆₇	—	8	4	no	—
234	C₂×C₆₇	#2143	1	2	yes	direct product factor
235	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
236	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
237	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#2143	C₂×C₆₇	1	8	yes	direct product factor
240	C₂×#359	C₂×C₆₇	1	4	yes	complemented
241	C₆₇	C₂×#2143	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²×#41	—	2	1	no	—
243	C₂×#135	—	2	2	no	—
244	C₂×#135	—	2	1	no	—
245	C₂×#135	—	2	1	no	—
246	C₂×#135	—	2	2	no	—
247	C₂×#135	C₄×C₆₇	1	2	yes	—
248	C₂×#135	C₄×C₆₇	1	4	yes	—
249	C₂×#135	C₂²×C₆₇	1	2	yes	—
250	C₂×#135	C₂²×C₆₇	1	4	yes	—
251	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
252	C₂²×SD₁₆	—	2	1	no	—
253	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
254	C₂²×M₁₆	—	2	1	no	—
255	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×M₃₂	—	2	1	no	—
257	C₂×M₃₂	—	2	1	no	—
258	C₂×M₃₂	—	2	2	no	—
259	C₂×M₃₂	C₂²×C₆₇	1	4	yes	complemented
260	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	complemented
261	C₂×#108	—	2	2	no	—
262	C₂×#108	—	2	1	no	—
263	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
264	C₂×#108	C₂²×C₆₇	1	4	yes	complemented
265	#359	C₂²×C₆₇	1	16	yes	complemented
266	C₂×#41	—	2	2	no	—
267	C₂×#41	—	2	4	no	—
268	C₂×#41	—	2	2	no	—
269	C₂×#41	—	2	4	no	—
270	C₂×#41	—	4	1	no	—
271	C₂×#41	—	4	2	no	—
272	C₂×#41	—	4	1	no	—
273	C₂×#41	—	4	2	no	—
274	C₂²×Q₈	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	—	4	1	no	—
277	C₂³×C₄	—	4	1	no	—
278	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
279	#135	—	2	4	no	—
280	#135	—	2	4	no	—
281	#135	—	2	4	no	—
282	#135	—	2	2	no	—
283	#135	—	2	4	no	—
284	#135	—	2	2	no	—
285	#135	—	2	4	no	—
286	#135	C₂×C₄×C₆₇	1	4	yes	—
287	#135	C₂×C₄×C₆₇	1	8	yes	—
288	#135	C₂³×C₆₇	1	4	yes	—
289	C₂×Q₁₆	—	2	2	no	—
290	C₂×Q₁₆	—	2	1	no	—
291	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
292	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
293	C₂×SD₁₆	—	2	4	no	—
294	C₂×SD₁₆	—	2	2	no	—
295	C₂×SD₁₆	—	2	2	no	—
296	C₂×SD₁₆	—	2	4	no	—
297	C₂×D₈	—	2	2	no	—
298	C₂×D₈	—	2	1	no	—
299	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
300	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	4	no	—
303	C₂×M₁₆	—	4	1	no	—
304	C₂×M₁₆	—	4	2	no	—
305	C₂²×C₈	—	4	1	no	—
306	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
309	C₂²×C₈	D₄×C₆₇	1	2	yes	—
310	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₈	D₄×C₆₇	1	2	yes	—
314	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
316	M₃₂	—	2	4	no	—
317	M₃₂	—	2	4	no	—
318	C₂×C₁₆	—	2	2	no	—
319	C₂×C₁₆	—	2	2	no	—
320	C₂×C₁₆	C₂³×C₆₇	1	4	yes	complemented
321	#108	—	2	2	no	—
322	#108	—	2	4	no	—
323	#108	C₂³×C₆₇	1	4	yes	complemented
324	C₂⁴	—	4	1	no	—
325	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
326	#41	—	2	4	no	—
327	#41	—	2	8	no	—
328	#41	—	2	4	no	—
329	#41	—	4	4	no	—
330	#41	—	4	4	no	—
331	#41	—	4	4	no	—
332	#41	—	4	4	no	—
333	#41	—	4	2	no	—
334	#41	—	4	2	no	—
335	#41	—	4	4	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	4	2	no	—
339	C₂×Q₈	—	4	1	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	—	4	4	no	—
344	C₂×D₄	—	4	2	no	—
345	C₂×D₄	—	4	4	no	—
346	C₂×D₄	—	4	2	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
356	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	D₈×C₆₇	1	2	yes	—
359	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	Q₁₆	—	2	2	no	—
364	Q₁₆	—	2	4	no	—
365	Q₁₆	C₂²×C₄×C₆₇	1	4	yes	—
366	SD₁₆	—	2	4	no	—
367	SD₁₆	—	2	8	no	—
368	SD₁₆	—	2	4	no	—
369	D₈	—	2	2	no	—
370	D₈	—	2	4	no	—
371	D₈	C₂²×C₄×C₆₇	1	4	yes	—
372	M₁₆	—	2	4	no	—
373	M₁₆	—	4	4	no	—
374	M₁₆	—	4	2	no	—
375	C₂×C₈	—	4	2	no	—
376	C₂×C₈	—	4	4	no	—
377	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
378	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
379	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
380	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
381	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
382	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
383	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
384	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
385	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
386	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
387	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
388	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
389	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
390	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
393	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
394	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
395	C₁₆	—	2	4	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	4	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	8	4	no	—
400	C₂³	#102×C₆₇	1	1	yes	characteristic
401	C₂³	#102×C₆₇	1	1	yes	characteristic
402	C₂³	#102×C₆₇	1	2	yes	—
403	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
404	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
405	Q₈	—	2	4	no	—
406	Q₈	—	4	4	no	—
407	Q₈	—	4	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	4	4	no	—
410	D₄	—	4	8	no	—
411	D₄	—	4	4	no	—
412	D₄	—	4	4	no	—
413	D₄	—	4	2	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	4	no	—
418	C₂×C₄	—	4	8	no	—
419	C₂×C₄	#102×C₆₇	1	2	yes	—
420	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
421	C₂×C₄	#102×C₆₇	1	2	yes	—
422	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
423	C₂×C₄	#102×C₆₇	1	2	yes	—
424	C₂×C₄	#102×C₆₇	1	2	yes	—
425	C₂×C₄	#102×C₆₇	1	2	yes	—
426	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
427	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
430	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
431	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
432	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
433	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
434	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
435	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
437	C₈	—	4	4	no	—
438	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
439	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
440	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
441	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
442	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
443	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	4	no	—
450	C₂²	—	8	8	no	—
451	C₂²	—	8	4	no	—
452	C₂²	C₂×#102×C₆₇	1	2	yes	—
453	C₂²	C₂×#102×C₆₇	1	2	yes	—
454	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
455	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
456	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
457	C₄	—	4	4	no	—
458	C₄	C₂×#102×C₆₇	1	1	yes	characteristic
459	C₄	C₂×#102×C₆₇	1	2	yes	—
460	C₄	C₂×#102×C₆₇	1	2	yes	—
461	C₄	C₂×#102×C₆₇	1	1	yes	characteristic
462	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
463	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
464	C₂	—	2	2	no	—
465	C₂	—	2	1	no	—
466	C₂	—	2	2	no	—
467	C₂	—	2	1	no	—
468	C₂	—	8	4	no	—
469	C₂	C₆₇×#2143	1	2	yes	direct product factor
470	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₆₇	C₂	semidirect
3. C₂²×#108×C₆₇	C₂	semidirect
4. C₆₇×#2143	C₂	direct (N × H)
5. C₂×#359×C₆₇	C₂	semidirect
23. C₂×M₃₂×C₆₇	C₂²	semidirect
24. C₂²×C₁₆×C₆₇	C₂²	semidirect
27. C₂×#108×C₆₇	C₂²	semidirect
28. C₂×#108×C₆₇	C₂²	semidirect
29. #359×C₆₇	C₂²	semidirect
84. C₂×C₁₆×C₆₇	C₂³	semidirect
87. #108×C₆₇	C₂³	semidirect
228. C₂×#2143	C₆₇	direct (N × H)
234. C₂×C₆₇	#2143	direct (N × H)
237. C₂²×M₃₂	C₂×C₆₇	semidirect
238. C₂²×#108	C₂×C₆₇	semidirect
239. #2143	C₂×C₆₇	direct (N × H)
240. C₂×#359	C₂×C₆₇	semidirect
241. C₆₇	C₂×#2143	direct (N × H)
259. C₂×M₃₂	C₂²×C₆₇	semidirect
260. C₂²×C₁₆	C₂²×C₆₇	semidirect
263. C₂×#108	C₂²×C₆₇	semidirect
264. C₂×#108	C₂²×C₆₇	semidirect
265. #359	C₂²×C₆₇	semidirect
320. C₂×C₁₆	C₂³×C₆₇	semidirect
323. #108	C₂³×C₆₇	semidirect
469. C₂	C₆₇×#2143	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂²×M₃₂×C₆₇	2
C₂²×#108×C₆₇	2
C₆₇×#2143	2
C₂×#359×C₆₇	2
C₂×#2143	67

Subgroups — C2×C67×#2143 #30712635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2143 #30712635